In post 13 I sketched proof by transfinite induction, and definition by transfinite recursion. Let’s take a closer look at that. By definition, < means ∈ for ordinals—treat that also as vernacular. Sometimes one symbol or the other makes the meaning clearer. We must keep in mind though that we have not shown that < is a well-ordering on Ω, or even a simple ordering. Continue reading
Monthly Archives: August 2025
Set Theory Jottings 16. Axioms of ZFC
The Axioms of ZFC
The language of ZF (ℒ(ZF)) consists of basic first-order syntax, with a single binary predicate symbol ∈. Here is a list of the axioms, with a tag-line (an imprecise description) for each. Continue reading
Filed under History, Set Theory
Set Theory Jottings 15. From Zermelo to ZFC: Formal Logic
The Role of Formal Logic Continue reading
Filed under History, Set Theory
Set Theory Jottings 14. From Zermelo to ZFC: Replacement and Foundation
Replacement and Foundation
Thirteen years after Zermelo published his axioms, Fraenkel pointed out that they weren’t quite strong enough. For example, you couldn’t prove the existence of the set ⋃n∈ω𝒫n(0). Continue reading
Filed under History, Set Theory
From Kepler to Ptolemy 20
The Whirlpool Force: Early Thoughts
In the Astronomia nova, Kepler introduced the whirlpool force this way: Continue reading
Diagonal Argument 