Infinity paradox? Help

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fishfry
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Re: Infinity paradox? Help

Postby fishfry » Tue Mar 20, 2012 5:30 pm UTC

arbiteroftruth wrote:"Wrong" with it? Nothing. As for counterintuitive and quirky, it is such because of the very situation I outlined in my first post. For something like the integers, it's easy to construct a set in which, for all finite sizes, that particular set does have similar properties to its members, but this principle does not hold when the construction is carried out infinitely.


So you are saying, as a general principle, that a set of things should have the same property as the things?

Forget numbers, infinity, in fact forget math. Just go to common sense. A house contains people but houses don't have arms and legs. A library contains books but libraries don't have numbered pages. You could sit here all day and make up examples.

Why should a set of anything have the same properties as the things the set contains? I'm not following this point at all. It fails the test of common sense. It's not a mathematical point, it's a semantic one. You have a bunch of things, and you put those things in a container. The container is not one of the things, it's a container.

Am I properly understanding your point? Because this seems pretty simple. A container is not the same as the things it contains. You know, a can of pork and beans isn't pork and it isn't beans. It's a can.

It occurs to me that perhaps we should talk about the meaning of the word "set." A set is just a container. And containers are typically different types of objects than the things they contain. You agree?

Timefly
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Re: Infinity paradox? Help

Postby Timefly » Tue Mar 20, 2012 6:46 pm UTC

fishfry wrote:Stuff...


I think what he's getting is the natural numbers as a set having the property such that Nr = r
Then, from this, reasoning that the last member in N is given by NINFINITY because every index has a corresponding member with the property already stated and there must be an 'upper bound'. Therefore infinity is a member of the set of natural numbers. The fallacy obviously occurs in assuming that the set of naturals has some highest value, or 'upper bound' if you will.

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Dopefish
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Re: Infinity paradox? Help

Postby Dopefish » Tue Mar 20, 2012 9:38 pm UTC

What I'm hearing at the moment is something along the lines of "My definition (which I'm not actually going to give, since you could 'obviously' make one) would be consistant because it would be defined in a way to make it consistant.", which is perhaps strictly speaking fine, but is a rather impractical way of thinking, and can lead you down a long road of incompatible stuff to the point that the object (or set of objects) you're attempting to define no longer has anything to do with your initial intuition.

You might be inclined to dismiss the issues as only arrising in higher maths, but the thing with math is that it's all consistant, and if it turns out that something is wrong with it at any stage (higher math or not), it breaks everything, so the statement 1+1=2 is equally valid to 1+2=pokemon and pi=e=i and you can't make any meaningful statements about anything in that system.

Yes, it is possible to define some nifty new systems with certain properties that you may find intutive, but doing so in a self-consistant way with the rest of math is very difficult, as theres all sorts of corner cases that are apt to break if you're not super careful, and theres a very good chance that the system will be useless in practice, and perform extremely unintutively in areas you weren't considering when you defined it. In practice, it's likely to be much easier (and most likely more useful) to simply adapt your intuition to be in line with existing math, rather then adapt math to your intuition.

Also, using 'obvious' is a very dangerous habit to get into, particularly in math. It's not a good practice in most fields for that matter, since I don't know any subjects that don't have any surprising or counter-intutive results that you might have otherwise said were 'obviously' false.

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Re: Infinity paradox? Help

Postby silverhammermba » Wed Mar 21, 2012 2:55 am UTC

Rather than continuing to beat this dead horse, I would like to see arbiteroftruth come up with some basic axioms.

Either they will be rife with contradictions, or the system will be basically useless, or he'll develop an entirely new, exciting branch of math. Regardless of the outcome, it will be at least somewhat interesting to watch.

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Re: Infinity paradox? Help

Postby jestingrabbit » Wed Mar 21, 2012 3:18 am UTC

silverhammermba wrote:Either they will be rife with contradictions, or the system will be basically useless, or he'll develop an entirely new, exciting branch of math


or reproduce a known nonstandard model of arithmetic.
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Re: Infinity paradox? Help

Postby skullturf » Wed Mar 21, 2012 4:12 am UTC

Yes.

I don't want to pile on to any negativity. I think it's good to play around with ideas, explore alternate definitions, alternate axioms, etc.

