Usually one defines the ordinal operations via transfinite induction:
Author Archives: Michael Weiss
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
From Kepler to Ptolemy 13
Latitudes
First thing to note about Ptolemy’s latitude theory: its decoupling from the longitude theory. For longitudes, one projects the orbits into the ecliptic plane. The actual speeds will differ from the projected speeds. However, the effect is small because the inclinations are fairly small (see the table below), and Ptolemy’s longitude computations ignore it. The latitude algorithms use the longitude as an input.
From Kepler to Ptolemy 12
Mercury
Mercury refused to cooperate with Ptolemy’s basic paradigm. You might guess that the fault lies with Mercury’s larger eccentricity, but studies show that bad data bears most of the blame. Mercury hugs the Sun, only appearing near the horizon close to sunrise or sunset, hardly ideal observation conditions.
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
Diagonal Argument 