Suppose β>1, and let ζ>0 be arbitrary. Then ζ has a unique representation in so-called Cantor normal form: Continue reading
Category Archives: Math
Set Theory Jottings 8. Ordinal Arithmetic
Usually one defines the ordinal operations via transfinite induction:
Filed under Set Theory
Set Theory Jottings 7. The (Cantor-Dedekind-Schröder)-Bernstein Theorem
The trichotomy of cardinals says that for any 𝔪 and 𝔫, exactly one of these holds: 𝔪<𝔫, 𝔪=𝔫, or 𝔪>𝔫. It’s equivalent to the conjunction of these two propositions, for any two cardinals 𝔪 and 𝔫:
Filed under History, Set Theory
Set Theory Jottings 6. Zorn’s Lemma
Zermelo’s 1904 proof of the well-ordering theorem got a lot of blowback, as we’ve seen. On the other hand, the very next year Hamel used it to prove the existence of a so-called Hamel basis. In 1910, Steinitz made numerous applications in the theory of fields. He wrote:
Filed under History, Set Theory
Nonstandard Models of Arithmetic 32
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Previous Paris-Harrington post
[Ed. note: This post was essentially ready two years ago, but I got distracted with other matters. If you’re seeing this for the first time, or want to refresh your memory, posts 8 and 9 introduced the Paris-Harrington theorem. Posts 21 through 24 continued the discussion, in a dialog with Bruce Smith. MW]
Filed under Conversations, Peano Arithmetic
First-Order Categorical Logic 12
MW: Last time we looked at the categorical rendition of “C is a model of B”:
- Functors B:FinSet→BoolAlg and C:FinSet→BoolAlg
- A natural transformation F:B→C
where B and C are hyperdoctrines, and
- B is syntactic: the elements of each B(n) are equivalence classes of formulas (which we agreed to call predicates);
- C is semantic: the elements of each C(n) are relations on a domain V.
(We’ve been saying that C(n) is the set of all n-ary relations on V, but I see no need to assume that.)
Filed under Categories, Conversations, Logic
Set Theory Jottings 5. Zermelo to the Rescue! (Part 1)
Ernst Zermelo is remembered today chiefly for two results. His 1904 paper “Proof that every set can be well-ordered” introduced the Axiom of Choice. His 1908 paper “Investigations in the foundations of set theory” led to the most popular axiomatization of set theory. He thus claims credit for two of the letters of ZFC: Zermelo-Fraenkel with Choice.
Filed under History, Set Theory
First-Order Categorical Logic 11
MW: Last time we justified some equations and inequalities for our adjoints: they preserve some boolean operations, and “half-preserve” some others. And we incidentally made good use of the color palette!
Filed under Categories, Conversations, Logic
Set Theory Jottings 4. Ordinals
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,…
∞∞,…; ∞∞∞…; ∞∞∞∞…
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
Diagonal Argument 