Do theorems like "omitting types theorem", "Extended completeness theorem" etc.. hold inside arbitrary countable transitive models of ZFC?
2026-04-08 18:44:55.1775673895
Some theorems of model theory
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Just expanding on the comments here:
Yes. When we prove theorems in model theory, we're doing just the same sort of mathematics we do when we prove theorems in group theory or real analysis. This mathematics can be formalized in ZFC, and the theorems in model theory, just like the theorems in any other branch of math, are theorems of ZFC. Hence they hold inside any model of ZFC.
Of course, sometimes people (especially logicians) study consequences of stronger systems than ZFC. But authors are usually very careful to point out when a theorem they are proving requires extra assumptions.