How do axioms specify a new system vs continuing an old one?

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Treatid
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How do axioms specify a new system vs continuing an old one?

We have two axiomatic systems, A and B.

We have two sets of statements, a and b.

a are true statements with respect to A that we are using as the axioms of B.
b are true statements with respect to B.

Given that b are true with respect to a, then b must also be true with respect to A.

This suggests that B can only be a subset of A, and not a system in its own right.

But that would make all axiomatic systems, one system.

How do we stop describing one system so we can describe the next system?

Twistar
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Re: How do axioms specify a new system vs continuing an old

You're right up until "But that would make all axiomatic systems, one system."

You've assumed that the axioms of B are true statements in A. However, we could take another set of statements, c, which are false in A as axioms of B. This means that statements made in B would be false with respect to A and true with respect to B but that is ok. There is no such thing as objective truth or falsehood of statements. True and false are always relative to some set of axioms which the observer accepts based on ways of knowing other than logic and rationality.

edit: typo

Treatid
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Re: How do axioms specify a new system vs continuing an old

Taking your argument at face value - you are arguing that there are precisely two axiomatic systems. The initial system and its inverse. Is that really your intention?

We'll keep your false axioms for B. Now let us introduce system C. the axioms of system C are c; these are false statements with respect to B.

We now have two negatives. True statements in system C are also true statements in system A.

So system C must be a subset of system A.

I'm not convinced that a simple inverse through a true/false axis actually gives you a different system. I think such an inverse will be very much like the original system. However, I don't feel the need to argue the point.

Nor do I think that many actual axiomatic constructions choose false statements as their basis. There is no question it is possible, of course.

That still leaves the substantive portion of the question open: How do you create a distinct system while axioms are an explicit bridge between the old system and the new system? Remove the axioms and we don't have anything upon which to build. Keep the axioms and the two systems are connected. Two systems connected together are otherwise known as one system.

Twistar
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Re: How do axioms specify a new system vs continuing an old

You said a lot of unclear things there but I'll do my best at responding.

We can also have a 4th system D which has d as it's axioms. d can include both true statements from A as well as false statements from A.

In fact, we can take ANY statements in the language that we are speaking in and call them axioms. Then, once we have our set of axioms, we can try to derive new statements from these axioms. We can also ask things like are these axioms complete? are these axioms consistent? etc. For mathematics, one of the goals (sort of) is to take axioms which are consistent i.e. you can't derive a contradiction.

Now, notice in the last paragraph I didn't use the words true or false. This is because axioms themselves are not true or false. They are just statements. We SUPPOSE they are true and see what happens. If you equate axioms with true statements you are going to get into trouble. Axioms aren't true and they aren't false, they're just statements.

edit:
Sorry, maybe I was responding to something you didn't specifically ask. But I don't really know what you're driving at.. yes. If the true statements of one system are axioms of another system then the latter system is in some way a subset of the former. But you can also have a system whose axioms consist of both true and false statements of another system. You can also have a system whose axioms aren't even statements in some other system.

edit2:
Also, Treatid, you've made a lot of posts here about this axiom stuff. Do you understand what I'm saying when I say that axioms aren't true or false but rather that they're just statements? Does that jive with your definition of "axiom" or no?

FancyHat
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Re: How do axioms specify a new system vs continuing an old

Good grief.

Treatid, I think you need to get a good textbook about this stuff, or something, and properly study this stuff first.

Because, right now, this is looking really

Dunning-Kruger.
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Treatid
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Re: How do axioms specify a new system vs continuing an old

Twistar wrote:Also, Treatid, you've made a lot of posts here about this axiom stuff. Do you understand what I'm saying when I say that axioms aren't true or false but rather that they're just statements? Does that jive with your definition of "axiom" or no?

Yes - I am completely happy with the idea that axioms don't need to be justified in any way except that we quite like them to form consistent systems. A statement is true, false or undefined with respect to a set of axioms - but the axioms themselves are assumed to be true.

Having said that - axioms must tell us something. At the very least, we need to know which particular symbol manipulations they allow. This is where some context is needed. Just symbols by themselves don't tell us anything. Axioms must have sufficient context for us to know how to interpret those axioms at some basic level. This is where the context of axioms is relevant. Whether ZFC or an informal language - we need some context in order to use axioms in a consistent way.

You suggested we could mix true and false statements from A to form a set of axioms. Given that A is the context for interpreting those axioms - mixing true and false statements will lead to a contradiction.

This isn't a question of whether the individual statements are true or false. In the context of A - treating a false statement as true will lead to a contradiction and vice versa (unless we invert all statements at the same time).

You are right that we are allowed to mix true and false statements (with respect to A) in a set of axioms. And we can assume those statements to be true. However, if we continue to interpret those statements in the context of A - then we will find that the system defined is inconsistent.

X = Y is true, and X != Y is false. We could use these statements as axioms. We would end up with X = Y and X != Y both being true. It doesn't matter which statement was originally true and which was false. In this context - those statements lead to an inconsistent system.

So - technically you are right - we can take any statements in the given language and call them axioms. But we shouldn't expect to form a consistent system when we mix true and false statements (with respect to system A).

As a rule, we would only expect to have consistent systems when we choose all true, or all false statements for our axioms. (Again - true and false are only with respect to system A).

So we have three scenarios:
i. We pick true statements from A to form the axioms of B. B is a subset of A.
ii. We pick false statements from A to form the axioms of B. B is an inverted subset of A.
iii. We pick a mix of true and false statements from A to form the axioms of B. B is inconsistent because we know right from the start that the axioms are inconsistent. A mix of true and false statements treated as true are inconsistent.

Which leaves B as being a subset of A, an inverted subset of A, or null.

None of these options make it clear that B is a distinct system from A.

FancyHat wrote:Good grief.

Treatid, I think you need to get a good textbook about this stuff, or something, and properly study this stuff first.

Forest Goose made the very good point that without understanding definitions and stating assumptions, one could get very confused.

A number of people have objected to my characterisation of axiomatic systems. Interestingly - no-one has offered up an official formal definition of axiomatic systems.

This is largely because axiomatic systems haven't been formally defined. The use of informal language to boot-strap axioms means that right at the beginning of axioms you have a bunch of undefined stuff and probably some unstated assumptions.

If everyone was working with a set of strict definitions and fully stated assumptions - you would have shut me down right at the beginning of the conversation. There would be a simple set of unambiguous rules about which there would be no argument.

As it is - there are unknowns within axiomatic mathematics. I'm assuming most people arguing against me assume those unknowns are sufficiently constrained that they don't impact the overall effectiveness of axiomatic mathematics. Whereas, I think those unknowns are critical and completely undermine axiomatic mathematics.

Please do point me at a formal definition of axiomatic mathematics. Please show me how any ambiguity within informal languages can be mitigated.

I am well aware that many people think they have worked with axiomatic mathematics - and that they feel that experience is compelling evidence that axiomatic mathematics works. By the same token, there are many people with direct experience of God who tell me that I should trust their experience and believe without evidence.

This, however, is mathematics. If you can't demonstrate it completely unambiguously - then you haven't demonstrated it at all.

As much as ridiculing me is high entertainment... it doesn't show that I am wrong. Showing me that I am wrong shows me that I am wrong.

To pretend that axiomatic mathematics is just fine in the face of The foundational crises is absurd. There is no question that a problem exists. Nor is that problem just a matter of justifying one set of axioms over another.

I'm attempting to illuminate and quantify that problem. My interlocutors are determined to deny that any problem exists despite the fact that its existence has been established for the best part of a hundred years.

lightvector
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Re: How do axioms specify a new system vs continuing an old

Treatid wrote:So - technically you are right - we can take any statements in the given language and call them axioms. But we shouldn't expect to form a consistent system when we mix true and false statements (with respect to system A).

As a rule, we would only expect to have consistent systems when we choose all true, or all false statements for our axioms. (Again - true and false are only with respect to system A).

Treatid wrote:iii. We pick a mix of true and false statements from A to form the axioms of B. B is inconsistent because we know right from the start that the axioms are inconsistent. A mix of true and false statements treated as true are inconsistent.

This is clearly wrong. Here's an explicit example:

Axioms of a ring:
There are two binary operations, "+" and "*", that we call "addition" and "multiplication", and there exist two distinguished elements that we label "0" or "zero" and "1" or "one", such that for all a,b,c:
1. Addition is associative: (a+b)+c = a+(b+c)
2. Zero is an additive identity: a+0=0+a=a
3. Addition has inverses: There exists d such that a+d = 0.
4. Addition is commutative: a+b = b+a
5. Multiplication is associative: (a*b)*c = a*(b*c)
6. One is a multiplicative identity: a*1=1*a=a
7. Multiplication distributes over addition: a*(b+c) = (a*b)+(a*c) and (b+c)*a = (b*a)+(c*a)

These axioms are consistent because there exist many collections of objects that actually satisfy these properties. For example:
1. The integers where "+" and "*" are integer addition and integer multipliation.
2. The integers modulo 5, where there are only 5 elements {0,1,2,3,4} and where "+" and "*" are addition and multiplication modulo 5.
3. 2x2 real-valued matrices where "+" is entrywise addition and "*" is matrix multiplication.

In this axiom system, the statement "There exists a,b,c such that (a*b)*c != a*(b*c)" is false because multiplication is associative. How about we form a new system consisting of this "false" statement along with some other "true" statements, namely the axioms 1,2,3,4,6,7?

We get another consistent system! These are called non-associative rings. We can see that they're consistent because we can actually find collections of objects that satisfy these properties. For example, the octonions.

We can also replace the axiom "One is a multiplicative identity" with the false-in-that-system statement "For all elements e, there exists x such that x*e != x" but keep all other axioms, and also get consistent systems. These are called "rngs" or "rings without unit".

In general, every time you have a consistent system A plus a statement S that is independent of A, you have an example, because A+{S} is consistent, but you will also have A+{not S} consistent despite having as its axioms a mixture of true and false statements from A+{S}'s perspective.

Whether you consciously think this or not, I'm guessing you may have an intuition that systems of axioms have to agree with one another somehow in some globally "coherent" way because you view axioms as declarations of truth that have to agree with some universal underlying truth. That's not quite the right intuition here - a more useful intuition is one of axioms as the rules of different possible games. You are free to create lots of different games with rules that imply mixtures of things both that would be legal in other games and illegal in other games. The restriction that a game itself cannot itself declare the same thing both legal and illegal (that a system of axiom needs to be consistent) is not as strong of a restriction as you might otherwise intuit.

Treatid wrote:As it is - there are unknowns within axiomatic mathematics. I'm assuming most people arguing against me assume those unknowns are sufficiently constrained that they don't impact the overall effectiveness of axiomatic mathematics. Whereas, I think those unknowns are critical and completely undermine axiomatic mathematics.

Treatid wrote:I'm attempting to illuminate and quantify that problem. My interlocutors are determined to deny that any problem exists despite the fact that its existence has been established for the best part of a hundred years.

Would you say the unknowns in the foundations and rules of Chess, Connect 4, or other games are critical and completely undermine these games and people's ability to play them?

The rules of Chess (axioms of a mathematical system) are simply the rules of a game, and the only thing that matters in practice is that they are communicated well enough that people can agree and play the same game together. Very few people care about how to specify the rules in a way completely formally and fundamentally free of ambiguity. It's just not a problem in practice, because while a few beginners (young mathematics students) might be initially confused sometimes, quickly everyone learns the rules well enough. It's laughable to suggest that ambiguity in the rules undermines the usefulness and applicability of a standard checkmate pattern (the usefulness and applicability of a mathematical theorem). Chess players (mathematicians) on the whole are very practical - almost nobody cares about the ambiguity problem you bring up because it doesn't stop people from learning Chess and playing and usefully applying their knowledge (learning mathematics and playing and usefully applying their knowledge to the real world).
Last edited by lightvector on Sun Feb 08, 2015 3:52 pm UTC, edited 1 time in total.

