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Your Axiom of Choice is SILLY

Posted: Sun Jun 08, 2008 4:07 pm UTC
by fortyseventeen
Am I the only one who thinks that this is stupid?

Wikipedia wrote:A choice function is a function f, defined on a collection X of nonempty sets, such that for every set s in X, f(s) is an element of s. With this concept, the axiom can be stated:

For any set of non-empty sets, X, there exists a choice function f defined on X.

Thus the negation of the axiom of choice states that there exists a set of nonempty sets which has no choice function.

Each choice function on a collection X of nonempty sets can be viewed as (or identified with) an element of the Cartesian product of the sets in X. This leads to an equivalent statement of the axiom of choice:

An arbitrary Cartesian product of non-empty sets is non-empty.


Not once in that article are there any references to attempted proofs of this "axiom", which I find astounding. Thus, I attempt to do so now:

Everyone knows that the size of a Cartesian product is the product of the size of the sets in the product, right? I shouldn't need to prove that one, but it's an easy induction. Now, since the size of a set is a natural number, we can trivially prove the AoC, since within the naturals, the product of two (or more, by extension) numbers is zero iff at least one of them is zero to begin with. Finally, in case there's any doubt that those two statements of the AoC are indeed equivalent, i.e. how to identify choice functions with tuples in the Cartesian product: you can create a relation from each set in X to each element of one of the Cartesian tuples, and that relation is in fact a function, because we constructed its domain from a set X, and thus it can contain no duplicates, making one-to-many impossible (although it may be many-to-one, since its range is a tuple).

Why, then, are we still calling it an axiom? I think I know why: because mathematicians are lazy, and don't want to be bothered with the idea that there may be a set of sets for which no choice function is definable, even if many may exist. So, if we call it an axiom, we can just ignore it wherever it makes things too complicated for our poor little brains. First of all, I think it's baloney, since a choice function would only be undefinable over X if X itself was not well-defined enough. Second, if it's really too much hassle, you can just ignore the theorem, but you don't have to call a theorem an axiom and point to Gödel when questioned just to get your way. You miss out on a lot of other interesting things.

Re: Your Axiom of Choice is SILLY

Posted: Sun Jun 08, 2008 4:21 pm UTC
by rhino
fortyseventeen wrote:*snip*
Not once in that article are there any references to attempted proofs of this "axiom", which I find astounding. Thus, I attempt to do so now:

Everyone knows that the size of a Cartesian product is the product of the size of the sets in the product, right? I shouldn't need to prove that one, but it's an easy induction. Now, since the size of a set is a natural number, we can trivially prove the AoC, since within the naturals, the product of two (or more, by extension) numbers is zero iff at least one of them is zero to begin with. Finally, in case there's any doubt that those two statements of the AoC are indeed equivalent, i.e. how to identify choice functions with tuples in the Cartesian product: you can create a relation from each set in X to each element of one of the Cartesian tuples, and that relation is in fact a function, because its domain (X) is a set, and thus can contain no duplicates, making one-to-many impossible.

Why, then, are we still calling it an axiom? I think I know why: because mathematicians are lazy, and don't want to be bothered with the idea that there may be a set of sets for which no choice function is undefinable, even if many may exist. So, if we call it an axiom, we can just ignore it wherever it makes things too complicated for our poor little brains. First of all, I think it's baloney, since a choice function would only be undefinable over X if X itself was not well-defined enough. Second, if it's really too much hassle, you can just ignore the theorem, but you don't have to call a theorem an axiom to get your way. You miss out on a lot of other interesting things that way.

Not every set is a finite set, so your "proof" fails instantly.

The reason there is no proof of AoC is straightforward: it has been proved that AoC cannot be proven using only the other axioms of Zermelo-Fraenkel set theory. That's right, the axiom of choice cannot be proven. If you think you have a set of axioms that lets you prove the axiom of choice, you will have already assumed something stronger or equally strong. Mathematicians are not "calling it an axiom because they can't be bothered to prove it". It is especially ridiculous to suppose that all the mathematicians of the 20th century would have failed to come up with your naive one-paragraph "proof" if it was in fact valid.

If your intuition screams at you that the axiom of choice is so obvious that it must be provable - this is why it is usually assumed. Also you should consider that formal set theory and your intuition of what sets should be like are two starkly different things.

