Timeline for the strength of saying "each sentence of true arithmetic has a recursive proof"

6 events

when toggle format	what		by	license	comment
Mar 11, 2017 at 15:09	history	edited	Haidar	CC BY-SA 3.0	added addendum on equivalence
Mar 10, 2017 at 15:10	comment	added	Haidar		Concerning (A): There are $PA_{\omega}$-proofs that aren’t recursive: ones with too large of an ordinal height. Also, I recognize that the issue raised in Q1 doesn’t perfectly match up with the title, which doesn't mention the recursive $\omega$-rule.
Mar 9, 2017 at 16:08	comment	added	Benedict Eastaugh		Concerning (B): this is Shoenfield's completeness theorem (Shoenfield, J. R., 1959, "On a restricted $\omega$-rule", Bulletin de L'Académie Polonaise Des Sciences: Série des sciences mathématiques, astronomiques, et physiques 7:405–407). Shoenfield's original bound was $\omega^\omega$, but this can be improved to $\omega^2$. For details see Franzén, T., 2004, "Transfinite Progressions: A Second Look at Completeness", Bulletin of Symbolic Logic 10(3):367–389.
Mar 9, 2017 at 16:07	history	edited	Noah Schweber		edited tags
Mar 9, 2017 at 15:57	comment	added	Gro-Tsen		(A) Isn't "recursive $\mathit{PA}_\omega$ proof" redundant since you already defined $\mathit{PA}_\omega$ to use the recursive $\omega$-rule? Did I miss something? • (B) Why is it that every sentence of true arithmetic (sanity check: this is the same as "true sentence of arithmetic", right?) have a $\mathit{PA}_\omega$ proof (let alone one of height $<\omega^2$)?
Mar 9, 2017 at 14:54	history	asked	Haidar	CC BY-SA 3.0