Math 310 Class Notes 4: The Well-ordering Principle
Well-ordering principle: Every nonempty subset Tof Nhas a least
element. That is, there is an m∈Tsuch that m≤nfor all n∈T.
Intutively clear as it may seem at the first glance, this principle turns
out to be logically equivalent to the mathematical induction, the fifth
axiom of Peano, which is quite surprising.
Theorem 1. The mathematical induction is logically equivalent to the
well-ordering principle.
Proof. Part I. We show the well-ordering principle implies the math-
ematical induction.
Let S⊂Nbe such that 1 ∈Sand k∈Simplies k0∈S. We want
to establish that S=Nby the well-ordering principle.
Suppose N\Sis not empty. Then by the well-ordering principle there
is a least element m∈N\S. Since 1 ∈Swe know 1 /∈N\S. Therefore
m6= 1 and so by one of the homework 2 problems there is some q∈N
such that m=q0=q+ 1, which implies q < m by the definition of
<. We conclude that q∈S; or else we would have q∈N\Sand so
mwould not be the least element of N\S, which is absurd. However,
since Shas the property that k∈Simplies k0∈S, we conclude that
m=q0∈Sbecause q∈S. This contradicts m∈N\S.
The contradiction establishes that N\Sis empty. Hence S=N.
Part II. We show the mathematical induction implies the well-ordering
principle.
Let S(n) be the statement: Any set of natural numbers containing
a natural number ≤nhas a least element. Consider the set
E={m∈N:S(m) is true}.
1∈Ebecause 1 is the least element of N(why?).
We next show m∈Eimplies m0∈E. Now m∈Emeans if Xis
a subset of Ncontaining a natural number ≤m, then Xhas a least
element. From this we want to establish m0∈E.
So let Cbe any subset of Ncontaining a natural number ≤m0. If C
has no element < m0, then m0is the least element of Cand we are done.
Otherwise, we can now suppose there is a natural number y∈Csuch
that y < m0. In particular, y≤mbecause by one of the homework 3
problems we know there is no natural number strictly between mand
m0. Therefore, Cnow has an element y≤m, so that the induction
hypothesis given in the preceding paragraph implies that Chas a least
element. In any event, we have proven Chas a least element, so that
m0∈E. Hence, the mathematical induction implies E=N.
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