Well – Ordering Principle
Every nonempty subset of Z
+
(the positive integers) contains a
smallest element. Essentially the set Z
+
is wellordered.
Well, DUH!!! This sounds like a really useless principle. BUT,
it can be used to prove anything that we can prove inductively.
First, I’ll show you another proof of why mathematical
induction works, using the wellordering principle:
Let an open statement satisfy the requirements I showed you
the other day for proving an open statement by induction, so
we have s(1)
∧
s(k)
⇒
s(k+1), for all positive integers k.
Now, let’s assume that these two conditions could hold while
the statement s(n) is NOT true for all positive integers n. This
means that we can create a set F = { t
∈
Z
+
 s(t) is false } that is
nonempty.
Since this set is a subset of Z
+
, it must contain a smallest
element. Let that smallest element be w. (Thus
s(w) is NOT
true is our assumption.) We know that w
≠
1 since that is the
first requirement for an inductive proof to hold. Thus, we
know that w > 1, which means that w1
∈
Z
+
.
But, we know that if w1
∈
Z
+
, AND that s(w1) is TRUE
because we assumed that w was the SMALLEST value for
which s(n) was NOT true. Combining that with the fact that
s(k)
⇒
s(k+1), for all positive integers k, then we can plug in k
= w1 to get:
s(w1)
⇒
s((w1)+1) or s(w1)
⇒
s(w), implying that s(w) is
true. But, this contradicts our assumption that s(w) is false.
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 Spring '09
 Mathematical Induction, Natural number, wellordering principle

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