 
Summary: 1. ntuples.
We let
N = {0, 1, 2, . . . , n, . . .}
be the set of natural numbers and we let
N+
= {n N : n > 0}.
For each n N we let
I(n) = {m N : m < n};
Note that I(n) has n members and that I(0) = .
If A and B are sets we say A is equipollent with B and write
A B
if there are f : A B such that f1
: B A. Note that the relation is reflexive,
symmetric and transitive. Whenever A is a set we let
A =
n if n N and A I(n),
A I(n) for no n N.
Suppose n N. An ntuple is, by definition, a function whose domain is I(n).
Thus the only 0tuple is the empty function which equals the empty set. Note that
if x is an ntuple then x = n; we call x the length of x. We say x is tuple if x