Gwydion
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Re: How do axioms specify a new system vs continuing an old

Treatid wrote:You suggested we could mix true and false statements from A to form a set of axioms. Given that A is the context for interpreting those axioms - mixing true and false statements will lead to a contradiction.

This isn't a question of whether the individual statements are true or false. In the context of A - treating a false statement as true will lead to a contradiction and vice versa (unless we invert all statements at the same time).
This is wrong. One can create a consistent system by negating only certain axioms within a system - the most recognizable example is negating the parallel postulate of Euclidean geometry while maintaining the rest, which yields non-Euclidean geometry. Negating the axiom of choice but leaving the rest of ZF may also produce some valuable information. Lightvector gave a couple more good examples above while I was typing this.
As much as ridiculing me is high entertainment... it doesn't show that I am wrong. Showing me that I am wrong shows me that I am wrong.

To pretend that axiomatic mathematics is just fine in the face of The foundational crises is absurd. There is no question that a problem exists. Nor is that problem just a matter of justifying one set of axioms over another.

We have shown you to be wrong multiple times, and each time you either ignore the response or reject it. This is not any way to learn - it is preaching, proselytizing but with pseudo-mathematics instead of philosophy or religion. "But I'm right" is not any way to structure an argument.

Twistar
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Re: How do axioms specify a new system vs continuing an old

Treatid wrote:
Twistar wrote:You suggested we could mix true and false statements from A to form a set of axioms. Given that A is the context for interpreting those axioms - mixing true and false statements will lead to a contradiction.

This is true only if you continue to interpret those axioms with respect to A. But you're not supposed to interpret them with respect to A now that you're working in system B. You're supposed to treat them as true and see what happens. As others have shown, it is possible and often done that we shift which axioms we are using and this can lead to multiple systems (based on different axioms) which are all consistent (at least we think they're consistent).

The previous posters bring up good points. Do you take issue with the axioms of chess?

z4lis
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Re: How do axioms specify a new system vs continuing an old

Treatid runs into an airplane factory...

Engineer: What? We've known about string theory for years. It has no effect on what we're doing here. Isn't it completely theoretical anyway?

Treatid: No, these equations are all incorrect!

Engineer: No, those equations are actually fine...

Treatid: But couldn't this equation be incorrect?

Engineer: I suppose it could be, but even if it were, the error would be so tremendously tiny that none of the calculations would really change...

Treatid: How do you know? The electrons in the plane's computers detangle and cause drastic damage!

Engineer: What are you talking about? That doesn't even make sense. I mean, I guess that could happen, but we've been building planes this way for years and it's never come up...

Treatid: Can you prove that none of the airplanes will crash because of your error?

Engineer: Isn't the fact that it's never happened convincing enough?

Treatid: No! You should all start reading about string theory before building another airplane! Stop the factory immediately!

Engineer: We're not stopping the factory because you just discovered string theory. Anyone who knows anything about string theory or airplanes knows that airplanes will keep flying regardless of what string theory says. But since you don't agree, can you come up with a specific equation that will actually change because of predictions from string theory or demonstrate how an airplane could fail because of it?

Treatid: Prove that the equations don't change! I can't believe all you engineers are just sitting around in the face of this unresolved paradox in airplane engineering!
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.

Forest Goose
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Re: How do axioms specify a new system vs continuing an old

Grindstone Cowboy wrote:A number of people have objected to my characterisation of axiomatic systems. Interestingly - no-one has offered up an official formal definition of axiomatic systems.

I had this whole neat write up about how to use axioms; I even included a little section on using relational calculus to whittle down the variables and how NBG can be finitely axiomatized (so as to avoid assumptions, as much as possible). But, then, something occurred to me: I'm not your fucking teacher.

Why do you get to spew nonsense, day and night, while I have to spend my time teaching you why you're wrong? How does my not wanting to do so somehow make you right and me wrong? If that's your point, it's a bad one.

By the by, you are aware that English is way more hand wavely defined than anything in math...yet, here you are, using it - please stop! Oh My! There's a foundational crises in Engliushfuin46y98y9 ,,,, (That last part wasn't intentional: the foundational crises in math and English came together and wrecked up both my language and the computer's logic circuits - happens all the time).

Grindstone Cowboy wrote:Whereas, I think those unknowns are critical and completely undermine axiomatic mathematics.

In short, to the point, sentences, devoid of ranting empty pseudophilosophy, state those unknowns for me, then we can discuss them.

Again, not looking for philosophical diatribes, nor your own versions of what you think - just the unknowns.

Grindstone Cowboy wrote:I am well aware that many people think they have worked with axiomatic mathematics - and that they feel that experience is compelling evidence that axiomatic mathematics works. By the same token, there are many people with direct experience of God who tell me that I should trust their experience and believe without evidence.

Fair point, I'm not omnilogical, thus, nothing is to be trusted, nothing can be proven, there is no certainty...oh wait, that's stupid and, obviously, something no sane person is insisting...well, you insist it, but I doubt your sanity.

But, seriously, that's as obnoxious a point as insisting that gravity might cease working because no law of physics knows if you prefer turkey or ham sandwiches on Mondays.

Grindstone Cowboy wrote:This, however, is mathematics. If you can't demonstrate it completely unambiguously - then you haven't demonstrated it at all.

Fine, I agree, mathematics is all wrong. We will now use Goosematics, it looks just like mathematics, but "demonstrate" is defined within the constraints of the subject so that, unlike that crazy ol' math, when I write a proof on the chalkboard, I mean that that is done so within the constraints of informality required to bootstrap logic and a minimum metalogic.

I shall, henceforth assume, that everyone uses the word "mathematics" to mean "Goosematics" and that all that appears to be math is goosematics.

Looks like I've solved the collapse of all knowledge in less words than you, by the way - does that prevent you from winning a Field's Crank's Medal?

Grindstone Cowboy wrote:As much as ridiculing me is high entertainment

High? You overestimate yourself - this is much more like fart humour: you can't help but laugh while engaged in it, then feel really sad that it exists after the fact. That your argument exists makes me sad, ultimately - as does wasting my time to mock it - and, yet, I can't resist. It is a paradox that cannot be resolved...one might call it "The Foundational Crises in Humour". Since you're a pro at solving those, please solve it too, for me, I'd appreciate it.

Grindstone Cowboy wrote:To pretend that axiomatic mathematics is just fine in the face of The foundational crises is absurd. There is no question that a problem exists. Nor is that problem just a matter of justifying one set of axioms over another.

There is a lot of question that the problem exists. Here's one such question, "Didn't this get resolved for everyone who isn't an internet math crank, semi-philosopher, or trolling logician a long long time ago?". Here's the answer, "Yes".

Happy we cleared that up.

Grindstone Cowboy wrote:I'm attempting to illuminate and quantify that problem.

And I've been attempting to illuminify and quantitate* that you're obnoxious and absurd to you - looks like we both failed.

*I just won't "illuminate and quantify", I see how pretentious it makes you sound, so I refuse to do the same
Last edited by Forest Goose on Mon Feb 09, 2015 9:12 am UTC, edited 1 time in total.
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Forest Goose
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Re: How do axioms specify a new system vs continuing an old

Grindstone Cowboy wrote:A number of people have objected to my characterisation of axiomatic systems. Interestingly - no-one has offered up an official formal definition of axiomatic systems.

Last edited by Forest Goose on Mon Feb 09, 2015 8:36 am UTC, edited 1 time in total.
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FancyHat
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Re: How do axioms specify a new system vs continuing an old

Treatid is not persuaded by the arguments of others. Neither is a block of granite.
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Treatid
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Re: How do axioms specify a new system vs continuing an old

Gwydion wrote:
Treatid wrote:You suggested we could mix true and false statements from A to form a set of axioms. Given that A is the context for interpreting those axioms - mixing true and false statements will lead to a contradiction.

This isn't a question of whether the individual statements are true or false. In the context of A - treating a false statement as true will lead to a contradiction and vice versa (unless we invert all statements at the same time).
This is wrong. One can create a consistent system by negating only certain axioms within a system - the most recognizable example is negating the parallel postulate of Euclidean geometry while maintaining the rest, which yields non-Euclidean geometry. Negating the axiom of choice but leaving the rest of ZF may also produce some valuable information. Lightvector gave a couple more good examples above while I was typing this.

We have a number of true and false statements with respect to system 'A'.

If we assert that a false statement is true (or vice versa), then that newly true statement is a contradiction of what we have already established. We have created an inconsistency with respect to system 'A'.

Now, we take that inconsistency and formalise it as a set of axioms; 'a'. If our context for interpreting statements is still 'A' - then calling the statements 'axioms' hasn't changed anything. The inconsistency remains.

The only way for 'a' to be consistent is if we don't use 'A' as our context.

Inconsistent statements with respect to a system are always inconsistent with respect to that system.

If your axioms are consistent - then they must consist of only true, or only false statements with respect to WHATEVER system you decide you are actually using as the context for those axioms.

If we have no context - then we have no way to interpret the statements of 'a'. ('a' are not their own context - some people are saying some silly things in an attempt find fault - if statements have meaning independent of context then why are you bothering with axioms?).

Feel free to change your context as often as you like. It doesn't change the fact that for each context, inconsistent statements are always inconsistent.

Just bear in mind that the 'meaning' of axioms is tied to the context. Change the context and you change how the axioms should be interpreted. Change the context of the axioms and you change the system that those axioms describe.

Isn't the point of axioms to establish a known context so that axioms have a known meaning?

...
lightvector et al:

Thinking that a system is consistent does not prove that the system is consistent. Assuming consistency to prove consistency is a tad tautological.

Showing inconsistency, just once, trumps any number of apparently consistent operations.

You have shown me a set of axioms that are inconsistent with respect to system A. You've told me they are inconsistent with respect to system A. We don't have to go any further. The inconsistency is right there in the axioms - they are inconsistent with respect to system A. Any system they describe is inconsistent with respect to system A.

You could show me axioms that look similar but are defined with respect to some other systems. Those axioms might well be consistent with respect to system X. They are still inconsistent with respect to system A. And since you told me we were using system A - I'm going to do my interpreting based on system A.

You know that an inconsistent system can be made to show any statements as true. You can see whatever you want to see in an inconsistent system. You can construct any proof you care to construct; and any counter proof.

elasto
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Re: How do axioms specify a new system vs continuing an old

Treatid wrote:We have a number of true and false statements with respect to system 'A'.

If we assert that a false statement is true (or vice versa), then that newly true statement is a contradiction of what we have already established. We have created an inconsistency with respect to system 'A'.

Now, we take that inconsistency and formalise it as a set of axioms; 'a'. If our context for interpreting statements is still 'A' - then calling the statements 'axioms' hasn't changed anything. The inconsistency remains.

The only way you get an inconsistency is if you assert a statement to be both true and false within the same system. You get no inconsistencies if within one system you assert it true and in another you assert it false.

As others have asked you, please try to focus on a concrete example to help you understand this. My suggestion for a very simple example would be Euclidean vs non-Euclidean geometry.

We can take the following axioms:
1. A straight line may be drawn between any two points.
2. Any terminated straight line may be extended indefinitely.
3. A circle may be drawn with any given point as center and any given radius.
4. All right angles are equal.

