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.)
Monthly Archives: September 2019
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.)
JB: Okay, let me try to sketch out a more categorical approach to Gödel’s completeness theorem for first-order theories. First, I’ll take it for granted that we can express this result as the model existence theorem: a theory in first-order logic has a model if it is consistent. From this we can easily get the usual formulation: if a sentence holds in all models of a theory, it is provable in that theory.
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.)
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?
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.