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July 17

Large Recursive Ordinals and Large Cardinals

Let me preface this by saying: I'm just beginning, as in yesterday, to introduce myself to Ordinal analysis, and that this question comes more from a musing/stray thought than it does anything else. That said is there any sort of expected symmetry between large cardinal properties and large recursive ordinal properties - in other words, LCA some large cardinal axiom, does the structure of LCA tell us something about the nature of the proof theoretic ordinal of ZFC + LCA? Moreover, can we, from the relations of the various types of large cardinals, infer anything about the relations between their proof theory ordinals when added to ZFC? I apologize, in advance, if this question is not the most clear, or if it is stupidly glossing over something obvious. Thanks for any help/insight:-)Phoenixia1177 (talk) 06:58, 17 July 2014 (UTC)[reply]

It's not my field of mathematical logic, (except, or course, I knew ε₀ for PA), but the article ordinal analysis states: "Most theories capable of describing the power set of the natural numbers have proof theoretic ordinals that are so large that no explicit combinatorial description has yet been given." Most large cardinals require power set in their definition. — Arthur Rubin (talk) 07:23, 17 July 2014 (UTC)[reply]

Thank you for the reply - I realize that we have no description of these ordinals, but are we able to say anything about them otherwise? For partial analogy, as far as I know, we have no way of describing large cardinals directly, but we can say that if a Huge cardinal and a Supercompact cardinal exist, then the first huge is smaller than the first supercompact. In this case it need not be size comparisons, but is there anything that could potentially be said? --addendum: another analogy: we can't directly describe the Absolute Galois Group of the Rationals, but we can say things about the group itself; is there something that might be said/expected to be said about the overall collection of ordinals that would result? I realize that this is vague, my brain just keeps nagging me that there is something interesting here.Phoenixia1177 (talk) 07:38, 17 July 2014 (UTC)[reply]

This is a little off the track from your question, but the example that you mention is kind of an interesting one, because although the first huge cardinal comes before the first supercompact, the existence of a huge cardinal is a much stronger axiom, in terms of consistency strength, than the existence of a supercompact.

There are other anomalies like this, but as far as I'm aware, they're all based on LCAs that are not "local" — that is, you can't say that a cardinal is supercompact just because there's some sufficiently large α such that V_α thinks the cardinal is supercompact.

Non-local LCAs are a bit problematic because they rely on knowing stuff about the whole universe. In Woodin's work around 2000, he gave an abstract definition of large cardinal (not widely adopted, but interesting) that simply made locality part of the definition. I don't know of any counterexample to the claim that, when you restrict attention to local LCAs, the LCA with the larger first witness always has larger consistency strength. (However, a properly stronger LCA might have the same first witness as the weaker one.)

Also, the order of consistency strength on LCAs seems to be the same as how far up the Wadge hierarchy they prove regularity properties for sets of reals (again with possible collisions — for example, weak large cardinal axioms, those consistent with V=L, don't prove any regularity properties beyond ZFC itself). --Trovatore (talk) 11:32, 17 July 2014 (UTC)[reply]

Your "slight tangent" answers a question at least as interesting to me as my original, doubly so since I had no idea about the Wadge Hierarchy - thank you:-) Do you know of any good introductions to Woodin's work, or where to start with it? For some reason, I feel like there is some sort of connection between all the various hierarchies in computation, logic, and the transfinite, like they're all just instances of some underlying thing - like the difference between classical and modern algebraic geometry. Anyways, I digress. Thank you again for the interesting ideas:-)Phoenixia1177 (talk) 16:55, 17 July 2014 (UTC)[reply]

Don't know if this is exactly what you are looking for, but it is Woodin and it is around 2000:

Woodin, W. Hugh (2001a). "The Continuum Hypothesis, Part I" (PDF). Notices of the AMS. 48 (6): 567–576.
Woodin, W. Hugh (2001b). "The Continuum Hypothesis, Part II" (PDF). Notices of the AMS. 48 (7): 681–690.

YohanN7 (talk) 20:37, 17 July 2014 (UTC)[reply]

Yes, that's it exactly. Anyone reading it should be aware that not all of it has held up perfectly in the intervening years. A result that Woodin thought he knew (that there's a limit to the large cardinals you can have in HOD) fell through. I don't know how much of that paper depends on that. --Trovatore (talk) 21:29, 17 July 2014 (UTC)[reply]

Thank you:-)Phoenixia1177 (talk) 03:33, 21 July 2014 (UTC)[reply]