[FOM] Question about cardinal collapse
Robert M. Solovay
solovay at Math.Berkeley.EDU
Wed May 16 03:14:54 EDT 2007
Adding new functions does not lead to cardinal collapse. The simplest
example is to add to V a single generic subset of omega. Then it is easy
to see that for every infinile kappa and lambda > 1 there is a new
function from kappa to lambda. {Let it be the coordinates of the new Cohen
map from omega to 2 on the first omega elements of kappa and otherwise 0.]
But a basic result of Cohen is that no cardinals are collapsed. {Because
the forcing satisfies the countable antichain condition.}
--Bob Solovay
On Tue, 15 May 2007, Colin McLarty 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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