We can also have a fifth postulate which is independent from the others. That means you can produce one consistent system by assuming it true and another consistent system by assuming it false.

This axiom is:
5. For any given point not on a given line, there is exactly one line through the point that does not meet the given line.

- If you assume it true you get Euclidean geometry - which usefully models what happens in flat spaces - like drawings on a piece of paper.
- If you assume it false you get non-Euclidean geometry - which usefully models what happens in non-flat spaces - like structures on the surface of the earth

Which system is 'true' in some 'absolute sense'? Well obviously neither. Both are internally consistent and can produce endless proofs - but one or other may be more useful to you dependent on your particular circumstances.

FancyHat
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Re: How do axioms specify a new system vs continuing an old

Treatid, you've got things really muddled up. What you're arguing against isn't what others are actually saying, but what you mistakenly think they're saying. Because you've got it all muddled up, muddling up what seems, to you, to be what others are saying with your own misconceptions, you're basically arguing against nonsense that no one's actually saying anyway.

You really do need to properly study this stuff, from a decent textbook, or something, just so you actually know what others are saying. You're just not giving yourself a chance. You think you know what others mean by 'systems', 'axioms', 'statements', and so on, but you clearly don't. You're all over the place with this stuff, but you're just too ignorant to see it.

Again,

In system A, all the axioms, a, are taken as true. a can be a set of statements, {p, q, r, s}.

In system B, all the axioms, b, are taken as true. b can include some of the axioms of A, and can also have some axioms that, in A, would be false. But they are not taken to be false in B, whatever they'd be found to be in A. So, b could be a set of statements such as {p, not q, s, t}. q is true in A, but false in B.

There's nothing that ties such statements to specific systems, which seems to be what you're thinking. Just because statement q is used as an axiom in A, doesn't mean q is permanently tied to A, or that A is the only system or 'context' in which q can be interpreted, or that the truth or otherwise of q is determined in or by A when used in other systems. In B, the truth or otherwise of q is determined in B, not in A.

There is no contradiction here, just as there's no contradiction between the statement, 'the body was found in the library', being true in one novel and false in another. It's daft to say that the truth or otherwise of that statement has to be determined in the context of just one novel, once and for all, no matter what other novels we read. It's the same with systems and statements within them, including axioms.
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Forest Goose
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Re: How do axioms specify a new system vs continuing an old

Howdy, Lone Ranger of Logic, I see you're here to rustle us up some truths! Yeeeeehaw!

The Lone Ranger of Logic wrote:We have two axiomatic systems, A and B.

We have two sets of statements, a and b.

a are true statements with respect to A that we are using as the axioms of B.
b are true statements with respect to B.

Given that b are true with respect to a, then b must also be true with respect to A.

This suggests that B can only be a subset of A, and not a system in its own right.

But that would make all axiomatic systems, one system.

A swing and a miss! (Actually a couple swings and misses since this is in moonman, so some interpreting is needed)

Pedantry for this level of discourse
Spoiler:
Right off the bat there's a pretty big problem: you're using "true" in a really goofy way, for what is, supposedly, to be read as a real argument against mathematical logic (rather than crank postings). In short, you seem to be meaning that a is deducible from A, but if not, do you mean to say that a is satisfiable, that a is valid, or that a is true in some "intended" model that we are supposed to have (because this is your philosophical position)? The second, generally, can be assumed to be equivalent of "can be deduced", but that makes some assumptions on how we are deducing - the others are both possible, however...so, I'll just assume you mean that we are working in first order logic, everything is standard, and etc.

Quibble
Spoiler:
I'm taking you literally when you conclude that "B is a subset of A", I am not assuming you mean "B is a subsystem of A" (not that I'm saying that would work). For example, the theory of abelian groups is a subsystem of the theory of groups while the axioms of groups are a subset of the axioms of abelian groups.

More Problems
Spoiler:
"But that would make all axiomatic systems, one system" is your main conclusion. However, you seem to neglect the case of different signatures. Your whole argument is supposing that you are dealing with axioms A and B with compatible signatures, so even if true, at best, you could conclude that all systems in the same signature were one system.

I guess, if that all panned out, that would still be a problem, sure - however, the problem here is that you seem to lack an awareness that this is even a thing. In other words, that you don't get the subject is real real evident. Go figure.

But, really, an argument against mathematics, in general, and logic, in general, shouldn't make this kind of confusion.

Spoiler:
a is true with respects to A, b is true with respects to B, and b is true with respects to a; thus, b is true with respects to A, implying B is true with respects to A, so B is a subset of A.

Spoiler:
Let A consist of a single statement x, let B consist of x and any other statement y that neither follows from, nor is followed from by, x. Let a consist of just x, let b consist of (for fun) x AND x. Then, there is a proof of a from A, a proof of b from a, and a proof of b from B; and, yet, B is not a subset of A.

Spoiler:
a follows from A, everything in B follows from A, b follows from B, b follows from a; thus, B is a subset of A.

Spoiler:
First, this reading really cuts down B once A is specified; meaning: even if it worked, your conclusion wouldn't even sort of follow.

The simple solution is to let B just be the conjunction of anything in A with itself - so x is in A, then x And x is in B. Then, B is certainly not a subset of A; but, as usual, I'm betting you screwed up by using the word "subset".

So...what then? I guess, then, you're right: if you assume B can be derived from A, then it follows that you can derive B from A. Of course, none of that has a thing to do with your "proof", it's just that you assumed it (which, as observed, is limiting enough that your grand conclusion doesn't follow...).

The Lone Ranger of Logic wrote:We have a number of true and false statements with respect to system 'A'.

If we assert that a false statement is true (or vice versa), then that newly true statement is a contradiction of what we have already established. We have created an inconsistency with respect to system 'A'.

Now, we take that inconsistency and formalise it as a set of axioms; 'a'. If our context for interpreting statements is still 'A' - then calling the statements 'axioms' hasn't changed anything. The inconsistency remains.

The only way for 'a' to be consistent is if we don't use 'A' as our context.

Inconsistent statements with respect to a system are always inconsistent with respect to that system.

If your axioms are consistent - then they must consist of only true, or only false statements with respect to WHATEVER system you decide you are actually using as the context for those axioms.

If we have no context - then we have no way to interpret the statements of 'a'. ('a' are not their own context - some people are saying some silly things in an attempt find fault - if statements have meaning independent of context then why are you bothering with axioms?).

Feel free to change your context as often as you like. It doesn't change the fact that for each context, inconsistent statements are always inconsistent.

Just bear in mind that the 'meaning' of axioms is tied to the context. Change the context and you change how the axioms should be interpreted. Change the context of the axioms and you change the system that those axioms describe.

Isn't the point of axioms to establish a known context so that axioms have a known meaning?

WTF?

1.) You have no idea what true means, you seem to be talking as if Peano Arithmetic and the standard naturals (not that you mention them, but as an example) were the same basic thing.

2.) You re mixing levels all over the place - I should just drop the cowboy theme and start making DJ jokes instead.

3.) Nope, the point of axioms is not to "establish a known context so that axioms have a known meaning". That's either backwards, wrong, or the opposite of what every major result of metalogic has shown us to be the case. In short, you have no clue what the Hell you're trying to say.

4.) All this talk about asserting a false statement 'a' and needing a context and so on, lots and lots of problems there, Disco Logic DJ, lots of problems. For example, you are aware that axioms, statements, etc. are all strings in a specific signature; so, any FO wff with a 2-ary relation symbol can be an axiom of set theory to a theorem in the theory of equivalence relations to an axiom extending the theory of linear orders, and so on. Neat fact: a load of things about equivalence relations contradict a load of stuff about linear orders, and, yet, any statement in one is a statement in the other
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JudeMorrigan
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Re: How do axioms specify a new system vs continuing an old

Forest Goose wrote:
Grindstone Cowboy wrote:A number of people have objected to my characterisation of axiomatic systems. Interestingly - no-one has offered up an official formal definition of axiomatic systems.

I assume posts like this are why Treatid still has posting privileges. Well, that and for the lulz. Just wanted to say I appreciate your efforts in both regards!

gmalivuk
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Re: How do axioms specify a new system vs continuing an old

Treatid: So far every thread you've started or participated in has been a demonstration of your unwillingness to accept anyone else's comments about the problems with your ideas. Since you're not listening to what other people say in the first place, why are you even posting in a forum?

If your next post does not indicate that you are at least trying to understand and take to heart other people's really careful and helpful criticisms of your ideas, I'm going to take away your posting rights entirely.
Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true.
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Twistar
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Re: How do axioms specify a new system vs continuing an old

Treatid wrote:Isn't the point of axioms to establish a known context so that axioms have a known meaning?

No. What does it mean that knights are only allowed to move in an L pattern? isn't it strange that bishops can move diagonally but rooks can move in straight lines? shouldn't kings be able to move 2 spaces forward rather than one?

All of the italicized words indicate questions that you can't really ask about rules of a game. They are just rules. You can also change the rules and start playing a new game.

Also you completely ignored my previous post which I am pissed about because it serves as a direct response to your most recent post as well.

arbiteroftruth
Posts: 481
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Re: How do axioms specify a new system vs continuing an old

Treatid, if there's anything like the universal axiom system that you're arguing exists, it's something like second-order logic. The rules of logic can be built up from a small number of basic concepts. Things like direct logical implication, substitution of variables, and universal statements. Concepts like these are considered so basic that any competent mind can grasp them, and so we're comfortable leaving them informally defined. Likewise, the act of associating certain symbols with these concepts is considered so basic that we're comfortable needing to explain it informally. Then, with that kind of rudimentary language in place, everything else can be axiomatised within that language.

But here's the thing you're missing and that people have tried to explain in this thread. The fact that all our formal axioms can be expressed in this kind of language, and that the axioms of a given system are taken to be true within that system, doesn't mean all of the axioms of every system are simultaneously taken to be true. The axioms coexist in one system only in the sense of being valid statements in the language of choice. The axioms of the language establish that various statements are well defined, not whether they are true or false. The decision to suppose that some statement is true or false is independent of that, and the act of making that decision for certain statements is the act of choosing a set of axioms to work with. I can later change my mind, take some of those same statements as having the opposite truth value they did before, and suddenly I'm working in a different set of axioms. Again, they coexist only in the sense that I express them in the same language, not in the sense of simultaneously accepting all of them as true.

Here's another way to put it. The statements all are written within a given language for basic logic, and within that system of logic I decided to start deriving some tautologies. If I'm particularly interested in tautologies of the form "(some specific stuff)-->(some other stuff)", then the "some specific stuff" is the axiomatic system I'm working in. In that sense, all of math could be consisdered a subset of logic, but it's useful to categorize it more finely than that, by explaining what specific stuff I'm going to put on the left side of my tautologies.

z4lis
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Re: How do axioms specify a new system vs continuing an old

treatid, you have an extremely narrow (i.e. incorrect) idea about what axiomatic mathematics actually is, and I think your problem rests totally on

1) the application of the axiomatic method to set theory itself and/or
2) the adoption of set theory as a foundation for mathematics

The step that took me a long time to accept is that mathematics, as a whole, is completely independent of anything that has ever or will ever be done in the subject of set theory. If set theory were in any way necessary to do mathematics, why did we have numerous results about geometry, arithmetic, prime numbers, calculus, and topology before set theory was ever founded? Mathematics is NOT encompassed by set theory. It's the other way around entirely.
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.

Gwydion
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Re: How do axioms specify a new system vs continuing an old

Treatid wrote:We have a number of true and false statements with respect to system 'A'.

