(From chapter 4) Theorem (Euclid): The set of primes is infinite.

Proof: We restate the theorem in another way: For any finite set of
primes, there is a prime not in that set.  

So, let S be a finite set of primes, say p1, p2, p3, p4,..., pn. Let E
= (p1*p2*...*pn) + 1.  If we divide E by any of the pk, we get a
remainder of 1, so none of those primes divides E. But by the UFT, E
must be divisible by a prime (it has a prime factorization). So there
must be a prime not among those in S. So no finite set S of primes can
contain all the primes.

Chapter 5: Sets

Now we turn to sets. There is a strong connection with logic:

            AuB = {x | x in A v x in B}  by definition

            x in A u B iff x in A v x in B as logic connection between
	    u and v

            but beware: u connects sets and is not true
            or false, while v connects statements
            (and is T/F)

            (similarly for intersection)
   
            P(A) = powerset of A = {B | B subst of A} by definition

            B in P(A) iff B subst of A

            Empty set  notation (Scandinavian o, looks like phi), also
            written { }. It is not zero; it is not a number.
            Note: { }  =/= { { } }.

            Think of a set as a museum whose essential aspect is all
            the art objects in it.
            So { } has NO elements, while { { } } has one element: { }.

            Cardinality of finite sets: |S| = # elements of S.
            | { } | = 0; | { { } } | = 1. |{a}| = 1. A set {a} with one
            element is called a singleton set.