Skip to main content

Timeline for Does ZF have an initial model?

Current License: CC BY-SA 3.0

6 events
when toggle format what by license comment
Nov 29, 2013 at 15:20 history edited Andrés E. Caicedo CC BY-SA 3.0
added 25 characters in body
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
added 280 characters in body
Nov 29, 2013 at 6:08 history answered Andrés E. Caicedo CC BY-SA 3.0