Re: Your Axiom of Choice is SILLY

Posted: Sun Jun 08, 2008 4:27 pm UTC
by Owehn
If there were attempted proofs of it, it wouldn't be called an axiom would it? Seriously though, it's been shown that if the other axioms of set theory are consistent, then it's impossible to prove the axiom of choice from them. (And also impossible to disprove it.)

Your attempt does illustrate an important point: you assume that every set has an associated size, which is a natural number. That doesn't work for infinite sets (like the set of all natural numbers), so it isn't obvious that there's a way to associate something like "size" to every set. The standard way of doing so is with "cardinal numbers", which are meant to each be a representative set of a particular size, so that every set is in bijection with exactly one cardinal number. However, the construction of cardinal numbers doesn't guarantee that this is so, and in fact the statement that every set is bijective with a cardinal (i.e. that every set has a cardinality, or size) is equivalent to the axiom of choice.

So your attempted proof of the axiom of choice actually relies on it in a subtle way. There are several other statements that are equivalent to Choice, like

The well-ordering principle: Every set is in bijection with some ordinal number (and with a unique cardinal number).
The Hausdorff maximal principle: Every partially ordered set has a maximal linearly ordered subset.
Zorn's Lemma: If every linearly ordered subset of a partially ordered set has an upper bound, that partially ordered set has a maximal element.

Re: Your Axiom of Choice is SILLY

Posted: Sun Jun 08, 2008 4:41 pm UTC
by fortyseventeen
Hmm, okay, maybe I'm stupid. Of course, I still fail to see the usefulness in rejecting the axiom, because in the physical world, there is always a way to arbitrarily choose from a set. Then again, by the same logic, I fail to see the usefulness of a lot of set theory and classifications of infinity, so we should probably stop here. :)

EDIT: or maybe I'll just change my argument a bit: the axiom of choice seems too abstract to be an assumption, and so the statement that we call an axiom should be more fundamental. I would choose the well-ordering principle for this, since that's what I used in my proof. In other words, we should choose the one that makes further proofs simpler, or more understandable. Note that this is now an entirely subjective argument.

Re: Your Axiom of Choice is SILLY

Posted: Sun Jun 08, 2008 5:02 pm UTC
by auteur52
fortyseventeen wrote:EDIT: or maybe I'll just change my argument a bit: the axiom of choice seems too abstract to be an assumption, and so the statement that we call an axiom should be more fundamental. I would choose the well-ordering principle for this, since that's what I used in my proof. In other words, we should choose the one that makes further proofs simpler, or more understandable. Note that this is now an entirely subjective argument.


The well-ordering principle (theorem is more correct, the principle is usually referring to something else) is WAY more counterintuitive, at least in my opinion. Well-ordering says that any set has a well-ordering. As far as I know, we have no idea what a well-ordering for the reals would even look like, we just know that it would exist. Axiom of choice is much more easy to accept as an intuitive assumption (at least for most mathematicians).

Re: Your Axiom of Choice is SILLY

Posted: Sun Jun 08, 2008 6:10 pm UTC
by Torn Apart By Dingos
fortyseventeen wrote:I still fail to see the usefulness in rejecting the axiom

http://en.wikipedia.org/wiki/Banach-Tarski_paradox

Re: Your Axiom of Choice is SILLY

Posted: Sun Jun 08, 2008 7:25 pm UTC
by Cleverbeans
Torn Apart By Dingos wrote:
fortyseventeen wrote:I still fail to see the usefulness in rejecting the axiom

http://en.wikipedia.org/wiki/Banach-Tarski_paradox


Isn't that an argument for accepting it?

Re: Your Axiom of Choice is SILLY

Posted: Sun Jun 08, 2008 7:52 pm UTC
by antonfire
auteur52 wrote:Axiom of choice is much more easy to accept as an intuitive assumption (at least for most mathematicians).
Which is true, in fact, for precisely the reason that the OP mentioned. It seems obvious, as all axioms should. The axiom of choice is an axiom for precisely the same reasons that the axiom of pairing is an axiom. It seems like it should obviously be true, and it lets us do things we otherwise wouldn't be able to do.

Cleverbeans wrote:Isn't that an argument for accepting it?
What? How many times have you seen an apple get cut into 5 or 6 pieces and rearranged to make two apples of the original size? The axiom of choice leads to some very counterintuitive results, including the Banach-Tarski paradox. That is an argument against accepting it. I don't think it's a very convincing argument, but it is an argument against it nevertheless.

