Timeline for Does ZF have an initial model?
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Nov 29, 2013 at 15:20 | history | edited | Andrés E. Caicedo | CC BY-SA 3.0 |
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Nov 29, 2013 at 15:18 | comment | added | Andrés E. Caicedo | @JoelDavidHamkins Hi Joel. Yes, $T$ can be modified as you say in your first comment, at the cost of uniqueness. And yes, as you say in the second comment, I want a recursively enumerable $T$. | |
Nov 29, 2013 at 13:49 | comment | added | Joel David Hamkins | For your question at the end, I suppose that you want $Ta$ to be computably axiomatizable, since otherwise one can make examples by asserting that $V$ is a forcing extension of $L$ by a Cohen real whose first digit is $0$, second digit $1$, etc. (listing the digits of a particular generic real, plus no $L_\alpha$ is a model of ZFC. This has a unique model, but we've made the theory complicated. | |
Nov 29, 2013 at 13:30 | comment | added | Joel David Hamkins | Andres, one can modify your theory $T$ to say merely that $V$ is a forcing extension of $L$, and there are no $L_\alpha$ satisfying ZFC. This will still have your initial model, since all transitive models of $T$ will be forcing extensions $L_\alpha[G]$ of the minimal model; but meanwhile, $T$ now has many models. | |
Nov 29, 2013 at 7:51 | history | edited | Andrés E. Caicedo | CC BY-SA 3.0 |
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Nov 29, 2013 at 6:08 | history | answered | Andrés E. Caicedo | CC BY-SA 3.0 |