Timeline for How strong is separation + reflection without transitivity?
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| Dec 10 at 1:10 | comment | added | user76284 | @JoelDavidHamkins Free variables are allowed, though I’d be happy to hear about the other case as well. There are no other axioms, including extensionality. The motivation is simply curiosity about the strength of “axiomatically minimal” fragments of ZF. Like reverse mathematics, for set theory. | |
| Dec 9 at 0:59 | comment | added | Joel David Hamkins | If you have extensionality, then with reflection you are going to get adjunction and therefore arithmetic, and then a form of second-order arithmetic by reflection. Without extensionality, you are going to have to go through the interpretation of set theory in set theory without extensionality, which is already complicated in ZF-ext, but in your weak theory, this will be even harder to verify. I expect it probably works. | |
| Dec 9 at 0:37 | comment | added | Joel David Hamkins | Could you clarify whether in axiom 2 you allow $\phi$ to have free variables, and then you take the universal closure, or are you only asserting instances of reflection for sentences? Also, could you explain why you do not include extensionality and other ordinary axioms of set theory? After all, reflection is equivalent to replacement over the other axioms of ZF, but your theory is weak in a way that is irritating and seems pointless without explanation. | |
| Dec 8 at 14:53 | history | asked | user76284 | CC BY-SA 4.0 |