Re: Your Axiom of Choice is SILLY

Posted: Sun Jun 08, 2008 8:02 pm UTC
by fortyseventeen
auteur52 wrote:
fortyseventeen wrote:EDIT: or maybe I'll just change my argument a bit: the axiom of choice seems too abstract to be an assumption, and so the statement that we call an axiom should be more fundamental. I would choose the well-ordering principle for this, since that's what I used in my proof. In other words, we should choose the one that makes further proofs simpler, or more understandable. Note that this is now an entirely subjective argument.


The well-ordering principle (theorem is more correct, the principle is usually referring to something else) is WAY more counterintuitive, at least in my opinion. Well-ordering says that any set has a well-ordering. As far as I know, we have no idea what a well-ordering for the reals would even look like, we just know that it would exist. Axiom of choice is much more easy to accept as an intuitive assumption (at least for most mathematicians).


Oops, sorry, I keep forgetting that I think differently than most people. I don't suppose most mathematicians have autism-spectrum disorders, do they?

Anyway, on the subject of not knowing what things look like, do you even know what e^(4πi/3) looks like? Math is an imaginary science in the first place, but the axioms that we start with can be related to observations. I observe that I can choose an object from any set, even if I don't count or distinguish between any of the objects (v. my sock drawer), so I choose to subscribe to the AoC. I find that it's easier to reason that any set can be counted, even if it's impractical to do so (or I just don't have the time), but that's where my personal strangeness comes in.

antonfire wrote:What? How many times have you seen an apple get cut into 5 or 6 pieces and rearranged to make two apples of the original size? The axiom of choice leads to some very counterintuitive results, including the Banach-Tarski paradox. That is an argument against accepting it. I don't think it's a very convincing argument, but it is an argument against it nevertheless.


It's hardly a convincing argument! I think it's a rather neat demonstration of what's possible with AC. I'd be much more convinced if you found something useful like that (well, arguably useful) that depends on the negation of AC.

Re: Your Axiom of Choice is SILLY

Posted: Sun Jun 08, 2008 8:21 pm UTC
by Cleverbeans
antonfire wrote:What? How many times have you seen an apple get cut into 5 or 6 pieces and rearranged to make two apples of the original size? The axiom of choice leads to some very counterintuitive results, including the Banach-Tarski paradox. That is an argument against accepting it. I don't think it's a very convincing argument, but it is an argument against it nevertheless.


I guess I don't consider the result counterintuitive, just really awesome. There doesn't seem to be anything unnatural about an infinite object splitting up nicely into two objects of the same size. I normally argue for AC by assuming that all infinite sets are Dedekind-infinite since it seems so obvious, which is probably why I don't see Banach-Tarski as being a paradox.

Re: Your Axiom of Choice is SILLY

Posted: Sun Jun 08, 2008 8:25 pm UTC
by Ended
fortyseventeen wrote:I observe that I can choose an object from any set, even if I don't count or distinguish between any of the objects (v. my sock drawer), so I choose to subscribe to the AoC. I find that it's easier to reason that any set can be counted, even if it's impractical to do so (or I just don't have the time), but that's where my personal strangeness comes in.

Not every set can be counted. The AoC becomes much more unintuitive (for me, at least) when you consider uncountable collections of sets. For example, the set of all non-empty subsets of the real numbers. Even if you had an infinite amount of time, you could not 'choose', in the sock drawer sense, an element from each set , because there are uncountably many sets.

(Disclaimer: I don't actually think AoC is very hard to accept, I'm just playing devil's advocate).

fortyseventeen wrote:I choose to subscribe to the AoC

I see that. What you did there. :D

Re: Your Axiom of Choice is SILLY

Posted: Sun Jun 08, 2008 8:42 pm UTC
by rhino
Fact: for a very long time I thought it was called the axiom of choice because everyone got to choose whether or not they thought it was true.

Of course, now I realise that it simply *is* true and that's all there is to it!
</religious beliefs>

Re: Your Axiom of Choice is SILLY

Posted: Sun Jun 08, 2008 8:51 pm UTC
by auteur52
fortyseventeen wrote:Anyway, on the subject of not knowing what things look like, do you even know what e^(4πi/3) looks like? Math is an imaginary science in the first place, but the axioms that we start with can be related to observations. I observe that I can choose an object from any set, even if I don't count or distinguish between any of the objects (v. my sock drawer), so I choose to subscribe to the AoC. I find that it's easier to reason that any set can be counted, even if it's impractical to do so (or I just don't have the time), but that's where my personal strangeness comes in.