But it's also good to get into the habit of learning common conventions and definitions, including nonstandard conventions and definitions that have been proposed by others. As mentioned earlier by Dopefish, spend some time bending your intuition to agree with the math of others, as opposed to bending the math to agree with your intuition.

For example, two number systems that differ from the usual ones studied in an undergraduate math degree (and about which I know very little) occur in nonstandard analysis and the surreal numbers. I just give these as examples; I personally don't know much about them.

It's great to explore and to play around with ideas. But when you do so, there are two things that will happen very very frequently: (1) your ideas lead to contradictions or other severe stumbling blocks, or (2) your ideas lead to something interesting, but they overlap a lot with something that somebody else already described, perhaps in different language.

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Re: Infinity paradox? Help

Postby arbiteroftruth » Sat Mar 24, 2012 7:31 am UTC

gmalivuk wrote:Fine, but that only gets you out of that one particular paradox. What about the property, "is even (x)or odd"? 1 is even xor odd, 2 is even xor odd, ... n is even xor odd, but their limit is neither. In fact, I can think of very few typical properties of integers that could possibly be applicable to infinity. "Is 0 or 1 or strictly less than its square", "is 0 or has a unique prime factorization (modulo sign)", "is congruent to K modulo N" for any integer choice of K and any N with |N|>1, and on and on and on.


No contradiction arises from claiming that numbers remain "even or odd" when the digits are infinite. An infinite-digited integer, if it could be meaningfully written at all, would have to be written from right to left with either a repeating pattern indicated or a simple truncation. Either way, defining it as even or odd based on the value of its ones digit should be fine as always. And if the number cannot be meaningfully written, then whether it is specifically even or specifically odd cannot be determined, but there's no contradiction in saying that it satisfies the criteria "is even or odd".

And if you come at it from the angle of simply incrementing 1 and taking the limit of the "is even" property or "is odd" property, then it becomes obvious that that property simply doesn't converge. Sometimes that happens when taking a limit as something approaches infinity.

Similarly, there's no contradiction in claiming that the number always has a unique prime factorization. Want to actually compute that factorization? You can't, because it's infinite. Any property that only retains relevance when the number is computable becomes meaningless for the same reason that the "is finite" property becomes meaningless. The whole point of defining infinity's properties in terms of such limits is precisely because it defies normal computational methods. So properties that, like finiteness, are directly related to computability, become meaningless as an inherent consequence of the problem being addressed in the first place. Likewise with your other examples.

SunAvatar wrote:In addition to the above objections, even if you create some self-consistent system of arbiteroftruth-integers, some of which have infinitely many digits, I can still say "consider the set of all arbiteroftruth-integers with finitely many digits" and once again have an infinite set, all of whose elements are finite. Unless you want to posit that some finite numbers are also infinite (what would this even mean?), or alternately declare that certain sets are inadmissible. The latter is actually viable--it's more or less how the hyperintegers work--but it's up to you to give a precise rule for when a set is or isn't admissible. So this extension doesn't actually solve the problem (not that there is any problem to be solved in the first place).


This is the most relevant objection I've seen here. My system doesn't escape the counterintuitive issue, because the counterintuitive issue doesn't arise from the way we define integers, but from the entire concept of being "finite".

In any event, I have figured out a way to intuitively understand the situation. Normally, I think of finite numbers as being the more 'fundamental' concept, and infinity being a hypothetical extension of that(hence the entire basis of my hypothetical system). But things work much better if I think of infinite numbers as the more fundamental type of object, and define finite numbers as a subset. That is, I can think of a standard number as having an infinite number of digits both on the right and on the left of the decimal point. The finite numbers can then be defined simply as the subset of numbers that meet the criteria "has an infinite string of zeroes on the left side of the decimal". Likewise, rational numbers would be defined as having two infinite strings of zeroes, one on each side of the decimal, and integers would be defined as rational numbers where the right-side string of zeroes begins immediately after the decimal. In this way, finite numbers can be thought of as just a special case of infinite numbers, so the fact that they still retain some properties of infinite numbers (like lack of an upper bound) makes intuitive sense.

fishfry wrote:So you are saying, as a general principle, that a set of things should have the same property as the things?


Not at all. I'm saying that, when a set of things is designed specifically to share certain properties with the things, then intuitively one would expect the set to actually share those properties with the things. Sets in generally obviously can be completely unrelated to the things they contain, but I'm talking about a specific case in which that is explicitly avoided.