If we assert that a false statement is true (or vice versa), then that newly true statement is a contradiction of what we have already established. We have created an inconsistency with respect to system 'A'.

Now, we take that inconsistency and formalise it as a set of axioms; 'a'. If our context for interpreting statements is still 'A' - then calling the statements 'axioms' hasn't changed anything. The inconsistency remains.

The only way for 'a' to be consistent is if we don't use 'A' as our context.

Inconsistent statements with respect to a system are always inconsistent with respect to that system.

If your axioms are consistent - then they must consist of only true, or only false statements with respect to WHATEVER system you decide you are actually using as the context for those axioms.

If we have no context - then we have no way to interpret the statements of 'a'. ('a' are not their own context - some people are saying some silly things in an attempt find fault - if statements have meaning independent of context then why are you bothering with axioms?).

Feel free to change your context as often as you like. It doesn't change the fact that for each context, inconsistent statements are always inconsistent.

Just bear in mind that the 'meaning' of axioms is tied to the context. Change the context and you change how the axioms should be interpreted. Change the context of the axioms and you change the system that those axioms describe.

Isn't the point of axioms to establish a known context so that axioms have a known meaning?

No again. Let me be more explicit here - let's define an axiomatic system A as one in which a, b, c, and d are true. From these, we can deduce e, f, and g. Now, take a system B in which a, b, and c are true but d is false. Maybe here, e is true, f is false, but g is independent of the axioms of B. We can further define systems B1 (a, b, c, !d, g) and B2 (a, b, c, !d, !g) either of which might be more appropriate depending on the situation we are trying to model.

Neither A nor B assert that x and !x are true simultaneously (for any statement x). If they did, they would be inconsistent and thereby not useful. System C (a, b, c, !d, f) is a concrete example of such a system - can you see why? System D (a, b, c, d, f) is consistent, but since f follows from the others it doesn't need to be called an axiom. We typically try to choose axioms in such a way as to minimize the number of axioms (by choosing mutually independent statements), and to keep the statements as "simple" as possible (so it looks even cooler when we derive seemingly complicated results, and so we can more easily apply the models we create).

The point of axioms is to define the way a system works, and then we can explore the implications of these rules. If you don't think a particular system is a good model for whatever you're investigating, pick a different one! But keep in mind that system needs to be consistent, and you may want it to have other properties as well. Does this make any more sense? If not, with which part do you disagree?

Forest Goose
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Re: How do axioms specify a new system vs continuing an old

z4lis wrote:treatid, you have an extremely narrow (i.e. incorrect) idea about what axiomatic mathematics actually is, and I think your problem rests totally on

1) the application of the axiomatic method to set theory itself and/or
2) the adoption of set theory as a foundation for mathematics

The step that took me a long time to accept is that mathematics, as a whole, is completely independent of anything that has ever or will ever be done in the subject of set theory. If set theory were in any way necessary to do mathematics, why did we have numerous results about geometry, arithmetic, prime numbers, calculus, and topology before set theory was ever founded? Mathematics is NOT encompassed by set theory. It's the other way around entirely.

Depending on what you mean, I disagree. Yes, we did have results before set theory, but the scope and depth of those results was not anywhere near what we have demonstrated since then.

However, the major thing is that if set theory falls apart, it's taking lots and lots of proofs down with it; and lots of theorems would need to be recast in a totally different way (when you say "Let V be a vector space over a field k", how are you defining V and k? Do you have non-set theoretical definitions handy? Do you have proofs that those definitions don't contradict anything done with vector spaces, thus far, and, yet, won't also be called into question by a failure of sets - and those proofs don't depend on some theory of sets working? I don't - and this applies to almost everything in mathematics)***.

So, sure, we could still count, I'm sure we could find a way to do derivatives and integrals, and basic combinatorics, and muck around with equations, but beyond that, things are going to get hairy. Set theory failing is a lot like shutting off the lights: the room is still there, as is everything in it, but good luck navigating that room in the dark (and that's the best possible situation).

If you mean this all in the context of if ZFC fails*, and you are thinking in the sense of there being a problem at some huge cardinality, or that failure being something that wouldn't also break ZF, MK, or NF (or throw them into question); then, yes, that wouldn't be ultradevastating (it would still raise a lot of thorny questions, and it would still require going over lots and lots of mathematics with a fine tooth comb - and I would't really call that "set theory failing" as it is picking the most benign possible sense of that), but if set theory, in general, collapses, then there's going to be a lot of damage done and a lot of things drug down with it - the idea of sets, as foundational objects, instead of a specific theory is, essentially, more a matter of logic than of mathematics; so talking about sets failing sounds a lot like talking about a part of logic collapsing, and that's not so good for the whole enterprise (and it's certainly not something that would be a minor blip on the radar of mathematics).

*I am discussing the failure of set theory because if your post is accurate, then that failure shouldn't be a massive deal - and, also, since that is a lot of the topic, thus far, it seems the natural way of reading it, namely.

**(There is no ** in the text, this is just an add on) We also did some good physics before QM, but I think it would be a massive blow to physics if we found out that everything from wavefunctions to path integrals was just straight up flawed, not that some esoteric angle needed adjusted, but that if those things just went away - it would be a really really big deal.

***I really really want to stress the implications of this. If we can replace all of our definitions with equivalent ones from something else, so that nothing changes, then what exactly would have failed? In other words, if we can just swap over to something else, but no results need to change, then whatever happened doesn't sound like a failure since it didn't break anything, anywhere. So, that means that we are going to need something that is not equivalent to what we do with the things we have now - our "new" vector spaces would have to be different, in someway, than in our old. Maybe they can do less things, maybe they do different things, etc. Whatever the case, the ramifications of that will need to be sorted out, and they will need sorting all over the place, in every field - why does everyone always assume that all of our theorems would just keep working? I don't see anything that evidences that they would. It strikes as a bit naive that everyone uses set theory as their basic tool of reasoning and language, but then acts like all of this could easily be done without it, as if set theory was some neat esoteric subject, but of little consequence - again, despite being so built into mathematics that it is the basic language that everyone uses. (I don't mean any offense by this, this is just a really common opinion, and I have yet to see any justification of it; just because most people don't need to deal with the esoterica of foundations, doesn't mean that foundations aren't actually founding.)

JudeMorrigan wrote:I assume posts like this are why Treatid still has posting privileges. Well, that and for the lulz. Just wanted to say I appreciate your efforts in both regards!

Thank you very much:-)
Last edited by Forest Goose on Tue Feb 10, 2015 5:00 am UTC, edited 1 time in total.
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Treatid
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Re: How do axioms specify a new system vs continuing an old

gmalivuk wrote:Treatid: So far every thread you've started or participated in has been a demonstration of your unwillingness to accept anyone else's comments about the problems with your ideas. Since you're not listening to what other people say in the first place, why are you even posting in a forum?

If your next post does not indicate that you are at least trying to understand and take to heart other people's really careful and helpful criticisms of your ideas, I'm going to take away your posting rights entirely.

Thank you very much to everyone who has made a sincere effort to try and understand what I'm saying and responded with the intent of holding a constructive dialogue.

Despite appearances - I have been listening to everything that people say. I haven't found everything I have been told to be self evident. I tend to be pedantic. I like my justifications to be justified.

When axiomatic mathematics falls back on informal language, I feel that a justification is missing. This is reflected in the existence of the Foundational Crises in mathematics. There is a gap.

There is a piece of axiomatic mathematics that has to be taken on faith. For most people, this faith is justified by the fact that axiomatic mathematics appears to have been very successful.

So we are in a position where you have really, really strong evidence that axioms work - but can't quite define informal languages well enough to turn that evidence into a rock solid, unambiguous proof. If it was possible to prove axiomatic mathematics; there wouldn't be a foundational crises.

In mathematics, lots of evidence does not equal proof. A theorem may hold for almost its entire domain - yet if there is one exception - then the theorem is disproved.

You are providing me with lots of evidence in favour of the validity of axiomatic mathematics. But unless someone resolves the Foundational Crises - that evidence must fall short of outright proof.

...

It has been argued that complete and certain knowledge cannot be attained. That sooner or later we have to take a practical approach and simply accept a degree of doubt and uncertainty. One might argue that Quantum Uncertainty supports this view. This view might be correct. But what if it isn't correct and we give up looking for the alternative?

...

A popular refrain is that I don't understand axiomatic mathematics. I'm picking bits that I half understand and trying to slap them together. To an extent this is true. Nobody fully understands axiomatic mathematics. That is one of the issues with boot-strapping from informal languages. If informal languages are not 'formally defined and all the associated assumptions stated' - then no-one can be sure what the full definition of axiomatic mathematics is. To claim that my interpretation of axiomatic mathematics is wrong when you haven't fully defined axiomatic mathematics is rather akin to the pot calling the kettle black.

There is nothing shameful about having a partial understanding of axiomatic mathematics. It is impossible to have a full understanding of axiomatic mathematics. What is shameful is pretending that you know precisely what axiomatic mathematics entails.

...

That I require proof of the validity of axioms (rather than mere evidence) as a counter argument (to my position) may seem unfair. Except that there is no onus on you to demonstrate the validity of axioms.

Proving that a system is consistent is tricky at the best of times. It is accepted that there is a possibility that ZFC might not be consistent. However, until an inconsistency is demonstrated, ZFC is taken on faith. We'll deal with an inconsistency if it should arise.

I think that axiomatic mathematics is inconsistent. The onus is on me to demonstrate that inconsistency.

Based on the response to my last post, it appears that I don't understand the transition from statements in system A to axioms of systems B. I know that axioms are assumed to be true. What I'm not clear on is the mechanism that makes statements that are inconsistent with respect to system A, consistent when they become axioms of system B.

I assumed that system A provided the context for interpreting the axioms of system B. Is this not correct? How do we interpret the axioms of system B? Is there an alternate context?

z4lis wrote:treatid, you have an extremely narrow (i.e. incorrect) idea about what axiomatic mathematics actually is, and I think your problem rests totally on

1) the application of the axiomatic method to set theory itself and/or
2) the adoption of set theory as a foundation for mathematics

The step that took me a long time to accept is that mathematics, as a whole, is completely independent of anything that has ever or will ever be done in the subject of set theory. If set theory were in any way necessary to do mathematics, why did we have numerous results about geometry, arithmetic, prime numbers, calculus, and topology before set theory was ever founded? Mathematics is NOT encompassed by set theory. It's the other way around entirely.

I reference ZFC Set Theory as an example of an axiomatic system. ZFC is elegantly abstract and can be used to specify the axioms of almost every other axiomatic system. But then, any sufficiently sophisticated axiomatic system can specify the axioms of almost all other axiomatic systems.

If axiomatic mathematics was an entirely self contained system, I would be a happy bunny. The trouble is that axiomatic mathematics must be boot-strapped using informal language. A language which, by definition, is not formally defined.

arbiteroftruth wrote:Treatid, if there's anything like the universal axiom system that you're arguing exists, it's something like second-order logic. The rules of logic can be built up from a small number of basic concepts. Things like direct logical implication, substitution of variables, and universal statements. Concepts like these are considered so basic that any competent mind can grasp them, and so we're comfortable leaving them informally defined.

History is littered with self evident truths that turned out to be anything but.

The fact that we can't nail down definite definitions rings alarm bells for me.

Likewise, the act of associating certain symbols with these concepts is considered so basic that we're comfortable needing to explain it informally. Then, with that kind of rudimentary language in place, everything else can be axiomatised within that language.

