Timeline for Which models of set theory are locally presentable?
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| Feb 16, 2016 at 13:45 | comment | added | Joel David Hamkins | I suppose it was clear what you had meant, although I find that usage to be sloppy. If $B$ is a Boolean algebra, then a $B$-valued model (in a language) is a structure with a domain of names, and an assignment of instances of those names under the fundamental relations to truth values in $B$. If $V$ is the set-theoretic universe, then $V^B$ is the class of $B$-names in the forcing sense, and this is what you meant. But there are many other $B$-valued models, for example, $W^B$ for various inner models $W$, or $V^C$ for any complete subalgebra $C$ of $B$, or inner models of $V^B$, etc. | |
| Feb 16, 2016 at 8:12 | comment | added | Simon Henry | that is indeed what I mean, (I also thought it was a standard terminology but I am not a model theorist at all).In fact I realized that there is several results in topos theory that says that any boolean grothendieck topos can be written as localic over an atomic topos, which maybe can be translated into a statement saying that any such model (even not satisfying choice) can be obtained by taking a permutation model inside a boolean valued model... but there is a few details that need to be clarified and I can't assert just now... | |
| Feb 16, 2016 at 5:03 | comment | added | Mike Shulman | @JoelDavidHamkins I think he means the Boolean-valued model constructed from a Boolean algebra as at for instance en.wikipedia.org/wiki/…. (I thought this was fairly standard terminology?) | |
| Feb 15, 2016 at 22:18 | comment | added | Joel David Hamkins | Perhaps you can clarify your remark, "the only possibility is a Boolean-valued model," since of course every model of set theory can be seen as a Boolean-valued model, for any desired Boolean algebra. You mean something more specific. | |
| Feb 15, 2016 at 18:53 | history | answered | Simon Henry | CC BY-SA 3.0 |