Talk:Ordinal analysis: Difference between revisions

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This page would be greatly improved if there were a specific citation for the ordinal of each of the listed theories (e.g. where one can find the proof that I\Sigma_1 is w^w, etc). —Preceding unsigned comment added by 130.226.132.226 (talk) 10:02, 5 April 2008 (UTC)[reply]

proof-theoretic strength

Hi. I am curious about the redirect from proof-theoretic strength. I don't see why it is here. Sure, they are closely related, but from what I've seen, the proof-theoretic strength is not usually defined by the proof-theoretic ordinal. Rather, one logic is stronger than another if it proves more (when the other is interpreted in it). —Preceding unsigned comment added by 68.188.164.248 (talk) 00:29, 16 March 2009 (UTC)[reply]

There are several ways of defining "proof theoretic strength". One way is to say that T is stronger than S if T proves Con(S). Another is the say T is stronger than S if the ordinal of T is larger than S. These are very related but not identical. But if there is no better place for "proof theoretic strength" to redirect it may as well redirect here. — Carl (CBM · talk) 03:06, 16 March 2009 (UTC)[reply]

Notice that consistency strength redirects to equiconsistency; and is also related to large cardinal property. JRSpriggs (talk) 00:57, 17 March 2009 (UTC)[reply]

Hmm. Maybe it is worth starting an article on that topic, then. Do you have anything in mind for large cardinal axioms except that they tend to be linearly ordered by consistency strength? Because the consistency strength of subsystems of artithmetic is equally interesting in that case. — Carl (CBM · talk) 01:09, 17 March 2009 (UTC)[reply]

dynamic ordinal analysis

[1] I don't understand this well enough to add anything to the article about it yet, but it looks relevant. It is an approach to ordinal analysis on very weak (e.g. polynomially bounded) arithmetic systems for which the usual approach is too coarse. 67.122.211.205 (talk) 00:43, 7 September 2009 (UTC)[reply]

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 :::Hmm. Maybe it is worth starting an article on that topic, then. Do you have anything in mind for large cardinal axioms except that they tend to be linearly ordered by consistency strength? Because the consistency strength of subsystems of artithmetic is equally interesting in that case. &mdash;&nbsp;Carl <small>([[User:CBM|CBM]]&nbsp;·&nbsp;[[User talk:CBM|talk]])</small> 01:09, 17 March 2009 (UTC)
+== dynamic ordinal analysis ==
+[http://www-compsci.swan.ac.uk/~csarnold/publ/nfiles/doa.pdf] I don't understand this well enough to add anything to the article about it yet, but it looks relevant.  It is an approach to ordinal analysis on very weak (e.g. polynomially bounded) arithmetic systems for which the usual approach is too coarse. [[Special:Contributions/67.122.211.205|67.122.211.205]] ([[User talk:67.122.211.205|talk]]) 00:43, 7 September 2009 (UTC)