If it was genuinely that basic - then we would be able to express it in an unambiguous fashion. Arguments along the lines of "it is obvious" have come crashing down within mathematics time and time again. I see no reason to believe that this instance of "it is obvious" is any different to those instances where it turned out not only to not be obvious - but straight up wrong.

But here's the thing you're missing and that people have tried to explain in this thread. The fact that all our formal axioms can be expressed in this kind of language, and that the axioms of a given system are taken to be true within that system, doesn't mean all of the axioms of every system are simultaneously taken to be true.

I'm not missing this point - this is the point I'm questioning.

Individual axiomatic systems are assumed to be independent. But this view may not be consistent. We use (e.g.) ZFC to describe the axioms of some other system. There is an explicit relationship between ZFC and the system being described. If two things are connected, however indirectly, then they must be part of the same system.

Perhaps the argument is that it is the descriptions that are connected - but the subject of those descriptions are independent entities. But then you are arguing that individual statements are not part of the system.

The axioms coexist in one system only in the sense of being valid statements in the language of choice.

Yes, yes!. This is my point. The axioms do coexist within one system.

Your implicit assumption is that there is some other sense in which, not only do axioms not coexist within one system, but that this other sense somehow cancels out the first sense.

The axioms of the language establish that various statements are well defined, not whether they are true or false. The decision to suppose that some statement is true or false is independent of that, and the act of making that decision for certain statements is the act of choosing a set of axioms to work with. I can later change my mind, take some of those same statements as having the opposite truth value they did before, and suddenly I'm working in a different set of axioms. Again, they coexist only in the sense that I express them in the same language, not in the sense of simultaneously accepting all of them as true.

You talk as if the two are mutually exclusive. (although the particular property we are interested in immediately is consistency - not truth. Axioms are assumed to be true - they are not assumed to be consistent).

Here's another way to put it. The statements all are written within a given language for basic logic, and within that system of logic I decided to start deriving some tautologies. If I'm particularly interested in tautologies of the form "(some specific stuff)-->(some other stuff)", then the "some specific stuff" is the axiomatic system I'm working in. In that sense, all of math could be consisdered a subset of logic, but it's useful to categorize it more finely than that, by explaining what specific stuff I'm going to put on the left side of my tautologies.

Generally agree with this. I'll just expand that while categorisation can be useful - it doesn't necessarily represent an actual division. An element may occupy more than one category (e.g. a set of statement with respect to A, axioms of B). One category does not exclude the other category.

Twistar wrote:
Treatid wrote:Isn't the point of axioms to establish a known context so that axioms have a known meaning?

No. What does it mean that knights are only allowed to move in an L pattern? isn't it strange that bishops can move diagonally but rooks can move in straight lines? shouldn't kings be able to move 2 spaces forward rather than one?

It means that the knight moves in a particular way. I'm not asking for a justification of one choice over another.

You still have to specify the movement of the pieces. That movement is the meaning. I'm happy with really, really abstract axioms. But axioms have to tell you something. They have to tell you what symbol manipulation they allow, or forbid. It is this abstract 'meaning' that I intend.

It doesn't matter what the rules of chess are. But we do have to have some rules.

All of the italicized words indicate questions that you can't really ask about rules of a game. They are just rules. You can also change the rules and start playing a new game.

Completely agree. We are on the same page with regard to this.

I can see how my use of 'meaning' might lead you to believe that I meant some philosophical, navel gazing "meaning of life" thing.

That isn't what I meant. Axioms are a set of rules. Those rules can be anything you like; provided that they can be distinguished from some other set of rules.

Also you completely ignored my previous post which I am pissed about because it serves as a direct response to your most recent post as well.
Twistar wrote:
Treatid wrote:
Twistar wrote:You suggested we could mix true and false statements from A to form a set of axioms. Given that A is the context for interpreting those axioms - mixing true and false statements will lead to a contradiction.

This is true only if you continue to interpret those axioms with respect to A. But you're not supposed to interpret them with respect to A now that you're working in system B. You're supposed to treat them as true and see what happens.

We have to express the axioms of chess in a language. "Move forward two squares and left one square."

If we take away the language, then the rules disappear too.

We can't interpret a system in terms of itself. If we could do that - we wouldn't need axioms in the first place. System A is the language which allows us to interpret the symbols of system B. Throw away system A and all you are left with are a bunch of symbols that you don't know how to interpret.

As others have shown, it is possible and often done that we shift which axioms we are using and this can lead to multiple systems (based on different axioms) which are all consistent (at least we think they're consistent).

I agree that many people assume that axioms are independent of what has gone before. At the same time, those axioms must be part of a language in order to be able to do anything with them. You can freely choose whatever statements of that language will make up your axioms. You can't switch language half way.

The previous posters bring up good points. Do you take issue with the axioms of chess?

As noted - I don't take issue with the axioms of chess.

Gwydion wrote:No again. Let me be more explicit here - let's define an axiomatic system A as one in which a, b, c, and d are true. From these, we can deduce e, f, and g. Now, take a system B in which a, b, and c are true but d is false. Maybe here, e is true, f is false, but g is independent of the axioms of B. We can further define systems B1 (a, b, c, !d, g) and B2 (a, b, c, !d, !g) either of which might be more appropriate depending on the situation we are trying to model.

Neither A nor B assert that x and !x are true simultaneously (for any statement x). If they did, they would be inconsistent and thereby not useful. System C (a, b, c, !d, f) is a concrete example of such a system - can you see why? System D (a, b, c, d, f) is consistent, but since f follows from the others it doesn't need to be called an axiom. We typically try to choose axioms in such a way as to minimize the number of axioms (by choosing mutually independent statements), and to keep the statements as "simple" as possible (so it looks even cooler when we derive seemingly complicated results, and so we can more easily apply the models we create).

The point of axioms is to define the way a system works, and then we can explore the implications of these rules. If you don't think a particular system is a good model for whatever you're investigating, pick a different one! But keep in mind that system needs to be consistent, and you may want it to have other properties as well. Does this make any more sense? If not, with which part do you disagree?

I agree that this is a good illustration of axioms as they are practiced.

Neither A nor B assert that x and !x are true simultaneously (for any statement x).

This is where I disagree. I don't think A and B are distinct.

Your argument is that A and B are distinct systems. As such, a statement in B has no impact on A.

However, we are using A to specify how the axioms of B should be interpreted. B is not, in fact, distinct from A.

If we assert that B is not expressed in terms of A, then what is B expressed in terms of? Without some context, the axioms of B are just symbols. We don't know how to manipulate those symbols without a context. B cannot define itself. So we are left with A being an implicit component of B.

That is to say - when statements within system A become axioms of system B - they don't stop being statements within system A. They are both statements within system A and also axioms of system B.

I agree that the conventional view is that systems A and B are distinct.

I think that a careful examination shows that, in fact, B is a continuation of A and not a distinct system. That if we use the rules of A to specify B, then we must use all the rules of A.

Forest Goose
Posts: 377
Joined: Sat May 18, 2013 9:27 am UTC

Re: How do axioms specify a new system vs continuing an old

I see that you don't bother responding to me, what a shame. Oh well, I'm just going to point out the blatant over the top conceit that exists in your post.

DJ Disco Logic wrote: I haven't found everything I have been told to be self evident.

Why would you expect this? I've spent most of every day for most of my adult life studying this subject, I still feel humbled by it - you don't even know the terms, yet you're expecting it to be self-evident? Seriously, this is the height of arrogance.

DJ Disco Logic wrote:There is a piece of axiomatic mathematics that has to be taken on faith. For most people, this faith is justified by the fact that axiomatic mathematics appears to have been very successful.

So we are in a position where you have really, really strong evidence that axioms work - but can't quite define informal languages well enough to turn that evidence into a rock solid, unambiguous proof. If it was possible to prove axiomatic mathematics; there wouldn't be a foundational crises.

In mathematics, lots of evidence does not equal proof. A theorem may hold for almost its entire domain - yet if there is one exception - then the theorem is disproved.

Nope, sorry, you are confused as to what the term "certainty" means, I think.

In the same vein, you are using the standards of mathematics to apply to the definitions of it - that's like expecting an inductive proof of induction, then being dismayed that that can't work. You can't mathematically prove mathematics, the whole concept of that is quite off, to say the least. Part of math and logic is dealing with proofs, etc. why are you anticipating that we have those, for those notions, prior to having them? If you don't accept proofs, why would you accept a proof of proofs, anyway, even if that made sense? Maybe you should read a textbook and quit lamenting that no one seems to be able to give you the impossible (and in a self evident way!).

Just so I'm very clear: we don't have faith in mathematics because it works, mathematics is exactly what it is; as a whole, it can't not work. Math isn't trying to do something other than what it is; science is trying to figure out the world, that can fail, math is trying to be what it is defined to be, that cannot. You're point isn't even wrong, it's just that asinine - it'd be like pointing out that "ain't" should be considered an English word and, since some disagree, therefore we only have faith in poetry, but it really doesn't work...you have, quite literally, no idea, at all, about any of this.

I was wrong before, you're not the guy who read a Wikipedia article and felt dumb, you're the guy who disputes the existence of trucks because NASCAR races cars - sorry, I can't think up a ridiculous enough analogy to actually capture the ridiculousness of the reality of your arguments.

DJ Disco Logic wrote:It has been argued that complete and certain knowledge cannot be attained. That sooner or later we have to take a practical approach and simply accept a degree of doubt and uncertainty. One might argue that Quantum Uncertainty supports this view. This view might be correct. But what if it isn't correct and we give up looking for the alternative?

Translation wrote:Ha ha blah derp derp Quantum

Don't ever appeal to the uncertainty principle in debates like this, it makes you look really really backwards - seriously, that's one of the biggest red flags there is. You might as well post your proof of Fermat's Last Theorem that starts out with "Let x = 1/0, which I've totally defined!" or start your political screed with "Let me just compare this to Hitler and the nazis...".

DJ Disco Logic wrote:A popular refrain is that I don't understand axiomatic mathematics. I'm picking bits that I half understand and trying to slap them together. To an extent this is true. Nobody fully understands axiomatic mathematics. That is one of the issues with boot-strapping from informal languages. If informal languages are not 'formally defined and all the associated assumptions stated' - then no-one can be sure what the full definition of axiomatic mathematics is. To claim that my interpretation of axiomatic mathematics is wrong when you haven't fully defined axiomatic mathematics is rather akin to the pot calling the kettle black.

Ah, there it is, a real disproof of ZFC, there is so much conceit, smugness, and arrogance in this quote that it can only be measured by 0=1, or at least a reinhardt cardinal.

So, let me get this right, it's okay that you don't understand the subject because, by your arguments (about the subject, from you, who doesn't understand it), you've proven that it cannot be understood - it never crosses your mind that maybe, just maybe, you don't understand it and, as such, you should question your arguments, thus, concluding that you just don't understand it? Not even a little?

So, you, somehow, figured out the subject enough to blast it away with your powerful disco logic, thereby proving that your failure to understand the subject, or even its terms, is a result of the subject's failure...so much smugness...cannot continue insulting...I have never typed this acronym before, but all I can say is O M G Also, Holy Fuck! That's the most arrogant bullshit I've ever heard from another human being, if we were swapping fight stories over beers, you'd be claiming to have beat up the ocean in an MMA style fight at this point...

I have to stop here, lest I start replying in words that will earn me a ban for their nastiness...just wow!

