Page 341 - HOW TO PROVE IT: A Structured Approach, Second Edition

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The Cantor–Schr¨ oder–Bernstein Theorem 327
(1862–1943) gave a famous lecture in which he listed what he believed to be
the most important unsolved mathematical problems of the time, and the proof
or disproof of the continuum hypothesis was number one on his list.
The status of the continuum hypothesis was “resolved” in a remarkable way
by the work of Kurt G¨odel (1906–1978) in 1939 and Paul Cohen (1934– ) in
1963. The resolution turns out to require even more careful analyses than we
have given in this book of both the notion of proof and the basic assumptions
underlying set theory. Once such analyses have been given, it is possible to
prove theorems about what can be proven and what cannot be proven. What
G¨odel and Cohen proved was that, using the methods of mathematical proof
and set-theoretic assumptions accepted by most mathematicians today, it is
impossible to prove the continuum hypothesis, and it is also impossible to
disprove it!

Exercises

1. Prove that is reflexive and transitive. In other words:
∗
(a) For every set A, A A.
(b) For all sets A, B, and C,if A B and B C then A C.
2. Prove that ≺ is irreflexive and transitive. In other words:
(a) For every set A, A ≺ A.
(b) For all sets A, B, and C,if A ≺ B and B ≺ C then A ≺ C.
3. Suppose A ⊆ B ⊆ C and A ∼ C. Prove that B ∼ C.
4. Suppose A B and C D.
(a) Prove that A × C B × D.
(b) ProvethatifAandC aredisjointandBandDaredisjoint,then A ∪ C
B ∪ D.
(c) Prove that P (A) P (B).
5. For the meaning of the notation used in this exercise, see exercise 21 of
∗
Section 7.1. Suppose A B and C D.
A
B
(a) Prove that if A = ∅ then C D.
(b) Is the assumption that A = ∅ needed in part (a)?
6. (a) Prove that if A B and B is finite, then A is finite and |A|≤|B|.
(b) Prove that if A ≺ B and B is finite, then A is finite and |A| < |B|.
7. Prove that for every set A, A ≺ P (A). (Hint: See exercise 4 of Sec-
tion 7.2. Note that in particular, if A is finite and |A|= n then, by exercise
n
n
10 of Section 6.2, |P (A)|= 2 . It follows, by exercise 6(b), that 2 > n.
Of course, you already proved this, by a different method, in exercise 12(a)
of Section 6.3.)