Timeline for Forcing the consistency of $ZF$ from a fragment of $ZF$
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 17, 2017 at 1:39 | vote | accept | Thomas Benjamin | ||
| Jan 10, 2017 at 11:41 | history | edited | Thomas Benjamin | CC BY-SA 3.0 |
corrected vagueness in question
|
| Jan 10, 2017 at 7:19 | vote | accept | Thomas Benjamin | ||
| Jan 10, 2017 at 7:23 | |||||
| Jan 10, 2017 at 7:17 | comment | added | Thomas Benjamin | @AndreasBlass: I mean, "if $\mathfrak M$$\vDash$$T$, and therefore $T$ is consistent, then $\mathfrak M$$[$ $G$ $]$$\vDash$$T^{'}$, and therefore $T^{'}$ is consistent" (apologies all around for the vagueness of my question--I definitely concede that, given the helpful comments, the question could have been better written). It was my hope that one could use class forcing to show that Infinity was independent of $ZF$ $-$ Infinity ($ZF$ $-$ Infinity should be able to be proven consistent in $PRA$ $+$ $TI({\epsilon_0})$), but Noah's answer suggests that such a project is doomed to fail. | |
| Jan 9, 2017 at 17:50 | answer | added | Noah Schweber | timeline score: 2 | |
| Jan 9, 2017 at 16:50 | review | Close votes | |||
| Jan 10, 2017 at 22:46 | |||||
| Jan 9, 2017 at 15:48 | comment | added | Andreas Blass | If you're looking for a proof of the consistency of ZF from the consistency of ZF$-$Infinity, then there is no such proof (at least none that can be formalized in ZFC), because ZF proves the consistency of ZF$-$Infinity. As for proving the consistency, relative to ZF$-$Infinity, of ZF$-$Infinity$+\neg$Infinity, that should be possible by interpreting the latter theory in the former, but I don't see how to bring forcing to bear on that matter. | |
| Jan 9, 2017 at 15:43 | comment | added | Andreas Blass | I didn't downvote,but the question seems unclear to me. My first question concerns "$\mathfrak M\models T$ is consistent." This could mean that it's consistent that $\mathfrak M\models T$, or it could men that $\mathfrak M$ satisfies the formula "$T$ is consistent." Which of those do you mean? Or do you perhaps mean that $\mathfrak M\models T$, and therefore $T$ is consistent? Or something else? | |
| Jan 9, 2017 at 9:28 | comment | added | Thomas Benjamin | Why the downvote? If whoever downvoted my question can give a good reason for the downvote, I will delete the question. | |
| Jan 9, 2017 at 8:50 | history | asked | Thomas Benjamin | CC BY-SA 3.0 |