Timeline for Using consistency to create new axioms in set theory
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
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| Jan 25, 2010 at 6:20 | vote | accept | Ewan Delanoy | ||
| Jan 25, 2010 at 2:05 | comment | added | François G. Dorais | Well, proof theoretic ordinals and ordinal analysis only makes sense for recursively enumerable theories. I suppose you could go higher by relativizing, but proof theorists don't seem to like that sort of thing. | |
| Jan 25, 2010 at 1:44 | comment | added | Joel David Hamkins | Are you saying that in PA or ZFC both we use the notations up to omega_1^CK using Kleene's O (or whatever), but that the weaker theories don't always prove the notations are well-ordered? Do any proof-theoretic ordinals go beyond omega_1^CK? | |
| Jan 25, 2010 at 1:07 | comment | added | François G. Dorais | The notations are always the same. (In principle Kleene's O, but its often easier to work with different ones depending on context.) What differs is how well theories understand notations. Each notation t corresponds to an induction statement on notations: "induction on notations holds up to t" or simply "t is wellfounded". The proof theoretic ordinal of T is the "first" ordinal notation t such that T does not prove that t is wellfounded. | |
| Jan 25, 2010 at 0:39 | comment | added | Joel David Hamkins | Are the "representable ordinals of a theory the same thing as the proof-theoretic ordinal? Is this what the proof-theorists mean? | |
| Jan 25, 2010 at 0:25 | comment | added | François G. Dorais | PA is definitely easier to work with, but which ordinals are representable doesn't really change. (Albeit, some theories do much better than others at understanding certain types of notations, which does have consequences.) The point I was trying to make is to study notions in their appropriate settings. Con(ZFC) has no good semantics, so there is little point in studying it in the context of set theory. On the other hand, ZFC + "there is a wellfounded model of ZFC" is interesting to look at in set theory, but difficult to understand arithmetically. | |
| Jan 24, 2010 at 23:32 | comment | added | Joel David Hamkins | But so what? Any large cardianl consistency strength statement is also arithmetic (since it is a consistency statement), but we still analyze them in ZFC rather than PA. My sense of the question is that since we believe ZFC, we naturally also want to include Con(ZFC) etc., as among the strongest theories that we really believe in. Perhaps the situations is that it is the negative results (e.g. on which ordinals are not representable) are stronger/better for PA. Does that sound right? | |
| Jan 24, 2010 at 23:28 | history | edited | François G. Dorais | CC BY-SA 2.5 |
grammar
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| Jan 24, 2010 at 23:08 | history | edited | François G. Dorais | CC BY-SA 2.5 |
clarification
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| Jan 24, 2010 at 22:58 | comment | added | François G. Dorais | Yes, but you're still analyzing arithmetical statements. A more appropriate choice for T is PA + (all arithmetical consequences of ZFC), for example. | |
| Jan 24, 2010 at 22:52 | comment | added | Joel David Hamkins | But isn't it true that if the process makes sense with alpha using PA or even second-order arithmetic, then it will also make sense with ZFC for this same ordinal? | |
| Jan 24, 2010 at 22:29 | history | edited | François G. Dorais | CC BY-SA 2.5 |
correction
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| Jan 24, 2010 at 21:16 | history | answered | François G. Dorais | CC BY-SA 2.5 |