Gustavo Lacerda

Any two good axiom systems will agree about the truth-value of the Goldbach conjecture (whatever it is), though they might disagree about its provability

I don't understand what this means. If you have an axiom system that proves neither P nor not P, how can P be true or false according to this system?

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gwillen.livejournal.com

My intuitions for this sort of thing aren't great, but IIRC it's something like "if you can use some higher logic to demonstrate that system X plus not-P would be a contradiction, but using X itself you cannot prove P."

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This means you're talking about a system where the completeness theorem fails. Which seems at least a little bit odd of an axiom system to me.

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gwillen.livejournal.com

Hm, I conclude after brief review that I _still_ do not fully appreciate the implications of the various theorems with Gödel's name on them. :-)

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wjl.livejournal.com

is there some trick here where first-order logic doesn't count as an "axiom system" or something?

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If the Goldbach conjecture is false, there is a counterexample, which is constructible in every good axiom system.

If GC is true, then all good axiom systems will either prove it, or be independent of it.

I'm talking about an "objective" notion of truth. Maybe this is the same as truth in the minimal model of ZF.

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If GC is true, then all good axiom systems will either prove it, or be independent of it.

If T is an axiom system, and GC is independent of T, I don't think it makes sense to say that T even has a truth value for GC.

I'm talking about an "objective" notion of truth. Maybe this is the same as truth in the minimal model of ZF.

I think that will just push the problem to the metatheory.

Edited Date: 2009-07-13 07:46 am (UTC)

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I've weakened my claims / the way they are worded.

Does my post look ok now?

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I agree with the statement about GC. All statements about the integers is a little fishy though. I mean, the consistency of ZF is a statement about the integers.

Edited Date: 2009-07-13 08:34 am (UTC)

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easwaran.livejournal.com

But it seems quite plausible that there is indeed a fact of the matter about the consistency of any theory T - the fact that PA doesn't settle Con(T) just illustrates the weakness of PA. I suppose you could also take the position that Con(T) doesn't properly express the consistency of the theory T - after all, it depends on the arithmetization of the notion of proof, just as Turing's results depend on the formalization of the notion of computation. I'm working on a paper on this stuff right now, trying to show that people really have to accept that there's a fact of the matter at least about these sorts of statements.

Also, you might want to restrict your original claim to Π⁰₁ statements about the integers, rather than statements with arbitrary quantifier complexity. Though I do suspect that many further ones will have determinate facts of the matter.

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I have the same qualms about there being a unique "fact of the matter" about consistency claims as I do in the comment I made below.

I'm very probably comfortable with whatever arithmetization you might make of Con(T) as "properly expressing the consistency of the theory", though, for what it's worth.

It's just that Con(T) says that there's no finite proof of bottom from T, and I still have to wonder, "ok, what do you mean by "finite"?" and this is tantamount to asking what the structure of the natural numbers "really" is.

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It's just that Con(T) says that there's no finite proof of bottom from T, and I still have to wonder, "ok, what do you mean by "finite"? and this is tantamount to asking what the structure of the natural numbers "really" is.

Are you talking about nonstandard integers? That is an interesting question. But it seems like you always might have a metatheory infected with nonstandard integers, so I'm not sure how you'd deal with that.

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I'd never heard of them until now.

Looking it up, I found a book talking about the phenomenon that axioms make truths, which this post is about. See red highlight

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yes, exactly --- how do we determine which numbers are "standard" and which are "nonstandard"? Your metatheory is always at risk for being "infected" by the "nonstandard" ones, because you can't know ahead of time (in a uniform, computable way that would manifest itself as a decision procedure for which axioms to include or not) whether you are surprised to the point of rejecting some number (some axiom) as nonstandard.

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easwaran.livejournal.com

One way to determine which are standard and which are non-standard is by saying that the standard ones are the ones that are in the unique (up to isomorphism) model of PA that injects into all others. Or do you question the existence and uniqueness of such a model?

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the fact that PA doesn't settle Con(T) just illustrates the weakness of PA

Maybe I don't understand what you're saying, but it seems like Godel's second incompleteness theorem means you always have this problem.

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That's correct (though "problem" seems like a loaded word). Thence the Feferman hierarchy.

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As Kenny said: << it seems quite plausible that there is indeed a fact of the matter about the consistency of any theory T >>. This is what I'm claiming.

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And I think it's precisely what I'm disagreeing with :)

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Do you agree that there's a fact of the matter about the Goldbach conjecture? (I say yes, since \Pi_2 statements about the integers are refutable)

If you agree, then I don't see how you could disagree with my claim that there is a fact of the matter about the consistency of every theory T. If T is not consistent, then a brute-force search will find a proof of bottom.

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Hmm... I'm not entirely sure anymorre what my "true" [ :) ] beliefs are anymore, but let me say "no" for the sake of having an argument.

How could I doubt that there is a fact of the matter about the Goldbach conjecture?

Suppose I announce that I have a counterexample to Goldbach. You ask me what it is. I say, "well... it's very large! If you accept common mathematical notation, though, I can write it down as (2^2^2^234098^18^(47 * 1923498234 * 3498))!!!" and in that case (apart from the difficulty of checking that that number is not the sum of two primes!) you are entirely justified in believing that I have at least denoted a definite natural number, because the termination of exponentiation, multiplication, and factorial, are easily justified without going outside of PA.

But what if I say that my counterexample is the number of steps until you reach zero in the Goodstein sequence starting with "(2^2^2^234098^18^(47 * 1923498234 * 3498))!!!"? Perhaps this is even the first counterexample to the goldbach conjecture, so that if this expression doesn't denote a number according to PA (since PA doesn't tell us that Goodstein sequences always terminate) then Goldbach's conjecture is true.