My point is regarding your suggestion that the Well-Ordering Theorem is more intuitive than AoC. It seems like you might be misinterpreting the Well-Ordering Theorem. It says that for any non-empty set, a well-ordering exists, meaning each subset will have a least element. However, it is a non-constructive theorem, so even though we know an easy well-ordering for the natural numbers, one for the reals is much much more counterintuitive. So it's not just that we don't know what it is, but it's quite surprising that one should exist. And that's why it is a consequence of AoC and not the actual axiom we accept, because I doubt most people would be willing to accept it as something self-evident.

Sorry if that was kind of confusing, I'm not the best at explaining things like this. I'm hoping to improve before I become a TA next quarter...

Re: Your Axiom of Choice is SILLY

Posted: Sun Jun 08, 2008 9:25 pm UTC
by ThomasS
fortyseventeen wrote:I find that it's easier to reason that any set can be counted, even if it's impractical to do so (or I just don't have the time), but that's where my personal strangeness comes in.


There are sets that cannot be counted. Perhaps you find the set [imath]\{a,b\}[/imath] to be more physical that the reals or the power set of the integers, but the mathematician's definition of set doesn't differentiate. Do you know what a Newton looks like? A gram? Part of science is relating mathematical results to physical reality, but there is nothing absolute about this connection.

Mathematicians keep careful tract of what they assume and what they derive. Well, there are exceptions. On most days you might not care whether the axiom of choice is true. A book on analysis might simply assume the field properties of the rational numbers without delving into why they hold. However, within an particular field a lot of care is takes to keep tract of what is assumed and what is derived.

The care is important because there are a lot of surprising derived things in math, and because if you assume something that happens to be false, you have a contradiction and might accidentally prove, well, anything. e.g. somebody might think it is obvious that 1.999999... is different than 2. However, this is simply false, and assuming that it is true creates a contradiction. Similarly, if you assume that all sets are countable, you then find that all sets are finite. But even if the number of particles in the universe is finite, why would the number of possible positions of a particle be finite?

Re: Your Axiom of Choice is SILLY

Posted: Sun Jun 08, 2008 10:27 pm UTC
by kevbrown
I find it funny that you tried to use the WOP to derive the axiom of choice, because the axiom of choice was first introduced by Zermelo to prove the WOP, as mentioned in the wikipedia article you called stupid.

Re: Your Axiom of Choice is SILLY

Posted: Sun Jun 08, 2008 11:24 pm UTC
by antonfire
fortyseventeen wrote:I find that it's easier to reason that any set can be counted, even if it's impractical to do so (or I just don't have the time), but that's where my personal strangeness comes in.
This is perfectly normal. If you do much more mathematics, you'll almost certainly get over it. Rather, you'll understand better what it means to say that every set can be counted (with ordinals) and see how asserting this is equivalent to the axiom of choice.

That, or you'll turn into one of those crazy mathematicians who think that everything should be finite and we should treat the reals as one huge finite field and things like that.

Re: Your Axiom of Choice is SILLY

Posted: Mon Jun 09, 2008 1:05 am UTC
by Sunsnail
What's a set? I'm interested in learning mathematics on my own since I'm done with school. I always liked it.

Re: Your Axiom of Choice is SILLY

Posted: Mon Jun 09, 2008 4:01 am UTC
by ICDB
I second the "what's a (formal) set" question and add one of my own: Why can't the function in the OP take the set s and return the element in s with the smallest magnitude? I suspect this question has been answered but most of the above posts flew over my head.

Re: Your Axiom of Choice is SILLY

Posted: Mon Jun 09, 2008 4:27 am UTC
by NathanielJ
ICDB wrote:Why can't the function in the OP take the set s and return the element in s with the smallest magnitude? I suspect this question has been answered but most of the above posts flew over my head.


Heh, you're about to run in circles. Being able to define "smallest" magnitude on an arbitrary set it turns out is equivalent to the axiom of choice. However, as a simple example of why it certainly isn't trivial to be able to define "smallest" on an arbitrary set, consider the set of all numbers larger than 0. What is the "smallest magnitude" element of that set? Under the standard definition of "smallest", there isn't one (because for any number larger than 0, there is a smaller number that is also larger than 0). So you have to find another way of ordering that set of numbers that has a smallest element. Being able to do this requires that you are able to choose a smallest element, then choose a second smallest element, and so on (which is basically just the Axiom of Choice, right?)

