Collapsing Cardinals

Let be a Transitive Model of . Let be a Finite Function Forcing where

Let be a -Generic Filter over . By Model Extension by Finite Function Forcing, we find that

for some . In particular, , so is not a Cardinal in .

Corollary

Being a Cardinal is Downwards Absolute but not Upwards Absolute.

Proof

Let be a Countable Transitive Model of a Sufficiently Strong finite such that the Model Extension , where is again Sufficiently Strong and finite. Let be as above and find to be a -Generic Filter over (which exists as is countable) By above argument, being a Cardinal is not Upwards Absolute between and .

However, we can express ” is a Cardinal” as

which is a formula in Formula Hierarchy and thus Downwards Absolute.