Timeline for Is restricting Replacement and Separation enough to make $Q+I\Sigma_n$ bi-interpretable with Set Theory?
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 28, 2023 at 16:12 | comment | added | Fedor Pakhomov | @Chill2Macht It entirely depends on the version of first-order logic one uses. Usually, as I did in my answer, people use first-order logic that doesn't allow empty models and hence empty set axiom is not needed, since existence of some set is guaranteed by the underlying logic and next the empty set could be constructed by comprehension. However, in principle, one could have a version of first-order logic without this restriction en.wikipedia.org/wiki/Free_logic and then of course empty set axiom had to be added (or some other axiom implying the existence of at least one set). | |
| Jan 28, 2023 at 4:33 | comment | added | Chill2Macht | This is a very dumb question which is somewhat besides the point of this wonderful and thoughtful answer, but does the definition of $\mathsf{ZFfin}$ = $\mathsf{ZFCfin}$ also require the empty set axiom in order to exclude the empty model? (Because normally existence of at least one set can be seen to follow from the axiom of infinity, which is negated here, and then empty set follows by applying separation.) Or does our semantics / metatheory preclude ever using empty models, so we're fine? | |
| Feb 12, 2019 at 13:46 | vote | accept | Not_Here | ||
| Feb 12, 2019 at 7:26 | history | answered | Fedor Pakhomov | CC BY-SA 4.0 |