FOM: Re: 102:Turing Degrees/2

[Harvey Friedman]

Reply to Ketland 1:28AM 4/10/01.

I wrote:

>>The following is a restatement of a theorem from Turing Degrees/1.
>>
>>Let Z2 be the usual first order system of second order arithmetic. Let Z2+
>>be Z2 with a satisfaction predicate added and induction and comprehension
>>are extended to all formulas in the expanded language.
>
>Are the axioms you use for Sat(x,y) in Z2+ Tarski's inductive axioms?

Yes.

>Do you consider "self-applicative" (usually called Kripke-Feferman) axioms
>in this context over Z2?
>E.g., Things like the axiom
>T-rep:  T(A) --> T(T(A)))
>from Friedman/Sheard 1987?

No. I am interested in Z2+ as a basic benchmark for new statements about
Turing degrees. Issues about self application are out of context.

>More generally, if Ref(S) is Feferman's reflective closure operation on
>system S (Fefermann 1991: "Reflecting on Incompleteness" 1991), do you know
>how Ref(Z2) turns out? Proof-theoretic strength, interpretability?

Presumably, such things are somewhat stronger.

The striking thing about the new statements about Turing degrees is that
one is taking gigantic leaps like from roughly ACA0 to Z2 to Z to large
cardinals.