In the last post, I mentioned Zermelo’s 1929 paper “On the concept of definiteness in axiomatics”. By this time, people had suggested replacing “definite” with “definable in first-order logic”. Zermelo did not agree with this. Continue reading
First-Order Categorical Logic 14
JB: So, let’s think about how we can prove this generalization of Gödel’s completeness theorem. First, remember that a hyperdoctrine B is consistent iff B(0) has at least two elements, or in other words, ⊤ ≠ ⊥ in this boolean algebra. Second, let’s say a hyperdoctrine C is set-based if every C(n) is the power set of Vn for some fixed set V. We call V the universe. Third, let’s say a morphism of hyperdoctrines, say F: B → C, is a natural transformation whose components F(n): B(n) → C(n) are Boolean algebra homomorphisms obeying the Beck–Chevalley condition and maybe the Frobenius condition. (We’re a bit fuzzy about this and we’ll probably have to sharpen it up.) Continue reading
Filed under Categories, Conversations, Logic
Set Theory Jottings 11. Zermelo to the Rescue! (Part 2)
In 1908 Zermelo published his paper “Investigations in the foundations of set theory”. This contained the axiom system that eventually led to ZFC. Zermelo opens the paper with this rationale: Continue reading
Filed under History, Set Theory
From Kepler to Ptolemy 17
The Mysterium cosmographicum
The Mysterium cosmographicum (Cosmographical Mystery) boasts one of most celebrated illustrations in the history of science: the planetary spheres nested with the five Platonic solids. This picture graces nearly every history of astronomy. Who am I to break with tradition, here it is: Continue reading
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First-Order Categorical Logic 13
MW: It’s been a minute! Well, almost 60,000 minutes.
We left off with a question: does a natural transformation from a syntactic hyperdoctrine to a semantic hyperdoctrine automatically “respect quantifiers”? We saw that this amounts to a Beck–Chevalley condition. We wondered if we had to add that condition to our definition of a model, or if it came for free. Continue reading
Filed under Categories, Conversations, Logic
Set Theory Jottings 10. Axiomatic Set Theory
“An Axiom, you know, is a thing that you accept without contradiction. For instance, if I were to say ‘Here we are!’ that would be accepted without any contradiction, and it’s a nice sort of remark to begin a conversation with. So it would be an Axiom. Or again, supposing I were to say, ‘Here we are not!’, that would be—”
“—a fib!” cried Bruno.
“that would be accepted, if people were civil”, continued the Professor; “so it would be another Axiom.”
“It might be an Axledum”, Bruno said: “but it wouldn’t be true!”
—Lewis Carroll, Sylvie and Bruno Concluded
Filed under History, Set Theory
Set Theory Jottings 9. Cantor Normal Form
Suppose β>1, and let ζ>0 be arbitrary. Then ζ has a unique representation in so-called Cantor normal form: Continue reading
Filed under Set Theory
Set Theory Jottings 8. Ordinal Arithmetic
Usually one defines the ordinal operations via transfinite induction:
Filed under Set Theory
From Kepler to Ptolemy 15
The Planetary Hypotheses
In the Planetary Hypotheses, Ptolemy lays out his cosmology: that is, the structure and arrangement of the universe. This work answers the question, did Ptolemy believe in the physical truth of the Almagest’s celestial geometry?—with an unambiguous Yes. Contrary to an opinion often expressed by earlier historians, he did not regard it just as a calculational scheme for predicting planetary positions.
Diagonal Argument 