Timeline for Large cardinals arising from alternate set theories
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| S Oct 22, 2015 at 22:16 | history | bounty ended | CommunityBot | ||
| S Oct 22, 2015 at 22:16 | history | notice removed | CommunityBot | ||
| Oct 22, 2015 at 7:35 | comment | added | Thomas Benjamin | (cont.) get $NF$ to satisfy the large cardinal axioms you seek (if, in fact, the the results of that paper need the Ultrafilter Lemma to obtain its results)? The cite for that paper is as follows: Annals of Pure and Applied Logic 41 (1989) 93-106. | |
| Oct 22, 2015 at 7:31 | comment | added | Thomas Benjamin | Reading through my copy of Quine's book "Set Theory and its Logic", I am struck by the following comment found on page 289: "The universe of $NF$ indeed affords the Boolean algebra of classes." If so, then though $AC$ is false in $NF$, could one prove the relative consistency of, say, $NF$+ '$U$ satisfies the Ultrafilter Lemma" ($U$={x| x=x}) assuming that $NF$ is consistent? I ask because if so, then could one possibly apply the results of the Apter, DiPrisco, Henle, and Zwicker paper "Filter Spaces: Towards a Unified Theory of Large Cardinal and Embedding Axioms " to | |
| Oct 18, 2015 at 23:33 | comment | added | Noah Schweber | @ThomasBenjamin That's a really interesting paper - thanks for the reference! It's still pretty close to ZF, though. I'm ultimately looking for something further from ZF. | |
| Oct 15, 2015 at 10:43 | comment | added | Thomas Benjamin | (cont.) "Axiomatization of Set Theory by Extensionality, Separation, and Reducibility" (found in the "downloadable" section of his Homepage). Would these set theories found in this preprint be sufficiently 'alternate' to constitute an answer to your question? | |
| Oct 15, 2015 at 10:37 | comment | added | Thomas Benjamin | Though you mention you are especially interested in model-theoretic large cardinal axioms over $NFU$ ($NFU$ being, according to your question ,"the most likely example" of an alternative set theory where such model-theoretic large cardinal axioms 'naturally hold' in the sense of $NFU$+ "Infinity"+"Large Ordinals"+Small Ordinals" being equivalent $MK$+"the proper class ordinal is a measurable cardinal"), you mention you would consider "set theories other than $ZFC$". Consider the set theories found in Harvey Friedman's preprint | |
| S Oct 14, 2015 at 20:30 | history | bounty started | Noah Schweber | ||
| S Oct 14, 2015 at 20:30 | history | notice added | Noah Schweber | Draw attention | |
| Oct 11, 2015 at 23:26 | comment | added | Noah Schweber | @ThomasBenjamin I read that section; however, if I am reading it correctly, those axioms don't climb past a measurable, which is what I ask for. (I guess this relies on my vague recollection that "'MK + 'Ord is measurable'' is consistent relative to 'ZFC + a measurable'" is correct, though.) | |
| Oct 11, 2015 at 23:21 | comment | added | Thomas Benjamin | A pathway to the answers you seek are in the "Strong axioms of infinity" section of (of all places) the Wikipedia entry "New Foundations". The author of this entry claims that these results are found in some unpublished work of Solovay. It would be nice if Prof. Solovay would provide us access to this unpublished work so we could check this out for ourselves.... | |
| Oct 10, 2015 at 23:37 | history | asked | Noah Schweber | CC BY-SA 3.0 |