Basic concepts of set theory
The Diagonal Argument (and the Hierarchy of Cardinalities)
A Proof of Cantor's Theorem
Some of you asked me to post a version of this proof.

Let S be some arbitrary set. Suppose for reductio that ¬ |P(S)| > |S| (that is, it is not the case that the cardinality of the powerset of S is greater than the cardinality of S). Then |P(S)| ≤ |S|. But note: there is a simple function that is on S and into P(S) and is one-to-one. This the function that relates each element x, where x ∈ S, to the set of just that element {x}; note that by definition of powerset, {x} ∈ P(S) because x ∈ S. Since such a function exists, this tells us that |P(S)| ≥ |S|. So if |P(S)| ≥ |S| and |P(S)| ≤ |S|, then |P(S)| = |S|. So by definition of cardinality, there exists a function f on S and onto P(S) that is one-to-one. Now consider the set t ∈ P(S), where t is {x | x ∉ f(x)}. This is a little tricky. What it says is, t is the set of all those elements of S that are related by the function f to a set in P(S) that does not contain that element. Such a set must be in P(S) because every possible combination of elements of S, including the empty set, is in P(S). Now observe, the function f cannot relate any element of S to t. For suppose for some x ∈ S, f(x) = t. Is x ∈ t? If it is, then this contradicts the definition of t. But if it is not, then by definition it should be the case that x ∈ t. We conclude that the source of this contradiction was the assumption that ¬ |P(S)| > |S|. Hence |P(S)| > |S|.
Galileo redux
What about the lines of different length? And the squares? Cantor's claim provides a quick solution to both of these puzzles by Galileo.

(Instead, the modern claim that a finite group of points cannot make up a volume or area; and also that all line segments and volumes have the same number of points (aleph-1), solve the problem of the soapdish and his claims about our inability to reason about infinitesimals.)

Ordinal Numbers
Sets that can be ordered can count as ordinal numbers. Cantor also developed a theory of transfinite ordinal numbers. We will not be devoting time to this notion, but suffice it to say it is also very controversial. The basic notion is that we define a successor to the naturals, called w (omega). Thus, one could refer to the omega-th element of some order. Then we can add to omega, and get the successor, w+1, and so on; there is also w+w. The math of transfinites is similar to that of finites, although in some systems the order (adding to the left as opposed to the right of the number) matters.
The Antinomy

Next, three schools of thought about mathematics: logicism, formalism, intuitionism.

[Revised 4 March 2016.]