See the following link for more information about this:
http://en.wikipedia.org/wiki/Well-ordering_theorem

Re: Your Axiom of Choice is SILLY

Posted: Mon Jun 09, 2008 4:46 am UTC
by TheCoelacanth
fortyseventeen wrote:I find that it's easier to reason that any set can be counted, even if it's impractical to do so (or I just don't have the time), but that's where my personal strangeness comes in.


That might seem more intuitive, but it is incorrect. Some sets are uncountable.

Re: Your Axiom of Choice is SILLY

Posted: Mon Jun 09, 2008 6:02 am UTC
by jestingrabbit
fortyseventeen wrote:
antonfire wrote:What? How many times have you seen an apple get cut into 5 or 6 pieces and rearranged to make two apples of the original size? The axiom of choice leads to some very counterintuitive results, including the Banach-Tarski paradox. That is an argument against accepting it. I don't think it's a very convincing argument, but it is an argument against it nevertheless.


It's hardly a convincing argument! I think it's a rather neat demonstration of what's possible with AC. I'd be much more convinced if you found something useful like that (well, arguably useful) that depends on the negation of AC.


How about the axiom L(R) which is "all subsets of the reals are Lebesgue measurable"? Its inconsistent with the full axiom of choice, though its consistent with dependent choice.

ICDB wrote:Why can't the function in the OP take the set s and return the element in s with the smallest magnitude?


Using the usual idea of magnitude, what is the smallest real number greater than 0? There isn't one, because if you think you've got one, I can just halve it and still be bigger than 0 with a smaller magnitude. So you then have to come up with a fresh order, [cue well ordering discussion from before].

Re: Your Axiom of Choice is SILLY

Posted: Mon Jun 09, 2008 10:15 am UTC
by Robin S
fortyseventeen wrote:Oops, sorry, I keep forgetting that I think differently than most people. I don't suppose most mathematicians have autism-spectrum disorders, do they?
Actually, a fair number of us do. I'm afraid your argument has nothing to do with your having autistic spectrum disorder. It's been put forward by plenty of people without it and rejected by plenty of people with it. You have issues with notions such as the concept of "axiom", which as antonfire said you will hopefully get over if you do enough Maths.

Re: Your Axiom of Choice is SILLY

Posted: Mon Jun 09, 2008 2:41 pm UTC
by SlyReaper
I haven't read the entire thread so I'm sure someone else has said this. Still...

It's an axiom. It is assumed to be true because it is basic and makes sense. Much like the axiom in logic that 0 is not equal to 1.

Re: Your Axiom of Choice is SILLY

Posted: Tue Jun 10, 2008 5:30 am UTC
by Pathway
antonfire wrote:
fortyseventeen wrote:I find that it's easier to reason that any set can be counted, even if it's impractical to do so (or I just don't have the time), but that's where my personal strangeness comes in.
This is perfectly normal. If you do much more mathematics, you'll almost certainly get over it. Rather, you'll understand better what it means to say that every set can be counted (with ordinals) and see how asserting this is equivalent to the axiom of choice.

That, or you'll turn into one of those crazy mathematicians who think that everything should be finite and we should treat the reals as one huge finite field and things like that.

Zeilberger!

Re: Your Axiom of Choice is SILLY

Posted: Tue Jun 10, 2008 8:43 pm UTC
by Tac-Tics
The Axiom of Choice is truly absurd! Functions which exist but can't be identified. Truly the claims of madmen!

From a computer scientist's view point, the idea of real numbers is pretty damned absurd too! Infinite precision? Transcendentals?? The great majority of which cannot be named!?? The cost we pay for algebraic closure is the cost of computability!

The nice thing about axioms in any system, though, is that they are always true, modulo your formal system. Of course, sometimes, they're both true and false, and those kinds of systems aren't terribly useful, but decades of research hasn't shown any contradictions in ZFC.

Re: Your Axiom of Choice is SILLY

Posted: Thu Jun 12, 2008 2:07 am UTC
by Swap
fortyseventeen wrote:do you even know what e^(4πi/3) looks like?


Yes.

It looks like a dot on the Cartesian plane, distance of one away from the origin, making an angle of π/3 with the vertical axis.

Your point?

