Timeline for When does Vopěnka's principle hold?
Current License: CC BY-SA 3.0
5 events
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| Dec 8, 2019 at 7:25 | comment | added | Keith Millar | I'd just like to note that the kind of coding that is being done here is, I think, full embeddings from one category into another, since "VP for category C" is equivalent to "Ord cannot be fully embedded into C" which of course means that if A and B can be fully embedded in each other (like the groups can be with the abelian groups) then VP for A is the same as VP for B. | |
| Apr 23, 2015 at 5:04 | comment | added | Noah Schweber | @AdamPrzeździecki that's a fascinating result about abelian groups - it doesn't answer my question fully, but if you post it as an answer I'll vote it up. | |
| Apr 22, 2015 at 14:21 | comment | added | Adam Przeździecki | @Joel David Hamkins "[...] and I believe that one can also code structure into groups" -- An equivalent statement of Vopenka's Principle in the category of GROUPS is described in arxiv.org/abs/0912.0510 (Adv. Math. 225 (2010) 1893−1913) and in the category of ABELIAN GROUPS in arxiv.org/abs/1104.5689 (Adv. Math. 257 (2014) 527−545). The latter solved a problem of Isbell - it was unexpected that one could encode Vopěnka by ABELIAN GROUPS. Another comment - the very first formulation of Vopěnka was for graphs. | |
| Apr 22, 2015 at 13:38 | comment | added | Noah Schweber | Yes, the reason I picked linear orders is that it is unclear what kind of coding power they have - for example, if $X\subseteq\omega$ is computable in every copy of some countable linear order $L$, then $X$ is computable (Richter); on the other hand, isomorphism for linear orders is Borel complete (folklore?). | |
| Apr 22, 2015 at 13:15 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |