10 1. The Real Numbers
this set A. Note that (1,x1) is the only element of A which is not in the range of S.
This is because any other such element could be removed from A and the resulting
set would still contain (1,x1) and be closed under S.
To complete the argument, we will show that the set A is the graph of a function
from N to X that is, it has the form {(n, xn) : n N} for a certain sequence
{xn} in X. This is the sequence we are seeking. To prove that A is the graph of a
function from N to X, we must show that each n N is the first element of exactly
one pair (n, x) A. We prove this by induction.
The element 1 is the first element of the pair (1,x1), which is in A by the
construction of A. If there were another element x X such that (1,x) A, then
(1,x) would be an element, not equal to (1,x1), which fails to be in the range of S.
This is due to the fact that 1 is not the successor of any element of N by N3.
Now, for the induction step, suppose for some n we know that there is a unique
element xn X such that (n, xn) A. Then S(n, xn) = (s(n),fn(xn)) is in A.
Suppose there is another element (s(n),x) A with x = fn(xn) and suppose this
element is in the image of S that is, (s(n),x) = S(m, y) = (s(m),fm(y)) for some
(m, y) A. Then n = m by N4, and y = xn by the induction assumption. Thus
if (s(n),x) is really different from (s(n),fn(xn)), then it cannot be in the image of
S. Since (1,x1) is the only element of A which is not in the image of S and since
s(n) = 1, we conclude there is no such element (s(n),x). By induction, for each
element of N there is a unique element xn X such that (n, xn) A. Thus, A is
the graph of a function n xn from N to X.
This shows the existence of a sequence with the required properties. We leave
the proof that this sequence is unique to the exercises.
Note that the proof of the above theorem used all of Peano’s axioms, not just
N5.
Using Peano’s Axioms to Develop Properties of N. In this subsection, we
will demonstrate some of the steps involved in developing the arithmetic and order
properties of N using only Peano’s axioms. It is not a complete development, but
just a taste of what is involved. We begin with the definition of addition.
Definition 1.2.4. We fix m N and define a sequence {m + n}n∈N inductively as
follows:
(1.2.3) m + 1 = s(m),
and
(1.2.4) m + s(n) = s(m + n).
These two conditions determine a unique sequence {m + n}n∈N by Theorem 1.2.3.
Note that (1.2.4) is the recursion relation of the inductive definition. It tells us how
m + s(n) is to be defined assuming that m + n has already been defined.
By (1.2.3) of the above definition, the successor s(n) of n is our newly defined
n + 1. At this point we will begin using n + 1 in place of s(n) in our inductive
arguments and definitions.
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