The Whirlpool Force: Early Thoughts
In the Astronomia nova, Kepler introduced the whirlpool force this way: Continue reading
Category Archives: History
From Kepler to Ptolemy 19
Kepler’s Physics
Let me repeat the motto, already implicit in the Mysterium cosmographicum, but come into full bloom in the Astronomia nova:
Forces emanating from the Sun guide or drive the planets in their orbits.
Set Theory Jottings 13. From Zermelo to ZFC: Everything’s a Set!
Zermelo’s 1908 system differs from ZFC in three respects. (1) ZFC is a “pure” set theory, without atoms. (2) ZFC includes two additional axioms, Replacement and Foundation. (3) ZFC is a first-order theory. Zermelo’s axioms are not formalized in this way. In this and the next two posts I will go into details. Continue reading
Filed under History, Set Theory
From Kepler to Ptolemy 18
The Astronomia nova
The full title of the Astronomia nova is The New Astronomy, based upon causes, or celestial physics, treated by means of commentaries on the motions of the star Mars, from the observations of Tycho Brahe. This hits all the high spots: the treasure trove of Tycho’s observations, Kepler’s new physics, and the “battles with Mars”. Continue reading
Set Theory Jottings 12. Zermelo on “definiteness”
In the last post, I mentioned Zermelo’s 1929 paper “On the concept of definiteness in axiomatics”. By this time, people had suggested replacing “definite” with “definable in first-order logic”. Zermelo did not agree with this. Continue reading
Filed under History, Logic, Set Theory
Set Theory Jottings 11. Zermelo to the Rescue! (Part 2)
In 1908 Zermelo published his paper “Investigations in the foundations of set theory”. This contained the axiom system that eventually led to ZFC. Zermelo opens the paper with this rationale: Continue reading
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Set Theory Jottings 10. Axiomatic Set Theory
“An Axiom, you know, is a thing that you accept without contradiction. For instance, if I were to say ‘Here we are!’ that would be accepted without any contradiction, and it’s a nice sort of remark to begin a conversation with. So it would be an Axiom. Or again, supposing I were to say, ‘Here we are not!’, that would be—”
“—a fib!” cried Bruno.
“that would be accepted, if people were civil”, continued the Professor; “so it would be another Axiom.”
“It might be an Axledum”, Bruno said: “but it wouldn’t be true!”
—Lewis Carroll, Sylvie and Bruno Concluded
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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
Diagonal Argument 