We’ll begin with Bailee telling us about Cantor’s result, that for every set there is another set of larger cardinality (and so, iterating, there are “an infinity of infinities”).
We will then move on to Section 2.6. Read the first page of that section; but since (in the interests of time) we’ll only prove results concerning just the cardinalities aleph-null, aleph-one, two to the power aleph-null, and the successor of this last, there is no need to pay much attention to Lemma 2.34, or to finite colourings of uncountable sets. Everyone can skip those sections, or skim them if you wish.
Return to careful reading at the bottom of page 72, and continue through to two-thirds of the way down page 79. This encompasses the two results I would like to end with:
-> a two-colouring of the edges of the complete graph on aleph-null vertices does not necessarily contain a complete monochromatic subgraph on aleph-null vertices, and
-> the Erdos-Rado theorem, Theorem 2.38 (a positive result, salvaging the negative result just mentioned).
Presentations:
- Colin can present the proof of Proposition 2.36
- Casey can present the proof of the Erdos-Rado theorem, using Lemma 2.39 as a black box (and can tell us the statement of the lemma — note that missing from the statement in the book is that \(c\) is some fixed \(2\)-coloring of the pairs from a set whose cardinality is the successor of two to the power of aleph-null)
- Ted can present the proof of Lemma 2.39 (this will probably spill over to Friday).