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: Set Theory
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
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.
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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,…
∞∞,…; ∞∞∞…; ∞∞∞∞…
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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.
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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:
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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
