proof of partial order with chain condition does not collapse cardinals
Given any function purporting to violate the theorem by being surjective (or cofinal) on , we show that there are fewer than possible values of , and therefore only possible elements in the entire range of , so is not surjective (or cofinal).
Suppose is a cardinal of that is not a cardinal in .
There is some function and some cardinal such that is surjective. This has a name, . For each , consider
, since any two which force different values for are incompatible and has no sets of incompatible elements of size .
Now suppose that for some , in and for some there is a cofinal function .
We can construct as above, and again the range of is contained in . But then . So there is some such that for any , and therefore is not cofinal in .
|Title||proof of partial order with chain condition does not collapse cardinals|
|Date of creation||2013-03-22 12:53:43|
|Last modified on||2013-03-22 12:53:43|
|Last modified by||Henry (455)|