We saw how Cantor introduced ordinals originally as “symbols”,
0, 1, 2,…; ∞, ∞+1, ∞+2,…; 2∞, 2∞+1,…; 3∞,…; 4∞,…
∞2, ∞2+1,…; 2∞2,…; 3∞2,…; ∞3,…; ∞4,…
∞∞,…; ∞∞∞…; ∞∞∞∞…
Category Archives: Math
Set Theory Jottings 4. Ordinals
Filed under History, Set Theory
Set Theory Jottings 3. The Paradoxes
Frege added an appendix to volume II of his 1903 magnum opus Grundgesetze der Arithmetik (Foundations of Arithmetic). It began:
A scientist can hardly meet with anything more undesirable than to have the foundations give way just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was nearly through the press.
Filed under History, Set Theory
First-Order Categorical Logic 10
JB: Last time we saw how to get some laws of logic from two facts:
• right adjoint functors between boolean algebras preserve products (‘and’),
and
• left adjoint functors between boolean algebras preserve coproducts (‘or’).
Filed under Categories, Conversations, Logic
First-Order Categorical Logic 9
Filed under Categories, Conversations, Logic
First-Order Categorical Logic 8
MW: We’re reviewing hyperdoctrines, which are specially nice functors B: FinSet → BoolAlg. When we have such a functor, any map f of finite sets gives a homomorphism of boolean algebras, B(f). But we’ve seen this is a morphism and a functor. (“It’s a floor wax and a dessert topping!”) What do you think about the term “adjoint morphism”? It might help keep the two levels straight.
Filed under Categories, Conversations, Logic
First-Order Categorical Logic 7
MW: John, it’s been eons since we last discussed First-Order Categorical Logic: not since September 2019! (I read a lot of Russian novels during the break.) But New Year’s seems like a good time to resume the tale.
JB: Yes indeed! It’s been a long time, and it’s mostly my fault. Let’s see if we can get back up to speed.
Filed under Categories, Conversations, Logic
Set Theory Jottings 2. Cantor’s Paradise
Cantor’s Paradise
No one shall expel us from the Paradise that Cantor has created for us.
—Hilbert, “Über das Unendliche” [On the Infinite], in Mathematische Annalen 95 (1925)
I used to believe these myths about the history of set theory:
Filed under History, Set Theory
Set Theory Jottings 1: Philosophy and Naive Set Theory
These notes are not a systematic “Introduction to Set Theory”. I intend them as a
blend of history, intuition, and exposition, with an occasional dash of philosophy.
Filed under Set Theory
The One Change in the Remake of Mean Girls All the Other Websites Missed
Here it is:
Apparently two-sided limits are problematic in 2024.
Filed under Bagatelles, Math
Quadratic Reciprocity
Quadratic reciprocity has hundreds of proofs, but the nicest ones I’ve seen (at least at the elementary level) use Gauss sums. One variant uses the cyclotomic field ℚ(ζ), where ζ is a p-th root of unity. Another brings in the finite fields 𝔽p and 𝔽q.
I wrote up a long, loving, and chatty treatment several years ago, going through the details for several examples. Much longer than the proofs! The diagram up top may give you an inkling.
Anyway, here it is.
Filed under Number Theory
Diagonal Argument 