Timeline for Can one say that there are equal numbers of sets satisfying formulas in Second Order Arithmetic?
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9 events
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| Dec 13 at 19:18 | comment | added | Alexander Pruss | Yes, indeed, I forgot about that. But then it's natural to extend the question to asking about equicardinality for all sets, and not just definable ones. In other words, suppose SOA is enriched with two unary predicates $F$ and $G$ with second-order arguments, and then we ask whether we can find a $\phi$ such that for all models $M$ of SOA with unconstrained interpretations of $F$ and $G$ we have $M\vDash \phi$ iff $|F^M|=|G^M|$. | |
| Dec 13 at 18:10 | comment | added | Joel David Hamkins | Good question! However, if the perfect set property holds for every uncountable projective set (this is a consequence of PD), then every definable set you are talking about would be either countable or size continuum, in which case equicardinality would be expressible, but CH needn't hold. | |
| Dec 13 at 15:08 | comment | added | Alexander Pruss | Thanks again. I suppose there is still one question in the neighborhood: are there any models where CH fails and where SOA can nonetheless express equicardinality? It would be a neat thing if SOA could express equicardinality iff CH was true. | |
| Dec 12 at 23:08 | comment | added | Joel David Hamkins | Yes, that is right. You can express countability directly, in the style of my answer, and so if there is only one other cardinality, then you can also express that. If CH fails, however, then we cannot (provably) express the different cardinalities, since forcing can make them the same without adding reals and therefore without changing second-order arithmetic. | |
| Dec 12 at 19:50 | comment | added | Alexander Pruss | And I assume that on the other hand if the continuum hypothesis holds, we can express equicardinality, since we can encode a countable sequence of subsets of the naturals as a set of naturals? | |
| Dec 11 at 21:25 | comment | added | Alexander Pruss | Yup, that was easy! Thanks for your patience. | |
| Dec 11 at 19:29 | vote | accept | Alexander Pruss | ||
| Dec 11 at 18:04 | history | edited | LSpice | CC BY-SA 4.0 |
Missing word
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| Dec 11 at 17:59 | history | answered | Joel David Hamkins | CC BY-SA 4.0 |