Timeline for Can this reflective class theory interpret ZFC?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Dec 21, 2018 at 18:08 | comment | added | Fedor Pakhomov | The theory $(ZF−PowerSet)+TMR$ is mutually interpretable with second-order arithmetic $Z_2$. Second order arithmetic could be interpreted in $(ZF−PowerSet)+TMR$ in a standard manner (by natural numbers and sets of naturals). $Z_2$ could formalize notion of constructive set (see Simpson's book) and then it is easy to show in $Z_2$ that $(ZF−PowerSet)+TMR$ holds in $L$. | |
| Dec 21, 2018 at 17:55 | comment | added | Zuhair Al-Johar | @FedorPakhomov but what's the exact the strength of $(ZF-PowerSet)+TMR$? | |
| Dec 21, 2018 at 17:52 | comment | added | Fedor Pakhomov | Note that nevertheless it still implies that your system couldn't interpret $ZF$. In $ZF$ it is easy to show that $(L\cap H\omega_1)\models (ZF−PowerSet)+TMR$. Thus consistencies of both $(ZF−PowerSet)+TMR$ and your system are provable in $ZF$. And by a standard corollary from Gödel's 2nd incompleteness theorem, no consistent theory $T$ extending Robinson's arithmetic $Q$ could interpret its own consistency. | |
| Dec 21, 2018 at 17:37 | comment | added | Fedor Pakhomov | I wasn't careful enough yesterday. My claim that $ZF-PowerSet$ axiomatizes pure set-theoretic consequences of your system is wrong, even if we consider the version of ZF with collection. My argument actually just shows that the pure-theoretic part of your system is $(ZF-PowerSet)+TMR$; here $TMR$ is transitive model reflection, e.g. the scheme $\exists M \;(\mathsf{Trans}(M)\land \forall \vec{x}\;(\varphi(\vec{x})\leftrightarrow \varphi^M(\vec{x})))$. But $(ZF-PowerSet)+Coll$ doesn't $TMR$, see Theorem 1.2 from Friedman, Gitman, Kanovei arxiv.org/pdf/1808.04732.pdf | |
| Dec 20, 2018 at 13:02 | comment | added | Fedor Pakhomov | The version with collection. I suspect that the version of $ZF-PowerSet$ with replacement couldn't prove transitive model reflection. | |
| Dec 20, 2018 at 12:54 | comment | added | Zuhair Al-Johar | @FedorPakhomov but which $"ZF" - power set$ you mean? is it the one with collection or Replacement, here both are theorems? | |
| Dec 20, 2018 at 12:50 | comment | added | Fedor Pakhomov | Yes, for any pure set-theoretic sentence $F$ (and $V=L$ in particular) if we add it to your theory, then the resulting set of set-theoretic sentences is precisely $(ZF-PowerSet)+F$. This holds by the same argument that I have mentioned above. | |
| Dec 20, 2018 at 12:05 | comment | added | Zuhair Al-Johar | @FedorPakhomov would that still be the same if we add $V=L$ to this theory? | |
| Dec 19, 2018 at 18:45 | comment | added | Fedor Pakhomov | The pure set-theoretic consequences of your theory are precisely the consequences of $ZF-PowerSet$. To convert a proof in your theory to a proof in $ZF-PowerSet$ the constant $V$ could be interlreted by a transitive set that reflects enough first-order formulas | |
| Dec 18, 2018 at 20:02 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
deleted 16 characters in body
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| Dec 17, 2018 at 14:34 | history | asked | Zuhair Al-Johar | CC BY-SA 4.0 |