Martin’s axiom and the continuum hypothesis
If then , and in fact
, so , hence it will suffice to find an surjective function from to .
Given any subset we will construct a function such that a unique can be recovered from each . will have the property that if then for finitely many elements , and if then for infinitely many elements of .
is a partial function from to
There exist such that for each ,
There is a finite subset of , , such that
For each , for finitely many elements of
This satisfies ccc. To see this, consider any uncountable sequence of elements of . There are only countably many finite subsets of , so there is some such that for uncountably many and is the same for each such element. Since each of these function’s domain covers only a finite number of the , and is on all but a finite number of elements in each, there are only a countable number of different combinations available, and therefore two of them are compatible.
Consider the following groups of dense subsets:
for . This is obviously dense since any not already in can be extended to one which is by adding
for . This is dense since if then is.
By , given any set of dense subsets of , there is a generic which intersects all of them. There are a total of dense subsets in these three groups, and hence some generic intersecting all of them. Since is directed, is a partial function from to . Since for each , is non-empty, , so is a total function. Since for is non-empty, there is some element of whose domain contains all of and is on a finite number of them, hence for a finite number of . Finally, since for each , , the set of such that is unbounded, and hence infinite. So is as promised, and .
|Title||Martin’s axiom and the continuum hypothesis|
|Date of creation||2013-03-22 12:55:05|
|Last modified on||2013-03-22 12:55:05|
|Last modified by||Henry (455)|