To all:

After playing around for a while with the idea of extending standard arithmetic into infinite digits to the left, the results are interesting. I doubt I'm the first to do it, so this may already be well known, but the results end up having some similarities to the p-adics, though there are differences. In general, any infinite-digited integer (I'll call them 'infintegers') written as a repeating pattern to the left is equal to the negative equivalent repeating decimal. So ...999 = -0.999 = -1, ...111 = -0.111 = -1/9, and so on. Because such values can only be easily defined with repeating patterns, terminating decimals have no equivalent representation as infintegers, so the infintegers don't quite equal the rational numbers, as they miss any fractions with denominators that include any of the same factors as your base. But a terminating decimal can be approximated arbitrarily closely by using an arbitrarily long pattern. For example, ...5000050000 = -0.50000500005... to approximate -0.5 . Because it is only the repeating pattern that equates to a negative decimal, positive decimals can be achieved by having digits that don't fit the pattern, as these digits still function as normal integers. For example, since ...888 = -8/9, ...8889 = 1/9, as you've simply added 1(or alternately, multiplied by 10 then added 9, either way it works out). Addition, subtraction, and multiplication all work exactly as usual. Division works for any divisor that isn't a factor of your base, otherwise you're forced to either use a standard decimal place or use remainder notation.

An infinteger that isn't a repeating pattern, but can still be defined by some algorithm for computing the digits(such as simply writing the digits of pi in the opposite direction), might or might not be reasonably considered a number, but it couldn't be converted into any equivalent standard real number. Given the general rule that an infinteger equates to the equivalent negative decimal with the same pattern, a non-repeating infinteger would equate to a non-repeating negative decimal, except that the right-most digits of an infinteger pattern equate to the right-most digits of the equivalent decimal pattern, so a non-repeating infinteger written from right to left and truncated after n digits would tell us the "last" n digits of the non-repeating decimal. Perhaps that could be given some degree of meaning along the lines of surreal numbers, but it obviously can't be interpreted meaningfully as a real number.

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Re: Infinity paradox? Help

Postby lightvector » Sat Mar 24, 2012 10:09 am UTC

I think the problem is that nobody except you can figure out what the actual rules of your system are. For example, earlier you wrote:

arbiteroftruth wrote:I haven't rigorously proved this of course, but it seems to me it should be possible to create a consistent number system in which behavior "at infinity" is always by definition the behavior found by the limit of making the relevant variable arbitrarily large.


From this and some further statements you made after this involving integers and induction, I and apparently many others took this to mean that you wanted to create a system that contains the natural numbers and their standard operations and at least one additional new number "infinity" with the rule:

(1): If a property P holds true for every sufficiently large integer n, then P holds true for "infinity".

I assumed that your system would also contain some way to speak of the property of a number of being finite or infinite, such that standard natural numbers would have the property of being finite and the new object "infinity" would have the property of being infinite. I then pointed out that if this were the case, you would obtain a contradiction when P is the property "is not infinite". You then replied:

arbiteroftruth wrote:That's a matter of properly defining the operation of taking the limit. That is, by definition, taking the limit as the variable approaches infinity implies explicitly replacing the property "is finite" with the property "is infinite", and my system would propose that, in doing so, all other properties are determined by how they converge, if at all.


So it seems like you weren't actually proposing (1). Reading your reply, it seems like you are proposing:

(2): If a property P holds true for every sufficiently large integer n, then P holds true for "infinity" if you replace every instance of "is finite" with "is infinite" within P. (Presumably also treating "is not finite" as "not (is finite)" and replacing it with "not (is infinite)", etc...)

But it's difficult to tell if this is actually what you meant, or if there is another exception to the rule as well. In fact, this actually doesn't prevent the paradox I provided. So you probably meant this instead:

(3): If a property P holds true for every sufficiently large integer n, then P holds true for "infinity" if you replace every instance of "is finite" with "is infinite" and every instance of "is infinite" with "is finite" within P.

But just now you've started talking about whether properties are computable as additional criteria. Are these additional restrictions that you are adding to your rule? If "computability" is an additional criterion, then you should be precise about how to tell if a property is or is not computable, as not everyone may share your notion of "computable", and there are a *lot* of possible properties and ways to combine them. And how do all the infinite decimals and strings and representations of infinite numbers relate to the idea of limits in (3)? Have you thought about the possible contradictions that might arise? Seeing as you keep introducing and speculating about various new types of infinite numbers, are you actually trying to present some specific thing that is self-contained and coherent, or are you really just taking a random walk through the through the space of ideas relating to "infinity"?

