MW: Ok, let’s plunge into the construction of Trued(x). The bedrock level: True0(x), truth for (closed) atomic formulas. Continue reading
Category Archives: Logic
Non-standard Models of Arithmetic 15
The Truth about Truth
MW: A little while back, I noted something delicious about the history of mathematical logic:
- Gödel’s two most famous results are the completeness theorem and the incompleteness theorem.
- Tarski’s two most famous results are the undefinability of truth and the definition of truth.
Filed under Conversations, Peano Arithmetic
Non-standard Models of Arithmetic 14
MW: Recap: we showed that PAT implies ΦT, where ΦT is the set of all formulas
Now we have to show the converse, that PA+ΦT implies PAT. But first let’s wave our hands, hopefully shaking off some intuition, like a dog shaking off water.
Filed under Conversations, Peano Arithmetic
Non-standard Models of Arithmetic 13
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
First-Order Categorical Logic 6
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
First-Order Categorical Logic 5
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.
Filed under Categories, Conversations, Logic
Non-standard Models of Arithmetic 12
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
Algorithmic Information Theory
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
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.
First-Order Categorical Logic 4
MW: I made up a little chart to help me keep all these adjoints straight:
Filed under Categories, Conversations, Logic