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I lied, I have to respond to the bullshit math too, I just can't not, it galls me to know that it is just sitting there, smugly thinking I have nothing to say (not you, DJ Disco, literally, the bad math...yep, your math is so bad that it has gained sentience, just to mock and taunt from it's dark little realm...that got weird, allow me to continue)

Before that, though, let's play the game of sanity check, find the flaw, in your own words:

Presburger arithmetic is decidable, Peano arithmetic is not. Presburger arithmetic is just Peano arithmetic minus multiplication, but we can define multiplication using repeated additions; thus, Presburger arithmetic has multiplication, thus, it is both decidable and not, thus, Godel, and those other foundational fools, were wrong.

My Reply, Spoilered For Size Reasons
Spoiler:
DJ Disco Logic wrote:I think that axiomatic mathematics is inconsistent. The onus is on me to demonstrate that inconsistency.

1.) You keep mixing levels, do you think axiomatic methods are inconsistent, logic is inconsistent, or every system of axioms is inconsistent? You keep bouncing around to all of these and talking like they were all the same thing. In reality, you are the murderer from The Murders in the Rue Morgue, you sound like you're speaking at various levels, but really you're just making noises - except, here, no Dupin is needed, most anyone with a math background could call bullshit on you're nonsense.

2.) And you've failed, horribly, to meet that onus. My money is on your continued failure.

DJ Disco Logic wrote:Based on the response to my last post, it appears that I don't understand the transition from statements in system A to axioms of systems B. I know that axioms are assumed to be true. What I'm not clear on is the mechanism that makes statements that are inconsistent with respect to system A, consistent when they become axioms of system B.

You're talking in such a weird way, I'm not sure how to make the point clear to you - there is no such transitioning. If some statements together lead to a contradiction, they're inconsistent, if they don't, they aren't. If A + statement leads to a contradiction, then that statement isn't consistent with A - and if A leads to that statement, then A would be inconsistent. There is no transitioning to B, we just do the same thing with B, A doesn't factor in.

For example, "We don't play baseball in the rain" and "We're playing baseball" and "It is raining" are inconsistent, "We don't play baseball in the rain" and "We're playing baseball" and "It's not raining" are not inconsistent - there is no transition taking place from one to the other, they're just different collections of statements.

DJ Disco Logic wrote:If axiomatic mathematics was an entirely self contained system, I would be a happy bunny. The trouble is that axiomatic mathematics must be boot-strapped using informal language. A language which, by definition, is not formally defined.

So, your problem is what? That mathematics needs to be defined? Again, you mix levels, that's all you do - you're confusing doing math with defining math with the foundations of math, those are all separate things, and not all of them are on the same level of discourse. You can't expect proofs of proofs - especially if you don't accept proofs in the first place - that whole notion makes no sense.

DJ Disco Logic wrote:History is littered with self evident truths that turned out to be anything but.

The fact that we can't nail down definite definitions rings alarm bells for me.

We do have definite definitions, the problem is that you think informal language means uncertain language - as in we can't even define what a string of symbols is without formal language. The definition of a string of symbols is pretty definite, I can definitely tell you if you're wrong about what that is.

DJ Disco Logic wrote:If it was genuinely that basic - then we would be able to express it in an unambiguous fashion. Arguments along the lines of "it is obvious" have come crashing down within mathematics time and time again. I see no reason to believe that this instance of "it is obvious" is any different to those instances where it turned out not only to not be obvious - but straight up wrong.

So, let me understand this, you anticipate that the definition of a string of symbols will come crashing down? That's a possibility in your world?

Again, you do realize that the notion of that makes no sense, right? You realize that mathematics would still be so defined, even if a string of symbols weren't what we thought, right? See, mathematics doesn't define strings of symbols, those are defined, then used to start doing math.

Consider: it is discovered that we are living in the Matrix, was I wrong when I said "Grass is green"? Was I wrong when I said "The coffee maker is real"? Of course not, at worst, I've just learned that those terms have some different ultimate meaning, nonetheless, I could not have meant them outside of the universe that is, in terms of themselves, they mean exactly what they always did in relation to themselves - was I wrong about where my ass has been before anyone knew about elementary particles making up chairs, did chairs somehow change on that date?

DJ Disco Logic wrote:Individual axiomatic systems are assumed to be independent. But this view may not be consistent. We use (e.g.) ZFC to describe the axioms of some other system. There is an explicit relationship between ZFC and the system being described. If two things are connected, however indirectly, then they must be part of the same system.

Perhaps the argument is that it is the descriptions that are connected - but the subject of those descriptions are independent entities. But then you are arguing that individual statements are not part of the system.

Sets do get used a bunch, but you aren't aware of how, where, or in what way. Here's some axioms: "For all x For all y: xy = yx", "For all x For all y For all z: (xy)z = x(yz)". Where did I use ZFC?

You seem to be talking about model theory, but that's not what you are actually saying, so I'm not going to try and figure out a way to recast your argument as if it had merit - because you certainly didn't bother.

DJ Disco Logic wrote:Yes, yes!. This is my point. The axioms do coexist within one system.

Your implicit assumption is that there is some other sense in which, not only do axioms not coexist within one system, but that this other sense somehow cancels out the first sense.

"I love Sandy", "John knows Bill"..."knows" and "love" are both 2-ary relationships, so what? Is "love" now "knows"? You are saying the mathematical version of this...please stop.

DJ Disco Logic wrote:Axioms are assumed to be true - they are not assumed to be consistent

Fascinating, do tell me more. The best way I know how to read this is: "Axioms are assumed to have a model - they are not assumed to have a model", that seems rather wrong, please continue to redefine the terms of this subject so blithely.

DJ Disco Logic wrote:You still have to specify the movement of the pieces. That movement is the meaning. I'm happy with really, really abstract axioms. But axioms have to tell you something. They have to tell you what symbol manipulation they allow, or forbid. It is this abstract 'meaning' that I intend.

If I were nice, I wouldn't bitch about this, but since this is a disproof of all of logic, I feel you deserve it. Axioms aren't really telling you how to manipulate symbols, that's logic, axioms are telling you the starting point of proofs (if they tell you anything). "There exists x so that for all y not yRx" does not tell me how I can manipulate symbols, it does let me prove "There is x so that not xRx".

DJ Disco Logic wrote:That isn't what I meant. Axioms are a set of rules. Those rules can be anything you like; provided that they can be distinguished from some other set of rules.

Axioms are a set of strings, logic provides the rules. I know, I know, casually this isn't how it is usually discussed, but you're disproving the subject, you don't get to be casual.

DJ Disco Logic wrote:We can't interpret a system in terms of itself. If we could do that - we wouldn't need axioms in the first place. System A is the language which allows us to interpret the symbols of system B. Throw away system A and all you are left with are a bunch of symbols that you don't know how to interpret.

No, that's wrong. Axioms are not the language, the language is given by the signature and the logic, the axioms are just wff's made using those. The theory of equivalence relations and ZFC are both in the same signature, "For all x For all y: xRy -> yRx" is an axiom of one and something I can disprove with the other, neither is the "language" I am stating it in. Perhaps you should buy one of those handy 20+ textbooks I linked earlier.

DJ Disco Logic wrote:I agree that many people assume that axioms are independent of what has gone before. At the same time, those axioms must be part of a language in order to be able to do anything with them. You can freely choose whatever statements of that language will make up your axioms. You can't switch language half way.

This contradicts 95% of what you've said prior, but, hey, at least you're getting closer, sort of. I have no idea where this talk of switching comes from, though.

DJ Disco Logic wrote:This is where I disagree. I don't think A and B are distinct.

Your argument is that A and B are distinct systems. As such, a statement in B has no impact on A.

However, we are using A to specify how the axioms of B should be interpreted. B is not, in fact, distinct from A.

If we assert that B is not expressed in terms of A, then what is B expressed in terms of? Without some context, the axioms of B are just symbols. We don't know how to manipulate those symbols without a context. B cannot define itself. So we are left with A being an implicit component of B.

That is to say - when statements within system A become axioms of system B - they don't stop being statements within system A. They are both statements within system A and also axioms of system B.

I agree that the conventional view is that systems A and B are distinct.

I think that a careful examination shows that, in fact, B is a continuation of A and not a distinct system. That if we use the rules of A to specify B, then we must use all the rules of A.

So, you admit you don't understand the concepts involved, you clearly cannot use the terms, but you have carried out this careful examination, correctly...that's rather hard to believe.

Even harder to believe is that you are disagreeing with a definition...you can't disprove a definition. By definition, they are separate.

So, what you mean to say is: "upon careful examination, I've decided to redefine the terms in such a way that I'm correct, thus, you're all wrong".
Last edited by Forest Goose on Wed Feb 11, 2015 10:02 am UTC, edited 2 times in total.
Forest Goose: A rare, but wily, form of goose; best known for dropping on unsuspecting hikers, from trees, to steal sweets.

Twistar
Posts: 445
Joined: Sat May 09, 2009 11:39 pm UTC

Re: How do axioms specify a new system vs continuing an old

Edit: Disclaimer, I'm not an expert in set theory or anything so I might get some terms and distinctions incorrect. If you know better than me point out flaws. I think I have enough of the general idea to make my point and help Treatid get an understanding..

Ok, I read your whole post and I just have a few things to say. I think there's concrete headway to be made here. First:
Treatid wrote:There is a piece of axiomatic mathematics that has to be taken on faith. For most people, this faith is justified by the fact that axiomatic mathematics appears to have been very successful.

You could have stopped the post here. What you have said is correct. You understand that it is correct, what you don't understand is that it is not a problem. You think it is a problem that a piece of axiomatic mathematics has to be taken on faith. Namely, you think it is a problem the formal mathematics must be founded on informal language. What I have been trying to you (more explicitly than others I think..) is that this is not a problem and that in fact all knowledge is at some point based on faith if you keep asking "why why why" and apply rigorous logic. I encourage you to find a statement which can't be traced back to some evidence which must be taken on faith. stuff like "I think therefore I am" gets you in the ballpark of answers I'll accept..

Ok, the next thing I want to say has to do with this A and B business and I think we can make real headway here. I think you are misunderstanding the hierarchy of languages, theories, axioms etc. I'll draw a diagram for this but let me type it up in words first.
At the top of this hierarchy we have a language. Let's use English as our example
In English (L - for language) there are rules (S - syntax or grammar) which tell us how to put words (atoms*) together in legal ways to form sentences.

Now, I'm going to divide English into different forms of the English language. Let's narrow our focus down to forms of poetry. Let's look at Haikus (system A) and Acrostics (system B)**.
a - axioms of system A) a Haiku is a three line poem in which the first and last line have 5 syllables and the middle has 7.
b - axioms of system B) In an acrostic you pick a base word and each letter from that word corresponds to a line of the poem. The first word of each line must begin with the corresponding letter of the base word. The lines of the poem should somehow relate to the base word.

Haiku:
one plus one is two
that is what they say to you
yet i've not seen proof

Acrostic
Long I've tried to claim
Once axioms are set
Gripping to absolutes
Invites nothing but
Confusion and pain!

Ok that was fun but let me get back to the point here. My acrostic and haiku would be examples of derived statements (or theorems) within the subsystems of haiku or acrostic. Does this make sense? Notice how haikus and acrostics are at the same level of hierarchy with respect to the language, English. You seem to be insisting that system B should be interpreted in the context of system A. This is an unnecessary mixing of different levels of the hierarchy. what you should instead be thinking is that both system A and B should be interpreted in the context of the underlying language L. Furthermore, the axioms a and b of A and B respectively are also stated in the language L. Notice that in my English example I literally wrote out the axioms a and b in English. the axioms of a subsystem (or theory) must follow the syntax (S) of the language we are working in.

