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 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 = i∅ 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 μ(μ(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 γ.(α+β) = γ.α+γ.β. See the solutions (PDF file).
• Page 54, Exercise 2.12: Should be Exercise 1.17, not 1.16.
• Page 61, line 2: x(σ) = T should read v(σ) = T. (Spotted by Kirk Sturtz.)
• Page 85, bulleted list: the formulae equivalent to φ∧ψ and φ∨ψ 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∩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 α, since α 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(β)×s(β)| ≥ α > |s(β)|.
• Page 129, last line: assuming ZFC is consistent. (Spotted by Katrin Tent.)

Please mail misprint reports to pjc20(at)st-andrews.ac.uk.

Peter J. Cameron
20 September 2019.