# Filter quantifiers

I have been supervising an undergraduate student in an independent study in topology this semester. We have just finished the Stone–Čech compactification, and the semester is ending, so I want to end with an ultrafilter based proof of Hindman’s theorem. There are several good proofs of this sort online, including Hindman’s Theorem Via Ultrafilters (PDF) by Leo Goldmakher and Ramsey Theory (PDF) by Imre Leader (lecture notes transcribed by Stiofáin Fordham). The latter is particularly nice because it uses the formalism of filter quantifiers.

Filter quantifiers generalize the usual quantifiers $\forall$ and $\exists$. If $\mathcal{U}$ is a filter on a set $W$, and $P(x)$ is a property that elements of $W$ may or may not have, then $(\forall_{\mathcal{U}}\, x)P(x)$ holds if the set of $x \in W$ that satisfy $P$ is in $\mathcal{U}$. The dual quantifier $(\exists_{\mathcal{U}}\, x)P(x)$ holds if the set of $x$ that satisfy $P$ is stationary in $\mathcal{U}$. These quantifiers also generalize the infinitary quantifiers $\forall^\infty$ and $\exists^\infty$, the “almost all” measure quantifier, and more.

I was somewhat surprised at how difficult I found it to locate a basic undergraduate-level introduction to filter quantifiers. I have encountered this topic at various points, but I had no luck finding it in Google. So I have written a short undergraduate-level note, which you can download from the following link.

Here are my questions for you:

1. Is this material covered in a textbook somewhere?

2. In my note, I give an example of how the ordinary definition of convergence of a sequence of real numbers can be expressed using filter quantifiers and the Fréchet filter on $\mathbb{N}$. There is a broader sense of convergence in which we keep the definition the same (in filter quantifier notation) but change the filter. I would like to add an example of this to my note, but I also can’t seem to find it anywhere, apart from a few possible hits in papers by John W. Brace in functional analysis.

I would welcome any help with these questions, or other topics that I should include in the note, either in the comments below or by email.

Dave L. Renfro pointed out a post of his on sci.math. I have also found an article on ultrafilters on the Tricki. I still don’t know of an introduction I could give to a talented undergraduate.

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### 4 Responses to Filter quantifiers

1. Dave L. Renfro says:

What follows mostly repeats a comment I just made on your 1 December 2014 math StackExchange question about this issue. I’m putting my comment here for anyone who might be interested but doesn’t visit that site.

Possibly relevant is my October 2004 sci.math post “Generalized Quantifiers” (URLs below). FYI, the Math Forum version has a lot of strange formatting errors. See also Brian Thomson’s 1985 book “Real Functions”, and see Thomson’s earlier 2-part survey Derivation bases on the real line (which contain examples and side-detours not in his book).

Math Forum sci.math URL:
http://mathforum.org/kb/message.jspa?messageID=3556191

• Carl Mummert says:

Thanks. I’ll need to pull those books by Thomson to see what’s in there. The surprising thing to me is that this material is apparently not in any introductory logic textbooks.

2. Peter says:

Not directly related, but Andreas has written lots of stuff on filters and quantifiers, going all the way back to his thesis. His union ultrafilters paper from 1987 has a few comments on quantifiers, his 1993 introductory paper definitely has some treatment (and in my thesis, the definition of idempotent filters comes down to his supervision).

The book by Hindman&Strauss has a section on filters and compactifications in a later chapter — that might be suitable for 2).

• Carl Mummert says:

Thanks – I’ll look up some of those. I hope to go back eventually and expand the note some.