We are in a first order logical world.

Let $ sigma $ be a finished signature and $ T $ a coherent theory of $ sigma $.

Due to the LΓΆwenheim – Skolem theorem, we can consider the $ underline {set} $ of all the most countable models of $ T $.

Let $ mu $ be a non-main ultrafilter, denoted by $ Omega ^ T_ mu $ the ultra product of

all of these models when it comes to $ mu $.

Obviously, the theory of $ Omega ^ T_ mu $ are formulas that are true for almost all models of $ T $.

In particular, this theory, call it $ T ( Omega ^ T_ mu) $, at $ T $ as a subset.

My questions are:

- Does equality $ T ( Omega ^ T_ mu) = T $ hold on?
- Does it depend $ mu $? Is the intersection on all $ mu $ equal to $ T $?
- If they are not equal, what does the difference mean? In addition, we could consider a sequence of theories $ T subset T ( Omega ^ T_ mu) subset T ( Omega ^ {T ( Omega ^ T_ mu)} _ { mu & # 39;}) subset ldots $. Does it stabilize?
- Is $ T ( Omega ^ T_ mu) $ Completed $ T $ in the right direction?

I suspect that all of these questions are classic and have already been answered, but I don't know where

look for them. Just before the pandemic, I finished my first course in mathematical logic and I have been wondering about this object since.