P 10, beginning of second para. A `$Dz$' has crept into the deinition of transitivity.

page 13, exercise 1 part (iv): ``R intersection R^{-1} = 1'' should of course be ``R intersection R^{-1} \subseteq 1''. (The empty relation is antisymmetric!) Thanks to Kostas Tsaprounis

Page 13 para 4: "with a partial ordering" should be followed by the non-strict symbol.

P 13. Drop the footnote

p.21. I omitted three cases (there are of course eight not five). Can you see what the missing ones are? (It doesn't make much difference to the discussion actually!)

Page 17 penultimate para ``preorder'' not yet defined.

p. 26, second line after the formula in the middle of the page: `is wellfounded' should be `has the magic ingredient'.

p 29. first para of proof, second line: should be *nonempty* subset

p. 30, last line of first paragraph. `in' should be an ε.

p.32. A LaTeXed message from Chad Brown:

One problem is
clear so I'll just point it out: It seems that each $f_x$ has domain
$\{y|*R y x\}$. If that is true, and you take the union of the $f_x$'s,
then the domain of the resulting function will not be all of $X$ in general.
Simple example:

Let $X$ be a singleton $\{*\}$ and $R$ be the empty relation. $f_*$
will be the empty function and $f$ (the union of $f_*$) will also be
the empty function.

It seems easy enough to get around this by defining $f z$ to be something like
$g(z,(\bigcup_{x\in X} f_x)``\{y|*R (y z)\})$. That is, do one more '$g$' step at the end.

p 35, line 3. Th `n' and the `m' should be swapped round.

(thanks to Jeff Dalton)

p. 36. There is a passage missing, which should be inserted
after the full stop in the middle of line 11, in which it is
explained, dually, how any even position all of whose children are
labelled `II' can be labelled `II' and how an odd position even one
of whose children is labelled `II' can be labeled `II'/.

(thanks to Jeff Dalton)

p. 37. The first word on the page should of course be `natural' not `integer'.

Page 41, para after exercise 13, penultimate line: should be "circumstances"

Page 41. Following paragraph: should be ""with the aid of theorem 8 in section 3.1.1"

Page 42, para 2, line 4: should be "real numbers" (plural)

bottom p 43. The start state is not designated in the diagram. It is in fact the state with a smiley on it.

p. 47: in standard parlance a literal is a propositional letter or the negation of a propositional letter (not just a propositional letter).

bottom p. 48. My definition of *signature* is at odds with the rest of the literature. The usual definition includes within the signature the symbols for the constants and relations. Thus the signature for ring theory not only tells you you have two symbols for binary operations and two constant symbols but actually tells you what those symbols are. On this standard account an alphabetic variant of a signature is a different signature. Thanks to Peter Johnstone and also to Andrew Pitts, who pointed out that what I had in mind is the idea of *Logical Framework* which the Edinburgh crowd developed. It certainly seems odd to make signatures so sensitive to alphabetic variation. Humph

p. 48, last para: should be "algebraically"

p 50 `horn' should be capitalised.

p 52. penultimate line of proof: the lower case `a' before `since' should of course be upper case.

p 52. first line of last para: ``increasing'' should of course be ``order-preserving''.

p. 52, sec 3.1, line 5, the minus sign should be a plus sign.

p. 57 first line should be ``that this sup is itself fixed ''

p. 56. The last letter in the displayed formula in par (ii) of exercise 21 should be `A' not `X'. (thanks to Mat Henderson.)

p 59 second para line 3: `continuous lattice' should be `complete lattice'

On line 10 the bound variable in the set abstract should of course be *n* not *x*.

p 67. l 10. It's exercise 24 in section 3.1.3.

p. 69, Exercise 36 ii): `subset' should be `superset'.

p. 70 para 2 penultimate line: "literals" should be "letters". Likewise para 4 line 5.

p. 76, line 5: The second "substitution" lacks a "t".

p. 82 last para. capitalise `no'..

p. 92 line -3. It should be ``and P are in Gamma'', not ``and Q are in Gamma''

p 107 the Quine citation should be 1962 not 1961.

(thanks to Jeff Dalton)

p 112, penultimate line: the quantifier prefix of the second formula is wrong.

p 120. Third line of the proof of theorem 34. The second expression should be `L(T)' where the `L' is in calligraphic font. (God knows how that happened!)

p 122. There is an exercise missing. Prove that the property of being a simple group is not first-order.

p 123. There is an intrusive `m' in the formula nine lines from the bottom.

p 126. l 5 should of course be: function in intension).

p 127 This displayed product on line 7 is
of course a definition of bounded *universal* quantification, and
it is bounded *existential* quantification one obtains by
duality.

