Countable Transitive Model

For every finite there is a countable Transitive Model of .

Proof

Consider the Von Neumann Hierarchy with the set theoretic universe. Let be finite and a formula equivalent to (just all formulas) Then is true as . Thus find such that where is Absolute using Lévy Reflection Theorem. In particular But then is a Transitive Model of as each is a Transitive set.

Now apply The Downward Löwenheim-Skolem Theorem to construct a countable Substructure of such that all formulas in are Absolute between and . In particular .

Take countable from above. Note that is Well-Founded in (since Axiom of Foundation is true) WLOG assume that Axiom of Extensionality is in . By Mostowski’s Collapsing Theorem there is some where is Transitive. and since is countable, so is and

Remark

When we defined above, it wasn’t necessarily Transitive (so we needed the last step) Let be the statement ” is the smallest uncountable Ordinal” Then To see this note: Being an Ordinal is Absolute, so is an ordinal in Countability is so Upwards Absolute so

If , its countability is witnessed by surjective. But then so

In particular, we have . Let the set of all subformulas of . Now by construction, but is the only witness to so . But as is countable, so is not Transitive.

Corollary

“Being countable” and “being cardinal” are not Absolute statements between and . In particular, countable is and cardinal is .

Proof

Suppose from above is in where means ” is the smallest uncountable ordinal” Then:

therefore by Absoluteness

But any that is a witness:

necessarily has to be as is Transitive Then is a countable Ordinal. Note that is defined as the witness for . So whatever thinks that is, it is definitely not the same as what thinks. So Cardinals are not preserved.

And moreover "" is countable in , but not countable in .