 
Summary: 1. Introduction to the theory of infinite sets.
Theorem 1.1. N is infinite.
Proof. We have N N {0} by means of S. Thus N cannot be finite since, as we
have already shown, no finite set is equipotent with a proper subset.
Theorem 1.2. Suppose A N and A is infinite. Then A N.
Proof. We do this by defining a function by induction. Let G = {g : for some n N,
g : I(n) A} and define G : G A by letting G(g) be the least element of A
rng g for g G. One obtains a function f : N A such that f(n) = G(fI(n))
for each n N. We leave to the reader the straightforward verification that f is
univalent with range equal to A.
Remark 1.1. Note that in the above proof f carries n to the nth least element
of A. Alternatively, we could use the ordering on N to induce a well ordering on A
and then use our previous results about well ordered sets.
Theorem 1.3. Nn
N for any n N+
.
Proof. The statement is trivially true when n = 1 and will follow by induction on
n if we can show it holds for n = 2.
We now show that N N × N. For each n N we let L(n) = {(l, m)
N × N : l + m = n} and we let M(n) = {L(m) : m < n}. Note that L(n) is
