Libration Force
The Libration Force
Kepler coined the term “libration” for the oscillation of a planet’s distance from the Sun, approaching and receding. Continue reading
Set Theory Jottings 17. Ordinals Revisited
In post 13 I sketched proof by transfinite induction, and definition by transfinite recursion. Let’s take a closer look at that. By definition, < means ∈ for ordinals—treat that also as vernacular. Sometimes one symbol or the other makes the meaning clearer. We must keep in mind though that we have not shown that < is a well-ordering on Ω, or even a simple ordering. Continue reading
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Set Theory Jottings 16. Axioms of ZFC
The Axioms of ZFC
The language of ZF (ℒ(ZF)) consists of basic first-order syntax, with a single binary predicate symbol ∈. Here is a list of the axioms, with a tag-line (an imprecise description) for each. Continue reading
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Set Theory Jottings 15. From Zermelo to ZFC: Formal Logic
The Role of Formal Logic Continue reading
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Set Theory Jottings 14. From Zermelo to ZFC: Replacement and Foundation
Replacement and Foundation
Thirteen years after Zermelo published his axioms, Fraenkel pointed out that they weren’t quite strong enough. For example, you couldn’t prove the existence of the set ⋃n∈ω𝒫n(0). Continue reading
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From Kepler to Ptolemy 20
The Whirlpool Force: Early Thoughts
In the Astronomia nova, Kepler introduced the whirlpool force this way: Continue reading
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
Diagonal Argument 