Timeline for Downgrading from ZFC with universes to ZFC
Current License: CC BY-SA 3.0
18 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jun 19, 2011 at 0:25 | comment | added | Ali Enayat | @Joel: Thanks for the further clarification; Feferman's system uses a class of reflecting cardinals, by the way. Now that you mention it, I also remember seeing them in Leibman's paper. | |
| Jun 18, 2011 at 22:37 | comment | added | Joel David Hamkins | Yes, I mentioned that in my comment above (to the question), where David also links to Mike's paper, and Andreas says this also in his answer on the question to which I linked. I'm not quite clear on whether Feferman has just one $V_\theta\prec V$, or whether he has a proper class of them, but the idea is essentially the same. My student George Leibman made essential use of this theory in his work on versions of the Maximality Principle. | |
| Jun 18, 2011 at 18:26 | comment | added | Ali Enayat | Joel: That' right. One last point: Sol Feferman used a similar theory to yours to implement parts of category theory, see, e.g., Mike Shulman's 2008 paper Set Theory for Category Theory [arXiv: 0810.1279]. | |
| Jun 18, 2011 at 18:12 | comment | added | Joel David Hamkins | Yes, the same compactness proof gives this, and I suppose that one could even have GBC, plus $C$ is a GBC class. | |
| Jun 18, 2011 at 12:00 | comment | added | Ali Enayat | Joel: one more aspect of the theory you described is that it is conservative over $ZFC$ if you further add instances of replacement using formulas that mention $C$ to the theory. | |
| Jun 18, 2011 at 11:05 | vote | accept | porton | ||
| Jun 18, 2011 at 11:05 | |||||
| Jun 17, 2011 at 21:00 | comment | added | Joel David Hamkins | Yes, that's right. I suppose that if one wanted to handle $\varphi$ in the language with $C$, then you could do the same trick again, having a club $D$ of $\theta$ for which $V_\theta$ reflect truth of $\langle V,\in,C\rangle$, and so on, iterating transfinitely! | |
| Jun 17, 2011 at 20:50 | comment | added | Ali Enayat | The theory described by Joel has the curious feature that it provides a definition of " $\phi$ is true in $(V, \in)$ for standard sentences of set theory [as opposed to those with nonstandard length] with via "$\phi$ holds on a TAIL of the structures of the form $V_{\alpha}$', where $\alpha\in C$". This does not contradict Tarski's theorem on undefinability of truth since $\phi$ is in the language using only $\in$, and is not allowed to mention $C$. | |
| Jun 17, 2011 at 18:02 | comment | added | Joel David Hamkins | Yes, I have edited. | |
| Jun 17, 2011 at 18:01 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
Correction about rank of x being below theta
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| Jun 17, 2011 at 13:23 | comment | added | Andreas Blass |
In the displayed formulation of the reflection schema, $\theta$ should be big enough so that $x\in V_\theta$.
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| Jun 17, 2011 at 10:25 | comment | added | Joel David Hamkins | If $\delta$ is an inaccessible cardinal, then by a Lowenheim-Skolem argument you can find a club $C\subset\delta$ with $V_\theta$ elementary in $C_\delta$ for $\theta\in C$, and so $\langle V_\delta,{\in},C\rangle$ is a model of the theory. | |
| Jun 17, 2011 at 5:54 | comment | added | François G. Dorais | Does this theory have any wellfounded models? | |
| Jun 17, 2011 at 0:17 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| Jun 17, 2011 at 0:02 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| Jun 16, 2011 at 23:53 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| Jun 16, 2011 at 23:47 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |