Timeline for A question regarding Koepke' s Ordinal Computability in HOD
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Jan 22, 2013 at 11:31 | comment | added | Thomas Benjamin | @Francis: I definitely will. Thanks. | |
| Jan 21, 2013 at 18:41 | comment | added | François G. Dorais | Thomas, all of these questions have well-known answers. I think you just need to do a little more careful research. In addition to Kunen, check out what Jech (Set Theory) and Kanamori (Higher Infinite) say on L, HOD, and other inner models. | |
| Jan 21, 2013 at 14:04 | comment | added | Thomas Benjamin | to exist is to be able to be defined (eg. the productive sets)). Of course those who believe in a single background universe of sets will say that the existence of larger and larger cardinals, existence to just short of inconsistency, will allow for noncomputable sets for ordinal machines. So why is there a question regarding the existence of 0-sharp, when Solovay proved its existence assuming the existence of "at least one Ramsey cardinal"? | |
| Jan 21, 2013 at 13:30 | comment | added | Thomas Benjamin | the next question that presents itself is whether there are models of n'th-order ZFC, n>=2, that satisfy L 'is a proper subset of' HOD--if not then this is surely evidence that L=HOD). This is why I said that problem 22 of Kunen, chapter 6 is possibly a mitigating factor. On the other hand, if L 'is a proper subset of' HOD then one seems to be in the situation where if V=HOD (as I assumed in my question) then there are non-computable sets for ordinal machines which can be defined from a finite set of ordinal parameters (more on a par with ordinary recursion theory if one assumes that | |
| Jan 21, 2013 at 12:59 | comment | added | Thomas Benjamin | @To all who commented: Thanks. I will have to recheck my copy of Kunen's book. Nevertheless, one can easily see that from the comments there are models of ZFC where L=HOD and L 'is a proper subset of' HOD (eg. L[0-sharp]). Unless one is a believer in the set theoretic multiverse, either L=HOD or L 'is a proper subset of' HOD. If L=HOD for ZFC, this puts HOD for first-order set theory (and for first-order logic) on a par with its higher-order counterparts (consider Kunen's problem 22 for chapter 6--although it speaks to L defined for n'th-order logic n>=2 rather for n'th-order ZFC | |
| Jan 20, 2013 at 20:20 | comment | added | François G. Dorais | Comparing with the traditional notion of computable, it's hard to see this as a mitigating factor: there are plenty of easily definable sets that are not computable. In fact, it's interesting that there need not be any non-computable sets for ordinal machines. | |
| Jan 20, 2013 at 18:49 | comment | added | Andreas Blass | To amplify Gro-Tsen's comment: If $0^{\#}$ exists then $L[0^{\#}]$ satisfies V=HOD and $0^{\#}$ is definable in it but not constructible. Finally, concerning the last sentence in the question: What is being mitigated? I see nothing in need of mitigation except the mistranslation of Kunen's $\subset$ as "proper subset". | |
| Jan 20, 2013 at 18:46 | comment | added | Andreas Blass | The next sentence after Theorem 3.5 in the question is also wrong for the same reason. L is a subclass of HOD, but not necessarily a proper subclass. | |
| Jan 20, 2013 at 18:44 | comment | added | Andreas Blass | Theorem 3.5 as quoted is wrong, but Francois's comment explains at least part of what's wrong with it. L is a subclass of any such M, but not necessarily a proper subclass. In particular, M could be L. | |
| Jan 20, 2013 at 15:05 | comment | added | Gro-Tsen | I'm not sure I understand what the question is, but $0^\#$, if it exists, is a hereditarily ordinal-definable set of ordinals (indeed, a $\Pi_1$-definable set of integers) that isn't constructible (hence not ordinal machine computable). | |
| Jan 20, 2013 at 14:21 | history | edited | François G. Dorais |
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| Jan 20, 2013 at 14:18 | comment | added | François G. Dorais | Kunen does not use $\subset$ to mean "proper subset". | |
| Jan 20, 2013 at 13:45 | history | asked | Thomas Benjamin | CC BY-SA 3.0 |