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You can probably replace "the number of steps in Goodstein sequence" with Busy Beaver (BB) function (e.g. "the maximum number of symbols that can be written by a Turing Machine with 20 states").

Though I'm not aware of theorems about the unprovability of specific BB numbers in particular axiom systems. (Do set theorists ever talk about BB? All I can find is the Chaitin excerpt below)

I also want to say that there is a true value of BB(n) for all n (any guess that is too low will eventually be refuted). The correct axioms will uncover the reality of BB values...

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G.J. Chaitin - Computing the Busy Beaver Function

In T. M. Cover and B. Gopinath, Open Problems in Communication and Computation, Springer, 1987, pp. 108-112

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The big difference between what you're describing and what Jason is describing is that PA will prove that BB(n) always exists, even if it can't prove what it is. Whereas the length of Goodstein sequences can't even be proven to exist from PA.

Edited Date: 2009-07-24 05:06 am (UTC)

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I like your question. It's an interesting attempt to get incompleteness to screw up my Platonism.

So, supposing you had a proof that the first counterexample to GC is s, the number of steps until the Goodstein sequence starting with n terminates (or similarly, the first counterexample is BB(n) )...

Q: Does this imply that s doesn't denote a number?
A: No. In fact, I would consider your proof as indication that Goodstein(n) is computable.

(How can an expression not denote a number according to PA?)

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You should be careful. What you're claiming is probably sensitive to your metatheory. PA doesn't prove that *all* Goodstein sequences terminate. However, in the presence of a strong enough metatheory, for any specific n you can describe, it should prove that the Goodstein sequence for that n terminates. All of the concrete PA independence results rely on using models with nonstandard elements. So also, if you object to a stronger metatheory, you may find that you can't prove that PA can't prove Goodstein sequences terminate ;)

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I still don't understand why being a "Platonist" is even sufficient to be convinced that statements about natural numbers always have determinate truth values.

Suppose I put my Platonist hat on and say, okay, mathematical objects truly exist "out there". Nonetheless I can't insist that a statement about a mathematical object has a determinate truth value, unless I pick out one distinct mathematical object.

Example: I ask you, "is the group commutative"? You say, "what do you mean, 'the group'? There's lots of different groups. Some are commutative, some are not"

Correspondingly, when you ask "do the the natural numbers have arithmetic property X?" it sounds to my ears like "does the model of PA have property X?" and I go "what do you mean, 'the model of PA'? There's lots of models of that theory, some have property X, some don't"

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easwaran.livejournal.com

This is the point of view that I'm arguing against in the paper I'm writing. I'm glad to find someone who actually holds the view! (I had been trying to pin it on Saunders Mac Lane, but on re-reading the relevant papers of his, he actually seems to reject the view.)

Anyway, when I talk about the natural numbers, I'm not just talking about an arbitrary model of PA. If there are multiple such models that are all equally good objects of study when doing number theory, then I don't see how you can accept the Gödel arithmetization of the notion of proof - arithmetization in one model will say that a particular theory is consistent, while arithmetization in another model will say that this very same theory is inconsistent.

If you accept the claim that one particular model is the one that is right for arithmetization purposes, then that's the model I'm after.

Also, if you're interested in studying models of PA, then you must be admitting that PA is consistent. But clearly you didn't accept this because Con(PA) was proven from PA. Instead, you seem to be admitting a notion of truth that can apply to consistency statements that goes beyond provability from PA. (If PA isn't your base theory, I'll repeat all my arguments with whatever your base theory is.)

Also, I don't think being a "Platonist" is necessary for denying that all models of PA are equally good, and for accepting that there's a notion of mathematical truth that goes beyond mere provability from the axioms. The claim of the paper that I'm drafting is that any philosophical position on what mathematics is about must agree here, except perhaps for a very radical sort of view that would seem to get in the way of the applicability of mathematics to the world. (I suspect that Nelson's non-standard analysis based ultrafinitism is of this sort.)

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roseandsigil.livejournal.com

Every time someone privledges finite mathematics over infinite mathematics because of their intuitions, God kills a pony. Just sayin'

(More serious comments may come later.)

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fancybred.livejournal.com

If the Goldbach conjecture is false, all good axiom systems will agree about this. (You can replace "Goldbach conjecture" with any statement about the integers)

I've started reading Gödel's Theorem by Torkel Franzén, and he seems to express a similar viewpoint:

In a mathematical context, on the other hand, mathematicians easily speak of truth: "If the generalized Riemann hypothesis is true...," "There are strong grounds for believing that Goldbach's conjecture is true...," "If the twin prime conjecture is true, there are infinitely many counterexamples...." In such contexts, the assumption that an arithmetical statement is true is not an assumption about what can be proved in any formal system, or about what can be "seen to be true," and nor is it an assumption presupposing any dubious metaphysics. Rather, the assumption that Goldbach's conjecture is true is exactly equivalent to the assumption that every even number greater than 2 is the sum of two primes.

[...]

Note that this use of "true" extends to the axioms of a theory. It is sometimes thought, when "true" is used in some philosophical sense, that the axioms of a theory cannot be described as true, since they constitute the starting point that determines what is meant by "true" in later discourse. All such philosophical ideas are irrelevant to the mathematical use of the word "true" explained above, which will be adhered to throughout the book when speaking of mathematical statements as true or false. For example, that the axiom "for every n, n+0 = n" in PA is true means only that for every natural number n, n+0 = n.

I sort of understand the spirit of Franzen's remarks, but I wish he didn't resort to Tarski's theory of truth as a supposed explanation. Is there a better way to explain this viewpoint?

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