Timeline for Is this theory of well founded countable sets formalized in infinitary logic, complete and categorical?
Current License: CC BY-SA 4.0
6 events
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| Jun 12, 2023 at 15:19 | comment | added | Noah Schweber | @ZuhairAl-Johar That doesn't affect this answer. Different models of $\mathsf{ZF(C)}$ disagree about the properties of $\mathsf{HC}$. However, this doesn't affect the fact that $\mathsf{ZF}$ proves "Every model of your theory is isomorphic to $\mathsf{HC}$." Think by analogy about how different models of $\mathsf{ZF}$ disagree over the properties of the real numbers, but the second-order theory of the reals is still $\mathsf{ZF}$-provably categorical. | |
| Jun 12, 2023 at 15:10 | comment | added | Zuhair Al-Johar | there is a problem about the $\sf HC$ since its cardinality is not determined in $\sf ZF$? So if all models are isomorphic then what is their cardinality? Is it $\omega_1$? | |
| Jun 9, 2023 at 11:29 | comment | added | Zuhair Al-Johar | @PeterLeFanuLumsdaine, yes the underlying logic does presume the existence of at least one object. | |
| Jun 9, 2023 at 8:05 | comment | added | Peter LeFanu Lumsdaine | One very minor caveat: This theory also admits the empty model (since all its axioms are universals), unless you set up your logic to preclude the empty model entirely. | |
| Jun 9, 2023 at 5:39 | vote | accept | Zuhair Al-Johar | ||
| Jun 9, 2023 at 0:45 | history | answered | Noah Schweber | CC BY-SA 4.0 |