Let me make the analogy to math complete
In math the language L is most likely second order logic***. The rules, S, of our language L can be found on Wikipedia or an introductory book on formal logic. The rules are things like: binary connectors must be sandwiched between to well formed formulas which means something like "A and B" is a sentence but "and A B" is not a sentence. So far I've only talked about the formal language, no math yet. Now lets get to math. Math is a theory within formal logic, so that means it has axioms, a. Those axioms are stated in the language L and the most popular set of axioms of ZFC. Now we use ZFC to derive all of math. so ZFC is equivalent to the rules about haikus and acrostic and the fundamental theorem of calculus is equivalent to my haiku****. You should also be aware that there are other systems on the same rung of the hierarchy as ZFC, but we usually don't talk about them because I don't think people other than mathematical logic or set theory people think about them very often.. But you need to realize that it is ZFC and one of these equivalent level theories such as Morse-Kelley (MK) set theory constitute the A and the B you are talking about. It is INCORRECT to think second order logic is system A and ZFC is system B. I think this is where you're getting confused... I think this is what people are criticizing you for when they say you're "mixing levels" and it seems to indicate you don't correctly understand how the levels work.

So the TLDR version is that you don't need to worry about interpreting B in the context of A because you should instead interpret both A and B within the context of language L.

But let me anticipate your objection.
Treatid: What is formal logic ultimately based on?
Twistar: Informal language
Treatid: So the whole system stands on shaky ground, this is a major problem!
Twistar: Not quite. You said "If axiomatic mathematics was an entirely self contained system, I would be a happy bunny." Axiomatic mathematics is entirely self contained***** WITHIN THE CONTEXT of second order logic. So you should stop drawing issue with mathematics PERIOD. You are struggling with the gap between having an informal language and then jumping to a formal language. My response to you goes back to the beginning of this post. That leap from informal to formal is a leap of faith. It ultimately relies on faith regarding human communication and that is FINE. The reason that is fine is because ALL human knowledge require some leap of faith. The beauty of mathematics is that all of the amazing and useful mathematical results and almost all of physics are able to follow from some (in my opinion) very very tiny leaps of faith. Furthermore, we can pinpoint EXACTLY what that leap of faith consists of which is a lot more than can be said about other branches of human knowledge.

*atom is the name for the smallest unit of a language. In formal language atoms are usually symbolized by letters like "a" or "A".
**From my choice of types of poems you can tell it's been a while since I've thought about poetry!
***Don't worry, I'll get to the question of what second order logic is based on. Also, starting at this point in the post some of the specific details may need to be fact checked by an expert.
****I'll put that sentence on my resume.
*****rather it is possible that axiomatic mathematics is entirely self contained and consistent etc. within second order logic. There is a possibility that it is not as well, but this is unavoidable due to Godel's arguments. This footnote is technically irrelevant for the conversation we are trying to have though, hence its status as footnote.

FancyHat
Posts: 341
Joined: Thu Oct 25, 2012 7:28 pm UTC

Re: How do axioms specify a new system vs continuing an old

Let's see if I understand what you're trying to discuss in this thread.

You're looking at using system A to formally define system B.

Yeah?

Looking back at your original post, it wasn't at all clear that that's what you were using A for. You didn't originally say that that's the reason the axioms of B were also true statements in A. I think that might be where a lot of confusion has come from about why you seem stuck on this idea that A is some kind of special context in which B's axioms have to be understood.

I think what you're really trying to say is that since A is somehow used to formally define B, we have to take A as a consistent system, with its own axioms true in A itself, in order to then be able to rely on our formal definition of B. Basically, A's axioms have to be true (in A) for the formal definition of B to be correct/reliable, and for B's axioms to be true, in B, in turn, when doing stuff in B.

Have I got that right? Is that what you're trying to say?

B's axioms don't all have to be true statements in A. One of B's axioms, p, could be a statement in A that can't be shown, in A, to be either true or false - it would be undecidable in A. But it can still be used as an axiom of B.

Suppose the set of axioms of B, which you originally called a, is {p, q, r, s}. Suppose, in B, following from those axioms, we also have another set of statements, which are true in B, which you originally called b, and which is {u, v, w}.

So, while p is undecidable in A, a statement to the effect that, 'if p and r, then u', would be true in A if, in B, u follows from just p and r.

You can then have another system, C, in which (not p) is one of its axioms. That would be a different system to B, but there wouldn't be any inconsistency or contradiction in A.

You do understand the utter basics of conditionals, don't you?

Don't you?

Edit: corrected an omission.
Last edited by FancyHat on Tue Feb 10, 2015 7:50 am UTC, edited 1 time in total.
I am male, I am 'him'.

Gwydion
Posts: 336
Joined: Sat Jun 02, 2007 7:31 pm UTC
Location: Chicago, IL

Re: How do axioms specify a new system vs continuing an old

Treatid wrote:
Neither A nor B assert that x and !x are true simultaneously (for any statement x).

This is where I disagree. I don't think A and B are distinct.

Your argument is that A and B are distinct systems. As such, a statement in B has no impact on A.

However, we are using A to specify how the axioms of B should be interpreted. B is not, in fact, distinct from A.

If we assert that B is not expressed in terms of A, then what is B expressed in terms of? Without some context, the axioms of B are just symbols. We don't know how to manipulate those symbols without a context. B cannot define itself. So we are left with A being an implicit component of B.

That is to say - when statements within system A become axioms of system B - they don't stop being statements within system A. They are both statements within system A and also axioms of system B.

I agree that the conventional view is that systems A and B are distinct.

I think that a careful examination shows that, in fact, B is a continuation of A and not a distinct system. That if we use the rules of A to specify B, then we must use all the rules of A.

I'm going to let everybody else handle your responses to their comments, but this is… striking to say the least. Given how I've defined A and B above, you truly believe that B is "a continuation of A"? Let me describe A and B with real axioms instead of letter placeholders then.

A:
a) All right angles are equal to one another.
b) Any two points can be connected by a straight line.
c) A circle can be described by its radius and the location of its center.
d) A finite straight line can not be extended continuously in both directions.

B:
same except d) A finite straight line can be extended continuously in both directions.

From this, we have statement e) "Given a line and a point not on that line, there is exactly one line through the point that does not intersect the first line. In system A, statement e is false - there could be many lines through that point which do not intersect the first line, because in system A that line can't necessarily extend infinitely in a given direction. In system B, e is neither strictly true nor strictly false - it could be either because it does not follow from the axioms stated in B. This brings about B1 (Euclidean geometry) and B2 (non-Euclidean geometry). If you want to get fancier, B2' would include the axiom e' stating that all lines through a given point will intersect the initial line (elliptic geometry), and B2" would include the axiom e" stating that there are infinitely many lines through a given point that fail to intersect the initial line (hyperbolic geometry).

I propose that A, B, B1, B2, B2', and B2" are all reasonable axiomatic systems. They are all demonstrably distinct from one another, in that a given statement x can be true in one of these systems and false in another. I don't see the practical applications of A, though they might exist - I'm not a professional geometer. All of the systems that start with "B" share certain axioms but may vary in others. Anything that follows from B is true in all of the others. Anything that is demonstrably false in B is going to be false in all of the others. However, a statement could be true in B1 and false in B2 - many in fact are.

You are correct in stating that a statement x in system B is also a statement in A, but it could easily be true in one and false in the other. For example, statement e above is an axiom of B1 and assumed true as a result, but is still a statement in A (where it is false as shown above), in B (where it is independent), and in B2 (where it is false as the negation of one of the axioms). The language I'm using to describe these (English, though symbols work too) is common to all of these systems. The "truth" of a statement is independent of its words or symbols, and depends solely on the system in which that statement is interpreted. (Aside - to some degree this is false. Some statements are tautologies based on form alone, such as x->x, and some statements are necessarily false by form alone, such as x&!x. Let's forget about these, because they are largely meaningless in the current discussion. If you try to have any of the former as axioms they add nothing because nothing can follow from them, and if you add any of the latter your system is instantly meaningless because you are assuming a necessarily false statement to be true.)

Let's take an example statement, since symbol manipulation is hard to follow sometimes: "Given a quadrilateral with three right angles, the fourth angle must be a right angle." In Euclidean geometry (B1 above), this statement is true and can be proven with a combination of the above 5 axioms. Often, a proof like this utilizes a lot of theorems which have been themselves proven over the years from other theorems, but all of those come from the axioms or from statements that logically follow from them. Let's put this same statement into B2, however - because the parallel postulate (e) does not hold, the same proof falls apart, and in fact one can construct a quadrilateral with 3 right angles and an acute/obtuse angle in elliptic/hyperbolic geometry. Thus, the statement is false in B2 but true in B1, and it is independent of B.

Treatid, based on the above points, do you still believe that all of these systems are identical to one another? If so, maybe I don't understand how you define two "identical" systems - I would argue that two systems are identical iff all true statements in one system are true in the other, and all false statements in one system are false in the other, and all statements independent of one system are independent of the other. Do you disagree with this?

edited because I don't in fact know my abc's

Twistar
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Re: How do axioms specify a new system vs continuing an old

edit: Bah sorry about that. When you click on the diagram it's HUGE!! You'll have to zoom your browser out probably. This was my first time posting pictures, sorry.

Here are the promised pictures. Treatid, I feel like we ARE teaching you basic formal logic that you should be learning from a class or textbook and it's kind of annoying because that's not necessarily what we're trying to do on this forum. It's also annoying when you act like you're so smart and clever yet show signs of not understanding some of the basic fundamentals.

Just a chart explaining the analogy made in my previous post

A venn diagram showing how a formal language can sort of be envisioned. On the right side you have rules that define how to work within the formal system (note that these are different from axioms). On the left side you have the set of all possible statements you can make in the language and I've given you two examples of how you could choose axioms from the set of all possible statements and use those axioms to derive theorems thus constituting different theories within the language.

Now, at this point we can run further backwards and get upset and ask how second order logic is justfied! but I want to stress that asking this is IN EVERY WAY EQUIVALENT to asking how English is justified. I anticipate you saying something like "but english doesn't purport to be a formal language and prove things rigorously" to which my response is "you are right, but math doesn't purport to prove things beyond the scope of second order logic either." Some people may try to claim that math can or should reveal fundamental truths about our universe and the way things work but frankly those people are dead wrong.

Forest Goose
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Re: How do axioms specify a new system vs continuing an old

arbiteroftruth wrote:Treatid, if there's anything like the universal axiom system that you're arguing exists, it's something like second-order logic.

Would you mind elaborating on that? Full SO doesn't have a complete proof theory, you can't check second order proofs; I don't think anything Treatid said looks anything like SO. And, in any event, theories in SO would still have axioms, just second order ones, there still would not be a universal system, or anything like it - anything reasonably called "logic" is going to have axioms and theories.

@Twistar
I like, and agree, with the gist of everything you've just said, except, mathematics is not done in second-order logic; you can't even be sure you've proven something using SO. That's not to say that people don't investigate it, but it is definitely not what mathematicians are using - at the strongest, fragments of SO are used, but they are quite conservative ones.
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Twistar
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Re: How do axioms specify a new system vs continuing an old

Forest Goose wrote:@Twistar
I like, and agree, with the gist of everything you've just said, except, mathematics is not done in second-order logic; you can't even be sure you've proven something using SO. That's not to say that people don't investigate it, but it is definitely not what mathematicians are using - at the strongest, fragments of SO are used, but they are quite conservative ones.

