Timeline for The Kunen inconsistency and definable classes
Current License: CC BY-SA 3.0
7 events
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| May 10, 2013 at 19:36 | comment | added | Sam Roberts | Thanks, Andreas! I suppose it's a matter of familiarity - although one could simply work in MK but rely ultimately on a plural interpretation, it's also pretty straightforward to work directly in plural logic. There a (proper class sized) function will be some ordered pairs. In any case, the reason I brought up plural quantification was because I was wondering if the reluctance to employ MK class quantification was to do with ontological commitment to set-like entities which aren't actually sets. Whether or not plural quantification is natural, it at least doesn't suffer from that problem. | |
| May 10, 2013 at 14:41 | comment | added | Andreas Blass | To assert the existence of an elementary embedding we'd say "There are some ordered pairs such that ..." followed by a description of what it means for those pairs to constitute an elementary embedding. Part of my difficulty here may be simply that this stuff is familiar in the set-based context but not in the plural quantification context. But in addition to that, it seems to me that such use of plural quantification loses a lot of the naturality that makes plural quantification attractive in the first place. | |
| May 10, 2013 at 14:37 | comment | added | Andreas Blass | @Sam: I know very little about plural quantification, but my immediate reaction is that it's likely to introduce some awkwardness into these issues. Like MK, its formalization would presumably rely on the pairing function in $V$ in order to code elementary embeddings (class-sized functions) as sets. In the MK context, this is quite familiar --- some people will even tell you (whether in MK or ZF) that functions are sets of ordered pairs (as opposed to being coded as sets of ordered pairs), but in plural quantification, the same idea sounds a bit strange. (continued in next comment) | |
| May 10, 2013 at 10:01 | comment | added | Sam Roberts | Thanks, Andreas! If it's not too much trouble, it would help me to get clearer on what considerations are at play here to know what you think about plural quantification plato.stanford.edu/entries/plural-quant in this context. For instance, suppose that plural quantification allowed one to unproblematically simulate MK over V (not merely over some $V_\alpha$). Would you then rather work in that setting when considering Reinhardt, supercompact etc cardinals or the question how Kunen's result goes beyond Suzuki's? | |
| May 8, 2013 at 10:00 | vote | accept | Sam Roberts | ||
| May 8, 2013 at 10:00 | |||||
| May 3, 2013 at 13:29 | vote | accept | Sam Roberts | ||
| May 3, 2013 at 16:56 | |||||
| May 3, 2013 at 13:16 | history | answered | Andreas Blass | CC BY-SA 3.0 |