Kepler
Kepler wrote five major astronomical works. Chronologically: Continue reading
Monthly Archives: May 2025
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.
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
From Kepler to Ptolemy 14
Cycle Counts
You may have heard that Ptolemaic systems grew to have 80 spheres or cycles, while the Copernican system had only 34. This is a myth.
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
Diagonal Argument 