Peter Krautzberger on the web

Return to bloglife

Maybe it was the summer heat, maybe the summer break at the UofM or something else. In any case, I did not feel like blogging the last couple of weeks. But this must change! So to get me back to writing I’ll start with something small tiny.

Unforeseen

I have the great pleasure of spending my PostDoc at the University of Michigan. After spending a winter here 2/3 years ago, I knew a lot of things I could look forwards to — like the amazing grad students.

One of the unforeseen pleasures so far has been to meet Francois Dorais of MathOverflow-Admin fame. Last Friday we talked he told me about a proof by Michael Canjar (sorry for linking to a paywall) on Mathias forcing and there is this small observation that I think is really cool.

(non) P-points

I mentioned them before, but repetition is never a bad thing.

An ultrafilter p on \omega is called a P-point if for every f: \omega \rightarrow \omega there is A\in p such that f restricted to A is either finite-to-one or constant.

P-points are truly classical ultrafilters having been studied since the dawn of time ultrafilters. They carry interesting properties and Shelah proved that they might not exist (though they do under reasonably weak assumptions like very weak versions of Martin’s Axiom).

The property of P-points somehow tells us that functions drastically ‘changes speed’ on a set in the ultrafilter. If you take a function which is ‘nowhere’ finite-to-one, i.e., every point has an infinite preimage, then a P-point either slows it down completely (by making it constant on a set) or speeds it up extremely (by making it finite-to-one).

But the cool thing Francois showed me (from Canjar’s proof) is what non P-points (so possibly all ultrafilters) can do. They can force any function to slow down in a weird fashion.

Slowing to identity.

Even though the argument I want to mention holds for arbitrary functions, you should think of quickly growing functions, i.e., strictly increasing functions. So let us pick some g: \omega \rightarrow \omega .

Now if p \in \beta \mathbb{N} is not a P-point, then there exists a function f: \omega \rightarrow \omega which is not constant or finite-to-one on any set A \in p .

So what about I_g := \{ n \in \mathbb{N} \ \vert \ g(f(n)) < n \} ?

On this set, g \circ f is dominated by the identity. That’s slow!!! Just imagine g was the Ackermann function or faster thatn all recursive functions! Suddenly, its only as fast as the identity? Wow…

And now the crazy part.

I_g \in p .

That’s right! On a set in p , g\circ f slows down like that. That’s crazy!

Proof.

  • f is finite-to-one on \omega \setminus I_g .
    • For k\in \omega , f^{-1}(k) \cap (\omega \setminus I_k) = \{ i \in \omega \ \vert \ g(k)= g(f(i)) \geq i \}
    • But this is a finite set for any k .
    • In other words, f is finite-to-one.
  • Therefore \omega \setminus I_g \notin p .
  • Since p is a maximal filter, I_g \in p .

That’s all.

What you can do with this.

Michael Canjar used this fact to show that Mathias forcing with a non P-point adjoins a dominating real. This is not too difficult now since it is easy to see that a Mathias real will dominate all sets in the ultrafilter. But that’s all for today.