Basically, my point is that the manner in which you've presented your ideas makes it very difficult for others to figure out what you're trying to say. If you're trying to present new ideas, being precise and unambiguous in your communication is critical. Without a precise statement of the axioms you are proposing, people are having to guess to varying degrees what you are actually saying and what you've left out.

arbiteroftruth
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Re: Infinity paradox? Help

Postby arbiteroftruth » Sat Mar 24, 2012 5:13 pm UTC

lightvector wrote:I think the problem is that nobody except you can figure out what the actual rules of your system are. For example, earlier you wrote:

arbiteroftruth wrote:I haven't rigorously proved this of course, but it seems to me it should be possible to create a consistent number system in which behavior "at infinity" is always by definition the behavior found by the limit of making the relevant variable arbitrarily large.


From this and some further statements you made after this involving integers and induction, I and apparently many others took this to mean that you wanted to create a system that contains the natural numbers and their standard operations and at least one additional new number "infinity" with the rule:

(1): If a property P holds true for every sufficiently large integer n, then P holds true for "infinity".

I assumed that your system would also contain some way to speak of the property of a number of being finite or infinite, such that standard natural numbers would have the property of being finite and the new object "infinity" would have the property of being infinite. I then pointed out that if this were the case, you would obtain a contradiction when P is the property "is not infinite". You then replied:

arbiteroftruth wrote:That's a matter of properly defining the operation of taking the limit. That is, by definition, taking the limit as the variable approaches infinity implies explicitly replacing the property "is finite" with the property "is infinite", and my system would propose that, in doing so, all other properties are determined by how they converge, if at all.


So it seems like you weren't actually proposing (1). Reading your reply, it seems like you are proposing:

(2): If a property P holds true for every sufficiently large integer n, then P holds true for "infinity" if you replace every instance of "is finite" with "is infinite" within P. (Presumably also treating "is not finite" as "not (is finite)" and replacing it with "not (is infinite)", etc...)

But it's difficult to tell if this is actually what you meant, or if there is another exception to the rule as well. In fact, this actually doesn't prevent the paradox I provided. So you probably meant this instead:

(3): If a property P holds true for every sufficiently large integer n, then P holds true for "infinity" if you replace every instance of "is finite" with "is infinite" and every instance of "is infinite" with "is finite" within P.

But just now you've started talking about whether properties are computable as additional criteria. Are these additional restrictions that you are adding to your rule? If "computability" is an additional criterion, then you should be precise about how to tell if a property is or is not computable, as not everyone may share your notion of "computable", and there are a *lot* of possible properties and ways to combine them. And how do all the infinite decimals and strings and representations of infinite numbers relate to the idea of limits in (3)? Have you thought about the possible contradictions that might arise? Seeing as you keep introducing and speculating about various new types of infinite numbers, are you actually trying to present some specific thing that is self-contained and coherent, or are you really just taking a random walk through the through the space of ideas relating to "infinity"?

Basically, my point is that the manner in which you've presented your ideas makes it very difficult for others to figure out what you're trying to say. If you're trying to present new ideas, being precise and unambiguous in your communication is critical. Without a precise statement of the axioms you are proposing, people are having to guess to varying degrees what you are actually saying and what you've left out.


I think I see the misunderstanding here.

I was never saying there should be a singular new number designated "infinity". I said the system would take the same approach as calculus, and "infinity" is not regarded as a number in calculus. If you were to think of it as a number, it would be a number whose properties can vary wildly depending on the exact equation you're using as some variable within the equation becomes arbitrarily large.

This is also why the notion of computability comes up. This treatment of infinity isn't the definition of a new number, but a matter of taking a given process (like iterating an algorithm or computing values of a function) and extending it beyond what you could ever do with normal computational methods, because normal computational methods must necessarily use a finite number of operations and thus cannot deal with infinity without some special rule for doing so. Calculus handles this by taking the limit, and I proposed that the same notion could be applicable everywhere.

