[FOM] Question about cardinal collapse
Colin McLarty
colin.mclarty at case.edu
Tue May 15 09:15:30 EDT 2007
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.
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.
Are those two conditions equivalent? Does every extension of a
universe of sets which adds new functions necessarily make some
originally non-isomorphic sets isomorphic? If not, is there a standard
name in set theory for the
condition of adding new functions?
For all I have found so far, there may be some terribly easy way to see
that adding a new function (between sets in the original universe) to a
universe of sets always adeds at least one isomorphism between sets
that were not isomorphic in the original universe. Or there may be
some well known forcing extension, for example, that does add new
functions without cardinal collapse. Can anyone here tell me?
thanks. Colin
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