Timeline for Replacement in von Neumann hierarchy of sets
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
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| Sep 30, 2010 at 0:15 | comment | added | Andrés E. Caicedo | Martin, by "a model of ZF with standard $\omega$" I mean: In the model $(M,E)$ there are a set $A$ and a set $B$ such that $(M,E)\models "A=\omega,B=\in\upharpoonright A^2"$. In $V$, we identify $A$ with the set $\hat A$ of those $a\in M$ such that $(M,E)\models a\in A$, and similarly with $B$. Then $(\hat A,\hat B)$ is what $(M,E)$ thinks is $\omega$. An $\omega$-model or, equivalently, a model whose version of $\omega$ is standard, is one such that in $V$ we have: $(\hat A,\hat B)\cong(\omega,\in)$. One usually just codes this mouthful by saying that the model has no non-standard integers. | |
| Sep 27, 2010 at 15:42 | comment | added | Martin Brandenburg | What is meant by standard $\omega$ in ZF? | |
| Sep 27, 2010 at 15:05 | comment | added | Andrés E. Caicedo | Martin: Here is a good exercise for you to think about: 1. In ZF we cannot prove that there is a model of ZF. 2. In " ZF + there is a model of ZF " we cannot prove there is a model of ZF with standard $\omega$ (an $\omega$-model). 3. In " ZF + there is an $\omega$-model of ZF" we cannot prove there is a well-founded model. | |
| Sep 27, 2010 at 10:29 | history | edited | Joel David Hamkins | CC BY-SA 2.5 |
added 708 characters in body
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| Sep 27, 2010 at 10:26 | vote | accept | Martin Brandenburg | ||
| Sep 27, 2010 at 10:21 | history | answered | Joel David Hamkins | CC BY-SA 2.5 |