Awhile back, the BBC website History Extra had a post that included this tidbit:
AD 0… the date that never was
The AD years of the Christian calendar are counted from the year of Jesus Christ’s birth, and, as the number zero was then unknown to the west, Dionysius began his new Christian era as AD 1, not AD 0. …
This evoked the ire of the noted historian Thony Christie. In a post Something is Wrong on the Internet, he explained:
Filed under Bagatelles, History
MW: OK, back to the main plotline. Enayat asks for a “natural” axiomatization of PAT. Personally, I don’t find PAT all that “unnatural”, but he needs this for Theorem 7. (It’s been a while, so remember that Enayat’s T is a recursively axiomatizable extension of ZF.)
Filed under Conversations, Peano Arithmetic
MW: An addendum to the last post. I do have an employment opportunity for one of those pathological scaffolds: the one where B(0) is the 2-element boolean algebra, and all the B(n)’s with n>0 are trivial. It’s perfect for the semantics of a structure with an empty domain.
The empty structure has a vexed history in model theory. Traditionally, authors excluded it from the get-go, but more recently some have rescued it from the outer darkness. (Two data points: Hodges’ A Shorter Model Theory allows it, but Marker’s Model Theory: An Introduction forbids it.)
Filed under Categories, Conversations, Logic
[This post is available in pdf format, sized for small and medium screens.]
Lewis Carroll Epstein wrote a book Relativity Visualized. It’s been called “the gold nugget of relativity books”. I wouldn’t go quite that far, but Epstein has devised a completely new way to explain relativity. The key concept: the Epstein diagram. (I should mention that Relativity Visualized is a pop-sci treatment.)
JB: It’s been a long time since Part 11, so let me remind myself what we’re talking about in Enayat’s paper Standard models of arithmetic.
We’ve got a theory T that’s a recursively axiomatizable extension of ZF. We can define the ‘standard model’ of PA in any model of T, and we call this a ‘T-standard model’ of PA. Then, we let PAT to be all the closed formulas in the language of Peano arithmetic that hold in all T-standard models.
This is what Enayat wants to study: the stuff about arithmetic that’s true in all T-standard models of the natural numbers. So what does he do first?
Filed under Conversations, Peano Arithmetic
Back in the 60s, Kolmogorov and Chaitin independently found a way to connect information theory with computability theory. (They built on earlier work by Solomonoff.) Makes sense: flip a fair coin an infinite number of times, and compare the results with the output of a program. If you don’t get a 50% match, that’s pretty suspicious!
Three aspects of the theory strike me particularly. First, you can define an entropy function for finite bit strings, H(x), which shares many of the formal properties of the entropy functions of physics and communication theory. For example, there is a probability distribution P such that H(x)=−log P(x)+O(1). Next, you can give a precise definition for the concept “random infinite bit string”. In fact, you can give rather different looking definitions which turn out be equivalent; the equivalence seems “deep”. Finally, we have an analog of the halting problem: loosely speaking, what is the probability that a randomly chosen Turing machine halts? The binary expansion of this probability (denoted Ω by Chaitin) is random.
I wrote up my own notes on the theory, mostly to explain it to myself, but perhaps others might enjoy them.
Filed under Logic
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.
If you’re reading this blog, you probably know Stirling’s formula:
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).
John Baez has a post outlining another derivation of the full Stirling formula, using Laplace’s method. It looks a lot easier than Ahlfors’!
Filed under Analysis
MW: I made up a little chart to help me keep all these adjoints straight:
Filed under Categories, Conversations, Logic
MW: Time to start on Enayat’s paper in earnest. First let’s review his notation. M is a model of T, a recursively axiomatizable extension of ZF. He writes 
Filed under Conversations, Peano Arithmetic
Diagonal Argument 