Re: Your Axiom of Choice is SILLY

Posted: Thu Jun 12, 2008 2:10 am UTC
by Swap
Tac-Tics wrote:The Axiom of Choice is truly absurd! Functions which exist but can't be identified. Truly the claims of madmen!


I too find the axiom of choice mostly untenable. For finite sets you don't need it, for infinite sets you get nonsense, and you cannot describe a non-measurable set, because you cannot describe its choice function in a finite number of steps.

It's just an axiom that makes some theorems more tidy to state without having to worry about how badly infinite the sets the theorems are talking about can be, but it's not a "useful" axiom in the sense that it doesn't elicit any tangible objects to play with. What good is a non-measurable set that you can't describe in full detail?

Re: Your Axiom of Choice is SILLY

Posted: Thu Jun 12, 2008 2:45 am UTC
by antonfire
Er, the ability to describe it in partial detail, for example.

You can't describe every subset of the natural numbers in full detail either. Is the power set axiom also untenable?

Re: Your Axiom of Choice is SILLY

Posted: Thu Jun 12, 2008 1:56 pm UTC
by Tac-Tics
Swap wrote:
Tac-Tics wrote:The Axiom of Choice is truly absurd! Functions which exist but can't be identified. Truly the claims of madmen!


I too find the axiom of choice mostly untenable. For finite sets you don't need it, for infinite sets you get nonsense, and you cannot describe a non-measurable set, because you cannot describe its choice function in a finite number of steps.

It's just an axiom that makes some theorems more tidy to state without having to worry about how badly infinite the sets the theorems are talking about can be, but it's not a "useful" axiom in the sense that it doesn't elicit any tangible objects to play with. What good is a non-measurable set that you can't describe in full detail?

I think you missed my tone of sarcasm. The axiom of choice asserts the existence of things which are just out of reach. My point was that even in accepting the real numbers, you allow yourself to talk about things which barely exist in any meaningful sense. The real numbers are screwy in similar ways as the axiom of choice, but no one complains about them.

Re: Your Axiom of Choice is SILLY

Posted: Thu Jun 12, 2008 3:08 pm UTC
by fortyseventeen
Swap wrote:
fortyseventeen wrote:do you even know what e^(4πi/3) looks like?


Yes.

It looks like a dot on the Cartesian plane, distance of one away from the origin, making an angle of π/3 with the vertical axis.

Your point?


That is merely a representation of the number, but I digress.

Thanks, I think my head is sufficiently "around" this concept now. Also thanks for not ripping my head off over this (or yanking too hard). We're done here.

Re: Your Axiom of Choice is SILLY

Posted: Thu Jun 12, 2008 5:21 pm UTC
by Fafnir43
Tac-Tics wrote:The nice thing about axioms in any system, though, is that they are always true, modulo your formal system. Of course, sometimes, they're both true and false, and those kinds of systems aren't terribly useful, but decades of research hasn't shown any contradictions in ZFC.


Better - it's been proven that ZF is consistent with the axiom of choice if and only if it's consistent without the axiom of choice, and that you can't prove the axiom of choice from the other axioms unless ZF is inconsistent. That is, the axiom of choice isn't superfluous and it doesn't cause any problems unless there's also a problem with one of the other, more intuitive axioms.

Re: Your Axiom of Choice is SILLY

Posted: Thu Jun 12, 2008 5:58 pm UTC
by Robin S
Tac-Tics wrote:The real numbers are screwy in similar ways as the axiom of choice, but no one complains about them.
Untrue. One of my friends is unconvinced that arithmetic with the real numbers is consistent.

Re: Your Axiom of Choice is SILLY

Posted: Thu Jun 12, 2008 9:34 pm UTC
by Token
fortyseventeen wrote:
Swap wrote:
fortyseventeen wrote:do you even know what e^(4πi/3) looks like?


Yes.

It looks like a dot on the Cartesian plane, distance of one away from the origin, making an angle of π/3 with the vertical axis.

Your point?


That is merely a representation of the number, but I digress.

So... your argument is that we have no idea what numbers look like as long as we're not allowed to represent them by things which can be visualised.

Re: Your Axiom of Choice is SILLY

Posted: Thu Jun 12, 2008 11:40 pm UTC
by Mathmagic
Token wrote:
fortyseventeen wrote:
Swap wrote:
fortyseventeen wrote:do you even know what e^(4πi/3) looks like?


