[FOM] Question about cardinal collapse
Kenny Easwaran
easwaran at berkeley.edu
Tue May 15 19:24:38 EDT 2007
Standard Cohen forcing adding a single real preserves all cardinals,
but adds a new function from omega to 2. I believe that the fact that
it preserves all cardinals means that it doesn't add any isomorphisms
- if the ground model satisfies AC, then every set is isomorphic to a
cardinal, and those isomorphisms persist, so adding any new
isomorphisms should result in a collapse of some cardinal.
I believe the same is true for Cohen forcing adding any number of
reals, or subsets of any set, so that the proof of Easton's theorem
stating that the continuum function can be anything consistent with
Konig's theorem creates a model with exactly the same cardinals as the
ground model of GCH. If that's right, then there are lots of new
functions but no new isomorphisms. (I'm assuming that you mean
isomorphisms between sets that already existed, rather than the "new"
isomorphism between the set of reals and some larger cardinal than
before.)
Kenny Easwaran
On 5/15/07, 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.
>
> 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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