Monthly Archives: July 2019

Review: Smullyan & Fitting, Set Theory and the Continuum Problem

The first sentence of Pollard’s review sums up my feelings perfectly: “This rewarding, exasperating book…” On balance, I found it more exasperating than rewarding. But it does have its charms.

I participated in a meetup group that went through the first two parts of S&F. My fellow participants possessed considerable mathematical knowledge and sophistication, but had only slight prior acquaintance with mathematical logic and none with axiomatic set theory. (The opinions here are strictly my own, but they reflect my experience in the meetup.) If I had just skimmed the book, glancing at familiar material, I would probably have a more positive impression.

I wrote an extensive set of notes for the meetup. This post is basically the last section of those notes.

I will begin with the book’s minuses, so as to end on a positive note.

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Stirling’s Formula: Ahlfors’ Derivation

If you’re reading this blog, you probably know Stirling’s formula:

n! \sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n

It’s not hard to estimate n! to within a factor of √2; I wrote up a note on this and even easier derivations. It’s quite a bit harder to show that the ratio of the two sides approaches a definite limit as n→∞ and that this limit is 1. You can find a variety of proofs in a number of places, one being Ahfors’ Complex Analysis.  I wrote up a note about this too, expanding on some of the details.

Incidentally, the two sides are asymptotic not just for positive integers n. Replace n! with Γ(z+1) on the left, and both n‘s with z‘s on the right. Allow z to go to infinity in the complex plane, while staying at least a fixed finite distance to the right of the imaginary axis. Then the two sides remain asymptotic. Ahfors proves this stronger result, and uses it to derive the integral form for the Γ function.

Note that if you replace the n‘s with z‘s, you have zz on the right. So you’ve got to worry about branches of the complex logarithm (since zz is defined as ez log z). The note deals with this (and other things).

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