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Let's say we have an equation that has no end to its result. (Sorry I don't have an example to hand, and the value of Pi is still under question so I won't use that).

Can this value be considered absolute (like an absolute truth)? Or would it be undefined (even though we know what it "is")?

P.S. this is not so much about maths (I used a bad example). What I'm talking about is hard to put into a better question I guess.

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    @Roland not sure I understand the question. (Note that mathematically speaking there are many different orders of infinity.) – Joseph Weissman Jun 15 '11 at 23:32
  • @Joe try not to interpret the question from the stand point of "scholarly speaking" or "scientifically speaking" etc, but rather just from "laymanly" speaking ;). – RolandiXor Jun 15 '11 at 23:37
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    @Roland: Please don't write things like "the value of Pi is still under question". Are you talking about equation that define an infinite number of digits? For numbers like Pi, there are algorithms that calculate the digits, there is no conceptual problem with the definition. There are other numbers that do pose problems, but you should first understand that the value of Pi is not "under question". – Phira Jun 16 '11 at 8:42
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    The value of pi is absolutely known; it just can't be expressed as the ratio of 2 integers. This is only a conceptual problem if you don't understand irrational numbers. – Wooble Jun 17 '11 at 17:39
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    just the fact that a number can't be written down does not meant that they don't exist or is a second class numbers in any way, your question is too vague and has no mathematical nor philosophical basis. – Lie Ryan Jun 22 '11 at 11:56
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Given any concrete and realistic definition of "absolute" the answer must be no, there are infinities which are undefined. π is undefined as a ratio of integers, and the rabbit hole never ends. One example is Chaitin's Ω constant the knowledge of which would give the ability to answer all questions of computation. Also consider inaccessible cardinals which set theory can discuss although there is no accepted basis on which they can be proven to exist.

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  • Forgetting what 'absolute' is intended to mean...what do you mean by 'undefined' and 'exist'? As inscrutable as 'inaccessible cardinals' is to most people (even most mathematicians), the can be proven to exist just as easily as, say, negative numbers. – Mitch Jun 17 '11 at 2:48
  • New axioms are needed to justify new ideas (e.g. negative numbers or inaccessible cardinals) and the acceptance of new axioms requires a widely-held belief that they won't lead to a contradiction. e.g. the Axiom of Choice has passed this belief threshhold. The Riemann Hypothesis has not, although it is believable enough that there are interesting theorems contingent on it and some have applications in cryptography. So I think "defined" depends on language (and it is not as simple as having a distinguished name), and "existence" depends on beliefs. – Dan Brumleve Jun 18 '11 at 20:38
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    I don't see how this is an acceptable answer. The constant π is not "undefined as a ratio of integers"; it is simply not definable (nor expressible) as a ratio of integers, nor as the root of any polynomial over the integers. This does not make it undefined! Similarly, 1/3 is not definable or expressible as a finite decimal expansion: does that make it undefined? Furthermore, just because one must occasionally explore alternative axiomatic systems to consider new ideas, does not make the new ideas "undefined" or "less well defined". – Niel de Beaudrap Sep 6 '11 at 14:11
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If you can talk about it, then it has a finitary description, no? While the decimal representation of pi may be infinite, the phrase "pi" is quite finite.

As another example, calculus is based on infinitesimals. And it seems reasonable to call calculus "absolute".

Lastly, there are many non-computable numbers, which is perhaps the closest thing to "not having an absolute value" that I can think of. Merely having an infinite representation does not guarantee that a number is non-computable though.

So in short: no, numbers with infinitary representations not only can be talked about usefully, but are talked about usefully.

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  • I agree that calculus is the study of infinitesmals and attempts to approximate their quantities. I further agree that numbers with infinite representations can be and are talked about usefully. But I don't necessarily think it follows that they are absolute. Some number theorists would certainly argue that the quantities are indeed technically "undefined", but that doesn't mean we can't talk about them in a useful way. – Cody Gray Jun 16 '11 at 4:44
  • @Cody: perhaps I misunderstand what "absolute" means. If a program computing a limit halts, it will halt with a fixed value on its tape - in what sense is this value not "absolute"? It will never change, and all equivalent programs will halt with the same value. – Xodarap Jun 16 '11 at 16:00
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Absolutely. or at least to the same extent that finite things can be understood as 'absolute'.

To make it more explicit, what I think you mean by 'absolute' is 'wholly knowable' or complete in knowledge'.

Let's limit ourselves simply to number-like things. I'll presume that you accept finite integers as 'absolute' in this sense.

So what about a rational number, say 1/3? In terms of your question it is the solution to the simple equation "3 x = 1". Is it 'absolute'? I think you'd think so because it is the ratio of two finite numbers. But in some calculations, you may need to know its decimal representation, which is 0.333333..., an infinite sequence of threes. This representation is 'absolute' because we know that any particular digit can be finitely described (namely it's always 3).

What about 1/7? An exercise for the reader...it's not as simple as 1/3, but still has the same property, we can easily know any particular digit no matter how far out.

Rather than try to continue this reasoning for more complicated numbers (which would similarly work for irrationals like sqrt(2) and the mentioned pi, I think the point is made with rationals with no-terminating decimal representations: there exist infinite objects which are 'absolute'.

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