[FOM] Question about cardinal collapse

[Harvey Friedman]

On 5/15/07 9:15 AM, "Colin McLarty" <colin.mclarty at case.edu> wrote:

> I have a question about cardinal collapse in set theory.  Let me make
> sure I have the standard definiton.  I take cardinal collapse in an
> extension of a universe of sets to mean: some two sets not isomorphic
> in the original universe are isomorphic in the extension.

The word "isomorphic" is not normally used. The usual situation is that the
ground model satisfies ZFC. kappa is a cardinal in the ground model. But
kappa is no longer a cardinal in the forcing extension. kappa is said to
have been collapsed.
 
> Unless I have badly misunderstood, it implies the following condition:
> some two sets S and S' in the original model gain at least one new
> function f:S-->S' in the extension, that is at least one function which
> does not exist in the original model.

It is much stronger than that. E.g., without collapsing any cardinals, you
can add lots of functions from omega to omega, no matter what the ground
model is - without collapsing any cardinals.

Harvey Friedman