Timeline for Formulating Kunen's inconsistency and Reinhardt cardinals in term of category theory
Current License: CC BY-SA 3.0
17 events
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| Jun 13, 2015 at 15:42 | comment | added | Tim Campion | I think that a $\Delta_0$-elementary embedding of the universe is implied by a measurable cardinal (according to Cantor's Attic, a measurable cardinal is equivalent to an elementary embedding from $V$ to some transitive class $M$, and the inclusion $M \to V$ is always a $\Delta_1$-elementary embedding). I would guess that a $\Sigma_n$-elementary embedding would be given by a logical functor that preserves classifying objects for $\Sigma_n$ formulas in the stack semantics. | |
| Jun 13, 2015 at 5:48 | comment | added | Thomas Benjamin | @TimCampion: The slide presentation of Shulman's you provided the link for clarifies matters nicely! Thanks. If a logical endofunctor of $Set$ is the same thing as a powerset-preserving $\Delta_0$-elementary embedding of the universe (Joel David Hamkins and Dustin Mulcahey--ask Prof. Hamkins for a copy of the draft of their paper), then how would one characterize $\Sigma_n$-elementary embeddings? Also, is the defintion of elementary embedding Kunen gives in his paper proving his inconsistency result the definition of $\Delta_0$-elementary embedding? | |
| Jun 12, 2015 at 15:37 | comment | added | Tim Campion | Lemma 3.3 basically says that equivalences of categories preserve and reflect the satisfaction of formulas in Shulman's language of categories. I would expect some sort of Lowenheim-Skolem theorem to show that the converse only holds when the category is finite. I haven't really studied the paper in detail, but judging from the result on this slide, I might guess that you want a logical functor which preserves classifying objects for formulas in the stack semantics. | |
| Jun 12, 2015 at 6:04 | comment | added | Thomas Benjamin | (cont.) def 3.1), then it seems that lemma 3.3 could give (possibly) a partial characterization of elementary embeddings of $Set$ into $Set$. Does this make sense? If so, then under what conditions would the converse of lemma 3.3 hold? If not, what sort of restrictions would be necessary to make 3.3 refer to elementary embeddings? | |
| Jun 12, 2015 at 5:57 | comment | added | Thomas Benjamin | @TimCampion: I am beginning to read through Mike Shulman's paper "Stack Semantics And The Comparison Of Material And Structural Set Theories" you provided the link to, and have a question regarding his lemma 3.3: "[Let F be a functor--my comment] If F is fully faithful and essentially surjective then A$\vDash$$\phi$ [where A, B are categories and $\phi$ is a sentence in the language of categories (def. 3.1)] iff B$\vDash$F($\phi$)". Now if one lets A=B=$Set$, then by Shulman's definitions (and if one can formulate the language of set theory in terms of the language of categories--i.e. | |
| May 26, 2015 at 13:45 | comment | added | Tim Campion | @ThomasBenjamin Anafunctors are a way to deal with a lack of choice in your meta theory. But in the Kunen inconsistency, it seems to be choice in the object theory which is used in an essential way. Only if you want to formulate a theory about an elementary embedding internally to your set theory will anafunctors become plausibly necessary. | |
| May 26, 2015 at 2:26 | comment | added | Thomas Benjamin | @TimCampion: What about the use of anafunctors as a means of eliminating the Axiom of Choice. Might these be of some use in 'breaking the [Axiom of] Choice barrier' (if in fact such exists)? | |
| May 17, 2015 at 14:03 | comment | added | Tim Campion | I have seen non-$\Delta_0$ formulas considered in a categorical setting in a few places, including here (and possibly in some of the theories that include proper classes in algebraic set theory). But I'm not sure I've seen the notion of functor preserving the meaning of these sentences considered, though it would probably be a pretty natural extension. | |
| May 17, 2015 at 13:58 | comment | added | Tim Campion | I think you might need to consider functors which are faithful but not full to get the full strength of these statements. Categorical logic includes many notions of functors between set-like categories preserving varying amounts of structure. The closest to an elementary embedding that I know of is a logical functor, but it only preserves the meanings of statements that fit into the usual internal logic, i.e. $\Delta_0$ statements. There's a note to this effect here. | |
| May 17, 2015 at 6:56 | comment | added | Thomas Benjamin | @TimCampion: Is it possible to construe an elementrary embedding from $Set$ into $Set$ as a particular type of endofunctor? I was looking at equivalances of the axiom of choice in ncatlab that would be of interest to category-theorists and found one in terms of full and faithful functors that might possibly imply Kunen's result--that is, the only elementary embedding from $Set$ into $Set$ (under $AC$, of course, so this would actually be a subcategory of $Set$) is the identity. If I am correct, this might be a starting point to obtain the answer I seek. | |
| May 11, 2015 at 14:42 | comment | added | Tim Campion | @AndreasBlass: Thanks! I've updated my answer to reflect your comments. | |
| May 11, 2015 at 14:41 | history | edited | Tim Campion | CC BY-SA 3.0 |
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| May 11, 2015 at 14:15 | comment | added | Andreas Blass | A special case of your item 4 is the topic of: Adelman, Murray(5-MCQR); Blass, Andreas(1-MI) Exact functors, local connectedness and measurable cardinals. (Italian summary) Rend. Sem. Mat. Fis. Milano 54 (1984), 9–28 (1987). | |
| May 11, 2015 at 14:13 | comment | added | Andreas Blass | Thanks for citing my paper, but, to give proper credit, I must point out that the main results were discovered earlier by Trnkova and Reiterman. See: Corrections to: “Exact functors and measurable cardinals” Pacific J. Math. 73 (1977), no. 2, 540. | |
| May 11, 2015 at 13:44 | history | edited | Tim Campion | CC BY-SA 3.0 |
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| May 11, 2015 at 12:27 | comment | added | Joel David Hamkins | Regarding your first point, every elementary embedding $j:V\to M$ into a transitive class $M$ (and this includes basically all the usual large cardinal embeddings) is also a $\Delta_0$-elementary embedding $j:V\to V$, simply because every transitive class $M$ is $\Delta_0$-elementary in $V$. The larger large cardinals then insist on greater absoluteness between $M$ and $V$, such as with strongness or supercompactness, and so the usual large cardinal development fits into the paradigm you describe. | |
| May 11, 2015 at 5:19 | history | answered | Tim Campion | CC BY-SA 3.0 |