Yes.

It looks like a dot on the Cartesian plane, distance of one away from the origin, making an angle of π/3 with the vertical axis.

Your point?


That is merely a representation of the number, but I digress.

So... your argument is that we have no idea what numbers look like as long as we're not allowed to represent them by things which can be visualised.

No no no. It's because e4πi/3 is an imaginary number. It doesn't actually exist, silly!

Re: Your Axiom of Choice is SILLY

Posted: Thu Jun 12, 2008 11:50 pm UTC
by Robin S
It's about as imaginary as e6πi/3, actually.

Re: Your Axiom of Choice is SILLY

Posted: Fri Jun 13, 2008 12:38 am UTC
by Mathmagic
Robin S wrote:It's about as imaginary as e6πi/3, actually.

Last time I checked, sin(4π/3) is NOT zero...

EDIT: Unless you mean 'imaginary' as in the 'not actually real' sense. Damn, I hate ambiguity.

Re: Your Axiom of Choice is SILLY

Posted: Fri Jun 13, 2008 4:06 am UTC
by Yakk
This needs a quote:
"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" — Jerry Bona

And yes, the Axiom of Choice is silly. At cardinalities that actually correspond to physical reality, you can get simpler axioms that are similar to the Axiom of Choice in power. But if you do that, you sometimes have to add some annoying clauses to theorems: so mathematicians assume the Axiom of Choice, because it makes some proofs really pretty.

You do run into the Banach–Tarski problem ("Banach–Tarski Banach–Tarski" is my favorite anagram of "Banach–Tarski"), and other similar strange things -- which often get patched over. The fear I have is that you take the "obvious and clear" axiom of choice, and you run into things like Banach–Tarski that you don't recognize as being dumb...

Re: Your Axiom of Choice is SILLY

Posted: Fri Jun 13, 2008 9:41 am UTC
by jestingrabbit
mathmagic wrote:
Robin S wrote:It's about as imaginary as e6πi/3, actually.

Last time I checked, sin(4π/3) is NOT zero...

and the last time I checked, cos(4π/3) isn't 0 either. You're confusing complex and imaginary.

Re: Your Axiom of Choice is SILLY

Posted: Fri Jun 13, 2008 1:00 pm UTC
by Mathmagic
jestingrabbit wrote:
mathmagic wrote:
Robin S wrote:It's about as imaginary as e6πi/3, actually.

Last time I checked, sin(4π/3) is NOT zero...

and the last time I checked, cos(4π/3) isn't 0 either. You're confusing complex and imaginary.

Mmm, I see... I guess I meant it has an imaginary part, but I didn't really think the distinction was that significant. In hindsight, I was obviously wrong. :razz:

Re: Your Axiom of Choice is SILLY

Posted: Fri Jun 13, 2008 4:14 pm UTC
by kevbrown
fortyseventeen wrote:That is merely a representation of the number, but I digress.


mathmagic wrote:No no no. It's because e4πi/3 is an imaginary number. It doesn't actually exist, silly!


It's not a matter of existence versus nonexistence. The numbers we call 'complex numbers' meet certain criteria, and those are useful, and that's perfectly good reason for existence. The natural numbers need not be defined in terms of counting things like apples or cell phones or hookers, it's just a neat way to describe them. It's like in high school, the answer to 'what's a vector?' is always 'something with a magnitude and a direction.' That's intuitive, but it's much more general than that. Does that mean that the vector space of kerblats and snoofles doesn't exist? Hell no, it's in my head. Likewise the question 'what's a number' might be thought of most easily at first in terms of apples and oranges counted up, but this isn't the whole story.
As for representing numbers, I'm of the opinion that there is no notion of 'number' outside of the things we represent it by. Number being things in my basket, or complex field element, or angle between me and that hot chick to the right, etc. So if you want to say that thing on the argand diagram is just a representation because it's equivalent to something else, I tend to think that's baloney. (Especially considering that we use the argand diagram both to 'represent' a complex number and use the complex numbers to represent angles. . ..)


In a somewhat similar fashion, saying that the AoC is silly or not silly becomes nothing more than a matter of taste. Saying that it should or should not be used only affects the validity of a proof or concept in the logical system it's embedded in (i.e. ZF or ZFC for example). You can say ZFC's a silly theory if you want, but that's a matter of personal taste, not absolute rightness.

Some mathematicians (notably Godel, I believe) disagree though.