Comments on: What Happens When You Iterate Gödel’s Theorem?

By: anon

anon — Fri, 03 Sep 2010 11:17:47 +0000

“However, we can make a reasonable definition for what it means for \mathrm{PA} (or any extension) to prove that a number a is an ordinal notation. (This is actually not trivial, since \mathrm{PA} can only talk about numbers, but the set of ordinal notations was defined to be the least set satisfying a certain property.) ”

This is surprising, considering that the ordinal notations do not form an arithmetical, or even hyperarithmetical set. Would you mind explaining how it can be done?

By: anonymous

anonymous — Thu, 22 Jul 2010 19:40:54 +0000

I think the forcing argument in SoSA that $WKL_0$ and $RCA_0$ have the same first order part (namely $I\Sigma_1$) can be used to show the same for $RCA$ and $WKL$.

By: mkoconnor

mkoconnor — Thu, 02 Apr 2009 23:53:18 +0000

In the second-to-last paragraph, everywhere I said “theory”, I meant “recursively enumerable theory”.

By: mkoconnor

mkoconnor — Thu, 02 Apr 2009 23:42:18 +0000

Todd: Thanks for the amusing link! Kenny: Thanks for your comment! First off, I made a mistake: I meant ACA, not ACA_0 (the former allows you to do induction on any formula, the latter only on formulas of the form "x is in S".) ACA_0 is actually conservative over PA, so it doesn't go anywhere up this hierarchy. Secondly, it's not that ACA is conservative over autonomous iterated reflection extensions, it's that it's an extension of all of them, with the additional property that we believe that it's true. If you had something that was conservative over some set of iterated reflection extensions, it would be reasonable to say that that theory was a reflection extension itself, and you could therefore take "one more step". So, it's hard to say that you have a theory which you know to be the union of all theories you know (or could know) to be reflection extensions. But there's no problem with saying that you have a theory (which you believe to be true) which you know to be stronger than all theories which you know (or could know) to be reflection extensions. With regards to your second question about where WKL is in all of this, I actually have no idea.

By: New to the blogroll « Secret Blogging Seminar

New to the blogroll « Secret Blogging Seminar — Thu, 02 Apr 2009 12:57:57 +0000

[...] from mathematical life, Michael’s blog is an excellent cure. Posts I particularly liked: What Happens When You Iterate Gödel’s Theorem?, How to Show that Games are Hard. Possibly related posts: (automatically generated)Carnival of [...]

By: Todd Trimble

Todd Trimble — Wed, 01 Apr 2009 15:50:18 +0000

On the matter of consistency, it seems fitting to link to an old post from Paul Taylor to the categories mailing list, exactly ten years ago today: http://www.mta.ca/~cat-dist/catlist/1999/zf-010499

By: Kenny Easwaran

Kenny Easwaran — Wed, 01 Apr 2009 15:17:17 +0000

That’s nice to see these thematic connections between things like ordinal notations and the theories used in reverse mathematics.

You say that ACA0 is conservative over autonomous iterated reflection extensions, and Pi1-1 CA is conservative over autonomous iterated truth extensions. Is WKL0 conservative over autonomous iterated consistency extensions?