Nonisomorphic free ultrafilters on $\omega$
Any bijection from $\Bbb N$ to itself transforms an ultrafilter on $\Bbb N$ to another (isomorphic) ultrafilter. Any two principal ultrafilters are isomorphic in that sense.
For free ultrafilters on $\Bbb N$, there are $2^{2^{\aleph_0}}$ of them. Since there are $2^{\aleph_0}$ bijections of $\Bbb N$ to itself, there are also $2^{2^{\aleph_0}}$ isomorphism classes of free ultrafilters on $\Bbb N$. So lots of free ultrafilters must be nonisomorphic to each other.
Question: Can you give an explicit example or construction of two free ultrafilters on $\Bbb N$ that are not isomorphic? Assume ZFC.
(Added at the suggestion of @bof in the comments below, in case the question proves too difficult to answer directly):
- Give an explicit example of two filters such that no free ultrafilter extending one of them can be isomorphic to a free ultrafilter extending the other.
- Can you state a property, preserved by isomorphism, possessed by some but not all free ultrafilters?
(1) is as good as the original question as far as I am concerned.
Solution 1:
The simplest property that I can think of (right now) that provably (in ZFC) distinguishes some non-principal ultrafilters on $\mathbb N$ from others is "weak P-point", which means "not in the closure in $\beta\mathbb N$ of a countable set of other non-principal ultrafilters." The existence of weak P-points is a theorem of Kunen; the existence of non-principal ultrafilters that are not weak P-points is trivial (take any countably infinite set of non-principal ultrafilters and take any other point in their closure).