Help me out, I'm just trying to talk about ZFC because I think that's common enough ground for this conversation. I know that leaves us in the early 1900s but I don't think we need to go any further than that for this conversation. What language can I do ZFC math in? My impression (self-taught) was that you could do ZFC in SO but I never quite understood all the distinctions with the different formal languages.

Edit: whoops, looks like it should be first order logic instead. I'll try to update that in the previous posts. Or maybe I'll just take the lazy route and correct it here.

Forest Goose
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Re: How do axioms specify a new system vs continuing an old

Twistar wrote:Help me out, I'm just trying to talk about ZFC because I think that's common enough ground for this conversation. I know that leaves us in the early 1900s but I don't think we need to go any further than that for this conversation. What language can I do ZFC math in? My impression (self-taught) was that you could do ZFC in SO but I never quite understood all the distinctions with the different formal languages.

ZFC is done in first-order, almost everything is done using FO in mathematics.

Second Order logic allows for sets of individuals as well as individuals, where as first-order just allows for individuals. A natural point of confusion is that in ZFC we are talking about sets, but sets are the individuals, and "an element of" is just a 2-ary relationship - this is why we can have countable models of ZFC (Lowenheim Skolem). Second Order logic quantifies over any subcollection of the universe - so, with ZFC, you can say things like "For all classes C that contain the ordinals...", where, here, class means any subcollection of the universe. (note: class, here, is, literally, any subcollection of the universe, whereas that word, usually, in regular ZFC is meant as a definable collection. The second order theory does not reduce down to the first).

Long story short, if you change "second order" to "first order", everything is fairly on the level (modulo extremely minor philosophical points that have no bearing on this and are, certainly, not universal).

-----

Here's an example of something that can be done with second order logic that cannot with first:

"There is R so that
1.) For all x, y, z: xRy and yRz -> xRz
2.) For all x: not xRx
3.) For all x there is y: xRy"

This sentence has as models every infinite set, and only those sets. (notice, since R falls under a quantifier, it is not like the successor S in arithmetic, nor like "an element of" in ZFC, it is not open to interpretation, it is not part of the signature). There is no first order sentence that can do this.

-----

First order logic is everything you can do with "=", "For all", "There exists", AND, OR, ->, etc. a countable collection of variables, and a set of constants, relation symbols, and function symbols (that are declared ahead of time).

For example, the theory of groups has a 2-place function symbol + and a constant 0. Any well formed formula using the logical connectives, quantifiers, equality, variables, and + is a valid statement. The axioms are the usual, "For all x, y, z: x + (y + z) = (x + y) + z", "For all x there is y so x + y = 0", etc. A model of this theory is any specific group: a set G with a function + and element 0 so that they obey the axioms. A statement (anything using + and 0 and logical symbols) is satisfiable if there is some model (group) it is true in; a statement can be proven if there is a proof starting with the axioms, using derivation rules, that leads to that statement.

Metalogically, if there is a proof of something, that something is true in all models; and if something is true in all models, there is a proof of it. So, for example, since we can exhibit abelian groups and nonabelian ones, "For all x,y: x + y = y + x" can't have a proof in the theory of groups, nor can its negation. On the other hand, if I want to show that inverse elements are unique, I can, being a bit informal, do so by considering a general group G, then reason about it as a set - for example: Let G be a group, let x + y = 0 = x + z, then y + 0 = y + x + z = 0 + z = z. Since it holds in any model, there must be a proof. (Yeah, not the best examples, I'm sleepy).

Finally, what is cool about FO is that I can check proofs, as in I can recursively verify that they were carried out correctly.

One big diff. between SO and FO is that if SO is sound and complete (proofs lead to logically valid results, logically valid results lead to proofs), then I cannot have an effective proof theory - I cannot recursively verify a proof. This is why full second order logic is not the main language of mathematics.

Sorry if this is didactic, it isn't meant for anyone in particular, just related to the above.

(I'm off to bed, I have not read the last section, excuse typos, errors, etc. I'm, literally, passing out on my keyboard).
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arbiteroftruth
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Re: How do axioms specify a new system vs continuing an old

Others have well covered the distinction between the language of logic and the axioms of a particular system within logic, and that's all my reply to Treatid would have consisted of, so I'll leave that alone. I'll just address this:

Forest Goose wrote:
arbiteroftruth wrote:Treatid, if there's anything like the universal axiom system that you're arguing exists, it's something like second-order logic.

Would you mind elaborating on that? Full SO doesn't have a complete proof theory, you can't check second order proofs; I don't think anything Treatid said looks anything like SO. And, in any event, theories in SO would still have axioms, just second order ones, there still would not be a universal system, or anything like it - anything reasonably called "logic" is going to have axioms and theories.

Yes, officially we like to say we're working in first order logic, to avoid those issues that second order has in general. But every now and then, we find we really need a second order sentence, but we get around it by instead calling it a "schema" for an infinite set of first order sentences. And if that's what we were really doing as humans, then ZFC really would be horribly defined, because it takes an infinite amount of time to write the axioms. Of course that's not actually a problem, because we're able to process an axiom schema as a single idea, but that means what we're really doing is understanding it as a second order sentence. So I'd say second order logic is the language of math, but we try to restrict ourselves to systems that could in principle be called first-order if we had an infinite amount of time to describe them, so that we don't run into the problems that second order logic can have in general.

No, nothing Treatid said directly sounds like second order logic. But he's mixing up the notions of axiomatic systems within a language and the "axioms"(syntax rules) of the language itself. In the sense that our axiom systems tend to coexist within a shared language of logic, he's correct that they are all just subsets of one system. His mistake is in not grasping the difference between syntax rules that say a sentence is well-formed vs. the act of assuming a sentence is true for the sake of using it as an axiom.

Forest Goose
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Re: How do axioms specify a new system vs continuing an old

Yes and no, we aren't using full second order semantics and we aren't working with the full power of second order logic when we do this, using schemas are, at best, an extension; not even that, nothing full out requires a theory have a finite number of axioms.

Moreover, axiom schemas use definable things, second order do not - that goes back to my above point about classes in first order logic vs quantifying over all classes in second order, the latter case in far stronger than the first, and very different.

Math, in general, is not using second order, we occasionally dip into it for a specific thing, even then, not at full power and not across the board.

*True Arithmetic is a first order theory consisting of all first order sentences true of the natural numbers; despite it being not possible to actually do anything with, it's still first order.
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arbiteroftruth
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Re: How do axioms specify a new system vs continuing an old

My point is that when we use something like the axiom schema of specification, what we're really doing is using a universal quantifier over predicates. We then observe that a universally quantified sentence can be thought of as a conjunction of all possible instances, and therefore we can also think of it as an infinite collection of separate first-order statements. That observation is useful because it means we can safely continue to rely on all the nice properties of first-order logic, but if we honestly describe what the human mind is actually doing, it's a sentence that quantifies over predicates. You say "nothing full out requires a theory have a finite number of axioms", and you're right that there's no explicit rule within formal reasoning that establishes such a requirement. But since we're talking at the foundational level about how logic and mathematics reflect human reasoning, in that sense a theory must have finitely many axioms, because the human mind that uses the system is finite.

It's fair to say that it's still not full second-order logic though, since the only second-order sentences we use are ones that could equivalently be thought of as infinitely many first order sentences, and that rules out existential quantifiers over predicates for example. Maybe call it first-and-a-half-order logic. Whatever you call it, again my point is that what the human mind is actually doing with an axiom schema is quantifying over predicates.

Forest Goose
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Re: How do axioms specify a new system vs continuing an old

But quantifying over predicates isn't all that second order logic is doing; and axiom schemas aren't doing it in the way second order does.

You can make a philosophical distinction, if you'd like, but I'm not sure what the value of that is, infinity isn't a problem just because it isn't finite. The fact that we are talking abou definable things in a first order way and, generally, in a way that is recursive, makes it all very much intelligible to the human mind. I wouldnt argue that since algorithms can spit out infinitely many values that they are incomprehensible, it's not the infinite part of functions that's inportant, it's if they're definable, if they're recursive, etc. - if they can be finitely represented and manipulated in a meaningful way. It might be of value to distinguish when first order theories are axiomatic in this way, and we do do so; that's why we don't talk about true arithmetic as a foundation, we can't really do anything with it, despite it existing.

As far as ZFC goes, NBG can be finitely axiomatized and every result it proves about sets can be proven in ZFC, and conversely; so you don't have to worry about schemas as far as that goes.
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arbiteroftruth
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Re: How do axioms specify a new system vs continuing an old

Forest Goose wrote:it's not the infinite part of functions that's inportant, it's if they're definable, if they're recursive, etc. - if they can be finitely represented and manipulated in a meaningful way.

And when they are, that finite representation of the infinite function amounts to using a quantifier over the inputs. In the case of an axiom schema, this input is a predicate. Thus an axiom schema amounts to using a quantifier over predicates. First-order logic cannot quantify over predicates. Hence using an axiom schema amounts to using something more than first-order logic. I originally called it "something like second order logic", and later agreed with you that it doesn't actually amount to full second-order logic.

Forest Goose
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Re: How do axioms specify a new system vs continuing an old

arbiteroftruth wrote:
Forest Goose wrote:it's not the infinite part of functions that's inportant, it's if they're definable, if they're recursive, etc. - if they can be finitely represented and manipulated in a meaningful way.

And when they are, that finite representation of the infinite function amounts to using a quantifier over the inputs. In the case of an axiom schema, this input is a predicate. Thus an axiom schema amounts to using a quantifier over predicates. First-order logic cannot quantify over predicates. Hence using an axiom schema amounts to using something more than first-order logic. I originally called it "something like second order logic", and later agreed with you that it doesn't actually amount to full second-order logic.

I can't think of anywhere I've ever seen that being called something more than first-order logic, though; and I'm not sure why it would be. You seem to be confusing the fact that it can be expressed as a single axiom in second order* and that we can nicely represent the expression of infinitely many axioms in first order. That second order can quantify over things to get the same (which, actually, is rarely the case, it's often a bit stronger since it isn't just "quantifying over predicates", anyway) doesn't somehow mean that first order can't do anything of the sort ever, in anyway - especially when it isn't purporting to do so in a single sentence to begin with.

Again, I stress: you are saying, essentially, "Yes, technically, first order schemas are infinitely many axioms, but if we set that aside and call it quantifying over predicates, since that could achieve the same end*, then it's doing what second order logic does, so it isn't first order". The problem is with that setting aside part, because it isn't quantifying over predicates; the metalanguage** can handle it just fine by recursively expressing the axioms, all of which are first order (that's the key part, right there: every axiom is first order).

*Second Order versions aren't, at least in my experience, quantifying over predicates anyway, they're doing something stronger from the get go. The second order versions of schemas aren't just more compact, they're also stronger. The point isn't that you are arguing for second-order logic, the point is that what you appear to be thinking of as something second-orderish really isn't. (nor is it quantifying over predicates either, see below)

**The axiom schema is being expressed in the metalanguage, as far as the theory is concerned, it has one axiom for every instance. In the same way that the number 3 doesn't know the difference between being a return value of some computable function -vs- being the return value of some non-computable function, 3 is 3 is 3 - each instance of the schema is an axiom, it has no idea how it arrived there.
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arbiteroftruth
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Re: How do axioms specify a new system vs continuing an old

All you've done is push the quantifying-over-predicates back to the metalanguage, which you can do if your only interest is in saying that the formal theory is technically first order. But I thought it was pretty clear this whole time that I'm talking about the way an actual human being actually understands things. And if you can show me an actual human being who can hold infinitely many distinct axioms in his head, rather than abstracting out the notion of "any expression can go here", I'll be very impressed.