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326 Infinite Sets
Thus, j − 1 is a natural number, but since j is the smallest element of S, j − 1 /∈
j−1
S. It follows that m + k ≤ x.
j
Let q = m + . Clearly q is a rational number, and since j ∈ S, q =
k
j j−1 1
m + > x. Also, combining the observations that m + ≤ x and <
k k k
y − x,wehave
j j − 1 1
q = m + = m + + < x + (y − x) = y.
k k k
Thus, we have x < q < y, as required.

Proof of Theorem 7.3.3. As we observed earlier, we already know that
+
P (Z ) R. But now consider the function f : R → P (Q) defined as fol-
lows:
f (x) ={q ∈ Q | q < x}.

We claim that f is one-to-one. To see why, suppose x ∈ R, y ∈ R, and x = y.
Then either x < y or y < x. Suppose first that x < y. By Lemma 7.3.4, we
can choose a rational number q such that x < q < y. But then q ∈ f (y) and
q /∈ f (x), so f (x) = f (y). A similar argument shows that if y < x then f (x) =
f (y), so f is one-to-one.
Since f is one-to-one, we have shown that R P (Q). But we also know that
+
+
Q ∼ Z , so by exercise 5 in Section 7.1 it follows that P (Q) ∼ P (Z ). Thus,
+
R P (Q) P (Z ), so by transitivity of we have R P (Z ). Combin-
+
ing this with the fact that P (Z ) R and applying the Cantor–Schr¨oder–
+
Bernstein theorem, we conclude that R ∼ P (Z ).
+
We said at the beginning of this chapter that we would show that infinity
comes in different sizes. We now see that, so far, we have found only two
sizes of infinity. One size is represented by the denumerable sets, which are all
equinumerous with each other. The only examples of nondenumerable infinite
sets we have given so far are P (Z ) and R, which we now know are equinu-
+
merous. In fact, there are many more sizes of infinity. For example, P (R)is
an infinite set that is neither denumerable nor equinumerous with R. Thus, it
represents a third size of infinity. For more on this see exercise 8.
Because Z ≺ R, it is natural to think of the set of real numbers as larger
+
than the set of positive integers. In 1878, Cantor asked whether there was a
size of infinity between these two sizes. More precisely, is there a set X such
that Z ≺ X ≺ R? Cantor conjectured that the answer was no, but he was
+
unable to prove it. His conjecture is known as the continuum hypothesis.At
the Second International Congress of Mathematicians in 1900, David Hilbert