Three Important Facts About 
There are three important statements about the set of natural numbers, , that turn out to be
equivalent to each other. As mentioned in class, minor variations of each one are also possible.
I. PMI (Principle of Mathematical Induction) Suppose W © Þ
If
then
a) " − W , and
b) a8 Ð8 − W Ê 8  " − WÑ
Wœ
II. PCI (Principle of Complete Induction) Suppose W © Þ
If
then
a8 ( Ö5 À 5  8× © W Ê 8 − W× Ñ
Wœ
ЇÑ
(Note: In class, I stated PCI as follows:
If
then
i) " − Wß and
ii) a8  " Ð Ö5 À 5  8× © W Ê 8 − W Ñ
Wœ
When 8 œ ", Ö5 À 5  "× œ g and, of course, g © W is true. So, if the conditional
statement Ð‡Ñ is true, it follows that " − WÞ There's no logical requirement to write separately (as
I did in class) that you need to check " − W when using CPI; writing that is redundant.
But when you use CPI to prove a specific theorem Ða8ÑT Ð8Ñ, you do need to verify that
Ð‡Ñ is true for all 8 (including 8 œ ")Þ And to do that, the case 8 œ " is often special: the reason
is that since Ö5 À 5  "× œ g, you have no “previously proved propositions T Ð5Ñ for 5  "” to
use in proving T Ð"Ñ, so T Ð"Ñ must be checked “from scratch.”Ñ. In practice, you often need to
start out by checking the 8 œ " case separately anyway.
To summarize: Listing “1 − W ” (as in class) as part of the statement of PCI is
technically redundant, but the redundancy is harmless and is often a useful reminder not to
forget about the case 8 œ ".)
III. WOP (Well-Ordering Principle) Suppose E © 
If E Á g
then E contains a smallest element.
For now, these three statements are treated as intuitively clear statements about  which can used
to prove more complicated facts. Later, we will see later that, in fact, that they are true about 
because they are “built into ” when  is constructed from set theory.
Even though PMI, PCI, and WOP are equivalent, sometimes one of them is more convenient to
use than the others. We have seen examples using each of them.
Perhaps you feel that WOP is easier to believe than the others. But (to repeat), PMI, PCI and
WOP are equivalent statements about : you must believe all three of the statements, or none of
them.
To prove that PMI, PCI, and WOP are equivalent, we will give three arguments: :
PMI Ê PCI
PCI Ê WOP
WOP Ê PMI.
Each proof follows on a separate page.
Prove PMI Ê PCI
Suppose W ©  and assume that PMI is true. We want to prove PCI:
If
then
a8 ( Ö5 À 5  8× © W Ê 8 − W× Ñ
Wœ
ЇÑ
To do this, assume Ð‡Ñ À
a8 ( Ö5 À 5  8× © W Ê 8 − W×ÑÞ
We need to show that W œ .
(Strategy: We will show that for every 8, Ö"ß #ß ÞÞÞß 8× © W . If that is true, then 8 − W
for every 8, which tells us that  © WÞ Since we already know W © ß we will conclude
that W œ .
Let's carry out the strategy.)
a) Since Ö5 À 5  "× œ g © W , (*) gives that " − W , so Ö"× © WÞ
b) Assume that for some 8, we have Ö"ß #ß ÞÞÞß 8× © W  that is, assume
Ö5 À 5  8  "× © WÞ By (*), we conclude that 8  " − WÞ Therefore Ö"ß #ß ÞÞÞß 8ß 8  "× © WÞ
Summarizing so far: " − W +n. Ð if Ö"ß #ß ÞÞÞß 8× © W , then Ö"ß #ß ÞÞÞß 8  "× © W Ñ.
Using our assumption that PMI is true, we conclude that for a8 Ö"ß #ß ÞÞÞß 8× © W . Therefore
 © W , so (as outlined in the strategy), W œ . ñ
Prove PCI Ê WOP
Assume PCI is true. We want to prove WOP:
Suppose E © Þ
If E Á g,
then E contains a smallest element.
(Strategy: We will the contrapositive of WOP instead. We assume that E contains no
smallest element, and prove that E œ gÞ
Let's carry out the strategy.)
Assume E contains no smallest element, and let W œ   EÞ
ÐÖ5 À 5  "×Ñ Ê " − W . ÐSince E has no smallest element, "  E and therefore " − WÞ Since
" − W is true, the conditional statement is true.)
Suppose, for some 8  " , Ö5 À 5  8× œ Ö"ß #ß ÞÞÞß 8  "× © WÞ Then "ß #ß ÞÞÞß 8  " are not in
E. Therefore 8 Â E (because 8 would be the smallest element in E). Therefore 8 − WÞ
Summarizing so far: If Ö5 À 5  8× © W , then 8 − WÞ
Using PCI, we conclude W œ . But W œ   Eß so therefore E œ g.
(See the strategy) ñ
Prove WOP Ê PMI
Assume that WOP is true. We want to prove PMI:
Suppose W © .
If
then
" − W and a8Ð8 − W Ê 8  " − WÑ
W œ Þ
(Strategy: We will prove an equivalent statement: PMI is equivalent to
If " − W and W Á , then µ Ða8ÑÐ8 − W Ê 8  " − WÑ
which, in turn, is equivalent to
If " − W and W Á , then Ðb8Ñ Ð8 − W • 8  " Â WÑ
We are using, from logic, that
T •UÊV
is equivalent to
T • µ V Ê µ UÞ
Let's carry out the strategy.)
Suppose " − W and W Á Þ Then   W Á g, so by WOP there is a smallest natural number, call
it 5 , in   W .
We know 5 Á " Ðsince 1 − W ) so 8 œ 5  " − . Then 8 − W (since 5 was the smallest natural
number not in W )Þ Therefore 8 − W but 8  " œ 5  W . (See the strategy). ñ