MATH 203, PROBLEM SET 4
DUE IN ASHER AUEL’S MAILBOX BY 5 P.M., FRIDAY , FEB. 17.
Note: The mid-term exam will be given in class on Feb. 6.
1. Suppose (U, ≤1) and (W, ≤2) are well-ordered sets. We will say (W, ≤2)≤(U, ≤1)
if there is an order preserving injection f:W→U. These sets have the same order
type if we can find such an fwhich is bijective. Suppose that (U, ≤1) and (W, ≤2)
are both equal the ordered set Z≥0={0,1,2, ...}of all positive integers with the
usual ordering. Show that there are uncountably many different order preserving
injections f:W→U, but that there is just one order preserving bijection.
Hints: To show there’s just one bijection f, show by induction on nthat f(n) = n
for all n. For the other part of the problem, show that there’s a bijection between the
set Sof order preserving injections f:W→Uand the set Tof infinite ascending
sequences a1< a2< a3<· · · of elements of Z≥0. How can you adapt Cantor’s
diagonal argument to show that Tis not countable?
2. Von Neumann’s theorem says that there is a unique way to assign to each well-
ordered set (U, ≤1) an ordered set (v(U),≤) together with an order preserving bi-
jection vU:U→v(U) such that for all y∈U,
vU(y) = {vU(x) : x∈U, x <1y}.
Here x <1ymeans that x≤1yand x6=y. In this case, we call (v(U),≤) an ordinal
number. Show that for q, y ∈v(U) we have
q < y if and only if q∈y.
Hints: If q, y ∈v(U), show there are unique elements q0, y0∈Usuch that q=vU(q0)
and y=vU(y0). Then show q < y if and only if q0<1y0, and use the definition of
vU(q0) and vU(y0).
3. The principle of transfinite induction says that if (W, ≤) is a non-empty well-ordered
set, and P(w) is a property of an element wof W, then P(w) is true for all w∈W
if the following condition is satisfied:
* For all elements yof W, if P(w) is true for all w < y, then P(y) is true.
a. Show that this principle follows from the fact that Wis well-ordered.
Hint: If P(w) is not true for some w∈W, argue that there must be a smallest
y∈Wfor which P(y) is false.
b. We can prove a statement Q(n) about positive integers nby showing that Q(1)
is true, and by then showing that if Q(n) is true for some n≥1 then Q(n+ 1)
is true. Notice that in this approach we had to show that Q(1) is true. In the
principle of transfinite induction above, we didn’t have to show P(w0) is true
for the smallest element w0of W. Is this a contradiction? Explain!
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