(thanks to Jeff Dalton)

p 134. The pseudocode is wrong: the second line should end `ASCII of h'

p136, the pseudocode should be: register_0(last(T(m, i, (μt)(current_instruction(last(T(m, i, t))) = HALT))))

p 140 defn 46 should be λ *x*. if *x*
ε A then 1 else 0

p 148. Second line of first para of 7.1: the word `be' is missing from near the end of the line.

p 150. Exercise 107. Insert ``in section'' after ``exercise 4'' (i.e., between that and ``3.3.2.1")

p 159. 2nd line after Ex 113. I meant: ...all OTHER infinite ordinals are singular...

Page 169, penultimate line of penultimate paragraph. P(y) should be P(x).

p. 177, paragraph for (iii) line 2: "so it must be": the "it" should really be `x'.

p. 171, line 6. Insert `years' after `100'.

p 184. Section 8.6.3.1, second para, first line. The n after the Pi should be a subscript not a superscript

Page 187, line 8: "usually".

Page 187, proof of corollary, line 2: should be "is a model". No subjunctive! line 5: the subscript should be alpha not omega.

Page 188, penultimate paragraph, last (long) sentence: wonky grammar. Suggest: "for games....Borel, there is..."

p 189. seven lines from the bottom: `A' and `B' should be exchanged in the definition of exponentiation.

p 190. Third line of the second paragraph of the proof: β' > γ should be β' > γ'. Similarly at the start of the second line of the next paragraph β' < γ should be β' < γ'.

p 193 four lines from the bottom. `boud' should be `bound'

Page 194, line above Remark 94: No apostrophe on Hartogs, or else add noun

p 199 line 2. I use the expression `strong limit' of a cardinal without explaining it. A cardinal α is strong limit if, for all β < α, 2-to-the-β < α.

p 199 In fact we do not need any AC to prove the existence of *H _{κ}*.
Nor do we need the full strength of the axiom of foundation: all we need is
Coret's axiom that every set is the same size as a wellfounded set. Small
prize for any student who can see how to do it. The proof is actually quite
hard, so I'll give the reader a hint: you need Scott's trick and a trick used
by Jech in his 1982 JSL article. If there is ever a second edition of Logic
Induction and Sets I might supply a proof.

p 202 Last line of the statement of Lemma 105. `NF' should be `ZF'. Also the expression *setlike* is not defined. A permutation is setlike if *j* of it is well-defined (defined on all arguments) and setlike...

p 202. Bottom. This is not so much a typo as an infelicity. I blithely assert that the fact that all stratified axioms are preserved ensures that the axiom of infinity holds. This is in fact correct, but it is not obvious. Many people think that the axiom of infinity is the assertion in the penultimate paragraph of p 180. And that isn't stratified! But it's easy to show in ZF that this axiom is equivalent to the other version that says that there is a set the same size as one of its proper subsets: a Dedekind-infinite set (And that assertion *is* stratified). I think in a second edition i will make this explicit, and add it an an exercise. One direction is easy. For the other direction, first show that if there is a Dedekind-infinite set then there is a countably infinite set, and then we obtain the von Neumann ω by replacement.

p 205. Least rational in every sock? This is an extreme ellipsis. What I was thinking was ``least rational in the sense of whatever canonical wellordering of the rationals we are deciding to use''. But the allegation is correct.

p 211. I think a tilde is missing in the URL

p. 211 Exercise 13 should be on p. 212 after Exercise 12

page 213. The `(iv)' refers to an exercise on p 20.

(thanks to Jeff Dalton)

Page 215, the answer to Exercise 41 is associated with Chapter 3, but should be Chapter 4.

Page 216, answer to Exercise 44, penultimate displayed expression should be prefixed by a turnstile.

third line should be decorated with`(0,5, MP)'

p 217 start of penultimate para. Not the bottom right-hand corner but the four corners.

p 219 part (vi). It's exercise 3 not 33. (It's in chapter 3)

p 312. line 1. The union sign should be a sup.

p 223. The first word of line 3 should of course be `decidable'. I was trying to consistently write `decidable' set instead of `recursive' set but this one slipped through the superego censor. *Mea culpa*

p 225, second line of answer to Exercise 90. `computable' should be `decidable'.

p.231. The title of the Bunder article is missing. It is ``The Logic of Inconsistency''

p236. The definition in the answer to exercise 108 is defective. There is a recursive definition, but it is not very smooth, so the exercise should be deleted.

Last updated: 26/iv/2010. Grovelling thanks to many contributors, particularly David Makinson.