[Tainnor]

> Take the collection of all true statements and make that your axiomatic system.

A complete proof system needs to be able to derive Γ |- φ for every pair Γ, φ such that Γ |= φ. Not just when Γ is the complete theory of some structure. Completeness of first-order logic (and its failure for second-order logic) is about the logical system itself, while the incompleteness theorems are about specific theories - people often mix these up, but they talk about very different things.

> Andreas Blass in the comments says that Incompleteness does not apply to PA_2.

He says something rather different, namely that its "meaningless". That's a value judgement. Incomplete proof calculi for second order logic do exist (e.g. any first-order proof calculus) and for those, what I wrote is true. Andreas Blass would probably just think of this as an empty or obvious statement.

[gedpeck]

You know more than me. That is certain.

However, my understanding is that the incompleteness results apply to only recursively enumerable axiomatic systems. I can find references for this. If I take the standard model of ZFC and collect all true statements in the one model of PA_2 and make that my axiomatic system then I have an axiomatic system that is not recursively enumerable and contains PA_1. It’s not a nice set of axioms. It’s not computable. But it shows that one can have an axiomatic system that contains PA_1 for which the Incompleteness theorems don’t apply.

Andreas wrote “meaningless” not “nonsensical”. I’m not a pedant but the former term evokes in me the idea of “does not apply in this situation becausethe hypotheses of the incompleteness theorem are not satisfied”.

From a mathematical logic book is the following. It’s the set up for the Incompleteness theorems.

Suppose that A is a collection of axioms in the language of number theory such that A is consistent and is simple enough so that we can decide whether or not a given formula is an element of A.

PA_2 is not such a system and as such the Incompleteness Theorems don’t apply. Maybe we are talking past each other. You know more than me.

[Tainnor]

> However, my understanding is that the incompleteness results apply to only recursively enumerable axiomatic systems. I can find references for this.

That's a matter of semantics as to what you consider the first incompleteness theorem to be precisely (of which there are several variants). Gödel's proof itself doesn't directly work for second-order logic. But the statement "if Γ is some axiomatic system that satisfies certain conditions, then for any sound proof calculus there is a sentence that isn't provable from Γ in this calculus" is true in second order logic too, it's just that the "failure" happens much "earlier" (and is in some sense obvious) than in the case of FOL.

> PA_2 is not such a system and as such the Incompleteness Theorems don’t apply.

I'm really not all that familiar with second-order PA, but it is my understanding that the set of its axioms is decidable. It consists of a finite collection of axioms plus one schema (comprehension axiom) which is valid when it's instantiated by any given sentence - but deciding whether something is a valid sentence is easy. Therefore, what you quoted applies to second-order PA too.

[gedpeck]

From what you and the other person on this thread has said and from what I've read it appears that perhaps the following is true:

1. The axioms of PA_2 are recursively enumerable. 2. The full semantics of PA_2 are what cause categoricity.

It seems to me then that the crux of the matter is that the full semantics of PA_2 prevent there being an effective deductive system. I think Z_2 is constructed to get around the non effectiveness of the full semantics of PA_2 and is a weaker theory.

Anyway, thanks for the enlightenment.

[Tainnor]

With the caveat that I don't really understand second order logic well enough to say all that much about it, there's a debate in the philosophy of mathematics as to whether second-order logic should count as the foundational logic, since on the one hand most first-order theories aren't categorical (due to Löwenheim-Skolem) and on the other hand, second order logic (with full semantics) already presupposes set theory.

In any case, the reason why PA_2 is categorical is because the second-order axiom of induction allows quantification over arbitrary sets which allows you to say that "0 and adding the successor function to 0 arbitrarily often already gives you all natural numbers".