The subject of doing arithmetic on infinite strings of digits is partly a new subject tangentially related to the subject of my hypothetical system, and partly a particular case of applying the methods of that system. That is, there are rules of arithmetic/digit manipulation that hold true for all numbers with a finite number of digits, so if we "take the limit" we conclude that the same rules apply to an infinite string of digits, and this leads to meaningful results.

One reason I regard that as a separate topic merely connected to the topic of definite infinities by taking limits is that it reveals one way in which my hypothetical system would break down. One of the things gmalivuk brought up as a property of integers that doesn't necessarily hold true at infinity is the property that an integer that isn't 0 or 1 is strictly less than its square. An infinite string of digits cannot retain both that property and the property of following normal arithmetic rules, because the extension of arithmetic to infinite strings allows positive fractions less than 1 to be represented as infinite integers, ...8889 evaluates to 1/9, for example. When the number of digits is allowed to be infinite, my system would be forced to choose between preserving the properties of arithmetic and preserving the properties of the numbers themselves, thus the system breaks down.

But that notion of applying normal arithmetic to infinite strings does interest me. If I were to simply choose to side with preserving the normal rules of arithmetic, and let things like squaring and the "less than" function fall where they may, I can't find any contradictions.

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Re: Infinity paradox? Help

Postby gmalivuk » Sat Mar 24, 2012 5:36 pm UTC

How do you order these new infinite-digit integer? As in, how do you know whether one is bigger than another?
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Re: Infinity paradox? Help

Postby arbiteroftruth » Sat Mar 24, 2012 6:23 pm UTC

gmalivuk wrote:How do you order these new infinite-digit integer? As in, how do you know whether one is bigger than another?


For now, I'll only address the case in which there is eventually a repeating pattern of digits to the left, since as I said I don't think a non-repeating case can be meaningfully defined, at least not to any equivalent position on the standard number line.

Given that, I'm not sure what the most formal and concise way to describe it would be, but an ordering is definitely possible, because the digits that create the repeating pattern will equate to a negative decimal with the same pattern(multiplied by the base of your system to the exponent of the position in which the pattern begins), and any digits that aren't part of the pattern are evaluated normally. The total number then is simply the sum of those two components, thus falls on the standard real number line, so these numbers can be ordered accordingly. To use the ...8889 example again, ...888 = -0.888 = -8/9, since the pattern begins in the tens position it instead corresponds to -80/9, then we add the 9 that isn't part of the pattern, giving us 81/9 - 80/9 = 1/9.

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Re: Infinity paradox? Help

Postby arbiteroftruth » Sat Mar 24, 2012 6:41 pm UTC

Given that we're now two degrees of separation from the original topic, perhaps it's best if I start a new thread for the subject of infinite-digit integers.

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Re: Infinity paradox? Help

Postby gmalivuk » Sat Mar 24, 2012 7:03 pm UTC

So your infinite-digit integers are actually not infinite in magnitude, but instead fall between other integers in magnitude? This removes well-ordering from the integers, which is kinda an important characteristic.

Also, is your system really different from the p-adics, which have already been mentioned here?
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Re: Infinity paradox? Help

Postby arbiteroftruth » Sat Mar 24, 2012 7:29 pm UTC

gmalivuk wrote:So your infinite-digit integers are actually not infinite in magnitude, but instead fall between other integers in magnitude? This removes well-ordering from the integers, which is kinda an important characteristic.

Also, is your system really different from the p-adics, which have already been mentioned here?


I don't follow. How does it remove well-ordering? Perhaps I'm not understanding something, but for any 2 numbers in this system, you can easily subtract them, see whether the result is greater than, equal to, or less than 0, and conclusively define which number is larger, if either.

It's not quite the same as the p-adics, at least as far as I understand the p-adics. Correct me if I'm wrong, but when converting between p-adics and standard numbers, moving more digits to the left in the p-adic numbers somewhat corresponds to moving more digits to the right in standard numbers, right? That's not the case in the system I'm looking at.

Also, I've started a new thread for this topic, as the original point of this thread has expired.

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Re: Infinity paradox? Help

Postby antonfire » Sun Mar 25, 2012 1:21 am UTC

Well-ordering and linear ordering do not mean the same thing. A set is well-ordered if every one of its (nonempty) subsets has a least element. The fact that the natural numbers are well-ordered is what allows you to prove things about all natural numbers by induction.
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