[FOM] Variants of ACA
pax0 at seznam.cz
pax0 at seznam.cz
Wed Nov 30 15:34:05 EST 2011
Hi Jeff
what is the typical theorem which can be proved in ACA but not in ACA0?More generally:
is there a list of theorems such that n-th entry can be proved in ACA with Sigma^0-n-induction
but not in ACA with (n-1)-Sigma^0 induction?
Thank you, Jan Pax
>
> On Nov 11, 2011, at 12:26 PM, pax0 at seznam.cz wrote:
>
> Is there a complete list of variants (together with definitions) of the second
> order fragment ACA.
> There are quite a few of them, including
> ACA^+,ACA_0,ACA,ACA^',ACA^*.
> Thank you, Jan Pax
>
>
> Hi Jan-
>
> Here's the short answer:
>
> The _0 indicates induction restricted to Sigma^0_1 formulas. No _0 indicates
> full induction.
>
> The ^' indicates that the n^th Turing jump exists for all n. The ^+ indicates
> the omega jump exists.
>
>
>
> So this explains these combinations:
>
> ACA_0 is RCA_0 plus comprehension for arithmetically defined sets.
>
> ACA is ACA_0 plus full induction.
>
> ACA_0^' is ACA_0 plus a formalization of "for every X and n, the nth Turing jump
> of X exists."
>
> Note: ACA_0^' is strictly between ACA_0 and ACA in strength. ACA^'
> would be ACA_0^' plus full induction,
> which is the same as ACA.
>
> ACA_0^+ is ACA_0 plus a formalization of "for every X, the omega-jump of X
> exists."
>
> ACA^+ is ACA_0^+ plus full induction.
>
>
>
> ACA^* is a new one for me. The * superscript usually indicates restriction of
> the induction scheme
> to Sigma^0_0 formulas and appending exponentiation axioms. (See section X.4 in
> Simpson's book.)
> Arithmetical comprehension allows reduction of formulas with many quantifiers to
> formulas with
> no quantifiers and one set parameter. In this setting, ACA_0^* would be ACA_0.
> Note that RCA_0^*
> and WKL_0^* are weaker theories than RCA_0 (resp. WKL_0).
>
> Simpson's book has definitions of ACA, ACA_0, ACA_0^+, and formalizations of
> Turing jumps.
>
> Marcone and Montalban recently published an article entitled "The Veblen
> functions for computability
> theorists" in the JSL (Vol 66 no 2 (2011) 575-602) that uses a very elegant
> formalization of the jump.
> That would be a good way to define ACA_0^' and ACA_0^+ (and equivalent to prior
> uses in the literature.)
>
> Hope this helps...
>
> -Jeff
>
>
>
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