- 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
*x*_{1}=*y*_{1}and*x*_{2}=*y*_{2}. (Spotted by Christopher Deeks.) - Page 12, line 13: for all
*distinct**p,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*(that is, the last symbol before the closing bracket is^{-1})g)*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*g*_{1}and*g*_{2}, not*h*_{1}and*h*_{2}. (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
*x*_{n+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:
*f*_{0}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*(Spotted by Sheila Willliams). Also, Chow Ka Fat points out that line -7 should say |_{(u,v)}*s*(beta) x*s*(beta) >= alpha > |*s*(beta)|. - Page 129, last line: assuming ZFC is consistent. (Spotted by Katrin Tent.)

Peter J. Cameron

17 March 2002.