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I know that forcing essentially does not change the ordinals, but by small I mean in comparison with other ordinals whose definition might not be stable under forcing, like the smallest uncountable ordinal … For example, given an ordinal in the ground model one can always make it countable in a forcing model, hence it can become smaller than $\omega_1$ after forcing (the $\omega_1$ of the forcing model of …

Are you familiar with this paper : Relating first order set theories and elementary toposes from Awodey,Butz,Simpson and Streicher. I haven't read in detail yet, but it really seems to provide a machi …

Here is a very brief sketches of the connection between this and forcing. … I'll describe you how I understand forcing, this is quite different from how it is generally described by logician, but this how peoples in topos theory/categorical logic understand it. …

If I work in a "ground" topos (whose object I call set), then a "forcing extention" would be just a Grothendieck topos, that is a topos of sheaves on a small site. … If you want to adopt an external point of view and start from an elementary topos $\mathcal{E}$ , then a forcing extension of $\mathcal{E}$ is a topos $\mathcal{F}$ that can be obtained as the category …

In short, this is just the result of me trying to make sense of this idea: Coming from topos theory, a "forcing extension" is for me a category of sheaves on a boolean locale, (that is a complete boolean … These are functorial on morphisms of boolean locales: That is given a morphisms of boolean locale $f:\mathcal{B'} \to \mathcal{B}$, I can think of $Sh(\mathcal{B'})$ as a further forcing extention of $ …