Sets, Logic and Categories: Misprints and further comments
Many thanks to those who have contributed to this list.
Page 4, Proof of Theorem 1.1: Yair Aviv suggests that the argument
here could be made clearer by writing it as follows:
For every set y, y is in S iff y is not in y.
In particular, when y is S,
S is in S iff S is not in S.
Page 7, Section 1.3, third line: should read
x1=y1 and
x2=y2. (Spotted by Christopher Deeks.)
Page 12, line 13: for all distinctp,q... (Spotted
by Sheila Williams.)
Page 18, line 15: of x; that is, lower-case x.
(Spotted by Christopher Deeks.)
Page 18, Proof of Lemma 1.1: the case Z = Emptyset has to be
considered separately. (Spotted by Katrin Tent.)
Page 19, line -9: x = g(y). (Spotted by Katrin Tent.)
Page 27, line 19: better "and so Q is at most countable. But Q is
infinite, so it is countable." (Spotted by Subhashis Mohanty.)
Page 28, Section 1.8: Kronecker really said "God made the integers ..."
(though he probably meant the positive integers). (Spotted
by Sheila Williams.) I am in good company here: Peter Høeg, in
Stories of the Night, page 2, makes the same mistake.
In the same vein but less pompously, E. Borel said,
"All of mathematics can be deduced from the sole notion of an integer;
here we have a fact universally acknowledged today."
Pages 29-30, definition of the rational numbers is confused. Sheila
Williams says, "You've changed horses in midstream in your definition of the
rational numbers. I think the easiest solution is to interchange a
and b, and c and d, in Page 29 line -6, and change
a to b in line -4 of the same page."
Page 33, Exercise 1.5(a): the second formula should be
mu(mu(x,g-1)g) (that is, the last symbol before the
closing bracket is g, not x).
Page 34, Exercise 1.9: "injective" and "surjective" should not be
reversed, despite what was said here before. Also, the two functions in
part (b) should be g1 and g2, not
h1 and h2. (Thanks to Subhashis Mohanty
for clearing up the confusion.)
Page 35, line -3: delete "zemph". (Spotted by Sheila Williams.)
Page 42, Lemma 2.6, should read "... then Y is a section of
X" (Spotted by Matthew Lewis.)
Page 53, Exercise 2.6(b): should read gamma.(alpha+beta)=
gamma.alpha+gamma.beta. See the solutions (PDF file).
Page 54, Exercise 2.12: Should be Exercise 1.17, not 1.16.
Page 61, line 2: x(sigma)=T should read v(sigma)=T.
(Spotted by Kirk Sturtz.)
Page 85, bulleted list: the formulae equivalent to phi and psi
and phi or psi should be swapped. (Spotted by Matthew Lewis.)
Page 86, line 3: first word should be "those" rather than "thoe".
(Spotted by Matthew Lewis.)
Page 91, 2nd line of Step 2: for "Compactness" read "Completeness".
(Spotted by Chiaka.)
Page 103: Katrin Tent points out that the statement here that Peano
arithmetic has models appears to conflict with the qualification "If Peano
arithmetic is consistent" in Theorems 5.8 and 5.9. Exercise: what
is going on here?
Page 106, line 11: (A5) should be (A4). (Pointed out by Chuks Kamalu).
Page 116, line 17 should read "...and so xn+1
in y intersection x, contradicting Foundation".
(Spotted by Matthew Lewis.)
Page 116, line 20: "for all n" (not x).
(Spotted by Matthew Lewis.)
Page 119, proof that AC implies ZL: there is no need to assume that
the partially ordered set is non-empty. The empty poset contains the
empty chain which has no upper bound! (Spotted by Katrin Tent.)
At the end of this proof: of course, it is not the set but
the class of all ordinals - this is the point of the contradiction.
(Spotted by Katrin Tent.)
Page 120, line 3: f0 is a bijection.
(Spotted by Katrin Tent.)
Page 127, line 16: If x and y are both finite, it is
not true that
|x|+|y|=max(|x|,|y|),
but it is still true that x+y is less than alpha,
since alpha is infinite. (Spotted by Katrin Tent.)
Page 127 line -11: the section P(u,v) (Spotted by
Sheila Willliams). Also, Chow Ka Fat points out that line -7 should say
|s(beta) x s(beta) >= alpha > |s(beta)|.
Page 129, last line: assuming ZFC is consistent.
(Spotted by Katrin Tent.)