Joe Shipman JoeShipman at aol.com
Tue Jun 4 18:28:35 EDT 2013

Would it be fair to say that the reason we talk about validities of SOL rather than axioms and theorems is that anything of interest is axiomatizable by a single sentence so we don't need an ontology, we just need to know what implications are valid?

In FOL we can similarly avoid ontology by replacing talk about ZFC proving X with talk about "Conjunction_of_GB_Axioms --> X*" being a tautology, where X* is the class-theoretic version of X that required all variables to be members of something rather than proper classes. For some reason people don't like to do this, maybe because Godel's completeness theorem gives us models so we don't mind ontological commitments or infinite axiom schemes.

Looked at this way, the SOL vs ZFC conflict simplifies to two fundamental points:

1) ZFC with FOL clarifies what is and isn't a theorem of mathematics as a whole, and the direction of foundational progress is just what set existence axioms to extend it with, but it is weak enough to leave propositions like CH not only unsettled, but fundamentally vague. It is great syntactically but its semantics are inadequate.

2) SOL takes the view that mathematical propositions like CH have independent and absolute meanings, but we can't have a way to generate all valid sentences and our epistemological tools peter out before ZFC does. Thus we can't "know" as much math as we can if we accept the ZFC axioms (for example, we can't reach the consistency of Zermelo set theory in a purely "logical" way, we need set existence postulates too strong for that, all we can say is ZF proves Con(Z).)

-- JS

Sent from my iPhone

On Jun 4, 2013, at 11:43 AM, jkennedy at mappi.helsinki.fi wrote:

From Jouko Vaananen:

The set of validities in second order logic is not Pi^1_2-complete, not even Pi^m_n-complete for any m and n, but it is Pi_2-complete in the Levy hierarchy (see Vaananen: "Second order logic and foundations of mathematics", Bulletin of Symbolic Logic, Volume 7, Issue 4, December, 2001).

Concerning your questions:

1) The only commonly used axiom system for second order logic that I know is the one given in Hilbert-Ackermann (1928) based on axioms of (impredicative) comprehension. The Soundness Theorem of this system is provable in ZFC, so anything that is valid in that system is provably valid in ZFC. One can easily formulate such new axioms for second order logic that cannot be proved valid in ZFC, for example the V=L, Continuum Hypothesis, etc. The question is then, is V=L a commonly used axiom of second order logic, I think not.

2) The obvious example is the following: Let A be the Hilber-Ackermann axiomatization of second order logic. Let B be the second order Peano. Let C be the number theoretic statement of consistency of A+B. Now ZFC proves the validity of the second order statement "B implies C". By Goedel's theorem we cannot prove in A+B the statement C, and hence not the statement "B implies C" in A.  (See also Vaananen: "Second order logic or set theory?" Bulletin of Symbolic Logic, 18(1), 91-121, 2012.)

Best Wishes

Quoting Joe Shipman <JoeShipman at aol.com>:

> The set of validities in Second Order Logic with standard semantics is Pi^1_2 complete, so we can't give a complete axiomatization. My question has two parts:
> 1) is there any commonly used system of axioms for second order logic that includes sentences which ZFC does not prove are validities?
> 2) is there any sentence which ZFC proves is a validity of second order logic, but which is not a consequence of any commonly used system of axioms for second order logic other than the system "enumerate sentences with ZFC-proofs of their validity"?
> -- JS
> Sent from my iPhone
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