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Re: ordinal strength of a theory - and large cardinals
Posted:
Feb 7, 2006 8:00 PM
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tchow@lsa.umich.edu in litteris
<43e7ed8d$0$570$b45e6eb0@senator-bedfellow.mit.edu> scripsit:
> For systems of axioms for set theory, rather than for arithmetic, I'd
> propose that "what arithmetical statements the system proves" may be a
> better concept for capturing what you're interested in. Empirically,
> this appears to create a linear order on all "natural" axioms for set
> theory, though this fact is somewhat mysterious. See for example
>
> http://www.cs.nyu.edu/pipermail/fom/2006-February/009669.html
Right: it is with these sort of ideas in mind that I was asking the
question: basically what I'd like to know is to what extent the
unexplainably-total (pre)order on mathematical theories by the
amount of arithmetical statements that they prove
(which you mention)
is reflected on the (*a priori* coarser)
(necessarily) total (pre)order of recursive ordinals up to which
they prove well-foundedness
(hence the proof-theoretic ordinal strength).
E.g., ZFC+IC (where IC = "there is an inaccessible cardinal")
certainly proves Consis(ZFC), so it is strictly stronger than ZFC on
the first scale, and I wonder whether it can be shown to be strictly
stronger on the second, ideally by explicitly giving some ordinal
notation which ZFC+IC proves is well-founded (is really an ordinal
notation, that is) but ZFC alone does not. Because I have trouble
seeing how a large cardinal can help proving well-founded of large
ordinal notations; and it seems even stranger to think that ZFC+IC
should prove exactly the same arithmetical sentences as ZFC + "o is
well-founded" (for some ordinal notation o). But, on the contrary, if
ZFC, ZFC+IC, ZFC+MahloC and so on all have the same ordinal, then this
also seems to be a rather remarkable fact.
Actually, the slides <URL:
http://www.mathematik.uni-muenchen.de/~aehlig/EST/rathjen4.pdf >,
which explain what the ordinal strength of Kripke-Platek set theory is
(the Bachmann-Howard ordinal), answer some of my questions - or at
least they make it seem reasonable that some large cardinal axiom
should prove the well-foundedness of huge ordinal notations. There
are also some apparently interesting remarks on <URL:
http://web.mit.edu/dmytro/www/other/OrdinalNotation.htm >, but I'm not
sure I can quite make sense of them...
From what I read here and there, I vaguely seem to understand that
there's a theory of "large computable ordinals" which precisely
parallels that of "large cardinals" (at least for small large
cardinals, like the inaccessibles and the Mahlo): however, I am
completely unclear on the details or on the relation between these
large computable ordinals and the proof-theoretic strength of various
systems.
--
David A. Madore
(david.madore@ens.fr,
http://www.dma.ens.fr/~madore/ )
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