# Errata

From Peano's Parlour

Errata for Kossak, Roman; Schmerl, James H. The structure of models of Peano arithmetic. Oxford Logic Guides, 50. Oxford Science Publications. The Clarendon Press, Oxford University Press, Oxford, 2006.

## Corrections

p. vii, l. -3: Should be: ``type $\omega+(\omega^{*}+\omega)\rho$..."

p. 21, Exercise 1.14.5: $b\in M$ should be $c\in M$

p. 22, l. -12: [70] should be [71]

p. 23, l. 19: [209] refers to a paper of A. Wilkie, it should refer to George Wilmer's thesis which is missing in the references.

p. 48, l. 3: the second instance of 2.3.2 should be 2.3.4.

p. 57, Proof of Corollary 3.1.17. Should be ``Theorem 3.1.16..." To finish the argument one also needs to evoke Theorem 2.1.1.

p. 85, Exercise 3.6.17: Delete the hint.

p.98, l.-3: In the first line of the displayed formula, $x\leq x'$ should be replaced

by $x < x'$ . The same change should be made in the second line.

p.106, paragraph starting with l. -9: Replace the third sentence with: ``Let $A = M (D)$ be the set of meet-irreducibles." Then at the end of the paragraph replace the last part of the last sentence starting with ``. . . containing $0_D$. . ." with ``. . . containing $1_D$ , which is $\{x \in M (D) : r \leq x\}$."

p. 109, Theorem 4.5.5: Definition of $\alpha^n$ is missing. Before the theorem insert: ``If $\alpha: L\rightarrow {\rm Eq}(A)$ and $\beta: L\rightarrow {\rm Eq}(B)$ are representations, the product $\gamma=\alpha\beta$ is a function $\gamma:L\rightarrow {\rm Eq}(A\times B)$ defined by $((a_1,b_1),(a_2,b_2))\in \gamma(r)$ iff $(a_1,a_2)\in\alpha(r)$ and $(b_1,b_2)\in \beta(r)$. Then $\alpha^1=\alpha$ and $\alpha^{n+1}=\alpha^n\alpha$."

p.110/l.15: Should be: $A =\{1, 2, 3, 4\}$.

p.110/l.18: Delete “$f (5) = 1,$”. p. 118, l. 12: Near end of last line of the Theorem, delete the extraneous “) ”

p.128/Lemma 4.7.4: Interchange “$\alpha_1 : D_1\rightarrow {\rm Eq}(A_1 )$” and “$\alpha_2 : D_2\rightarrow {\rm Eq}(A_2 )$”. p.152/l.21: Expression at end of line should be: $F_n (x_0 , x_1 , \dots , x_{n-1} , u) = $.

p. 159, l. -1. $f:{\mathbb Q}\rightarrow{\mathbb P}$

p. 159. Lemma 6.2.5. Delete the last part on the last sentence starting with ``and if..."

p. 177, l. -12: $f(t_{1})$ should be $f(t_{2})$

p. 179, l. -12: [132] should be [130]

p. 179, l. -8: should [166] be J. Schmerl, Peano models with many generic classes. Pacific Journal of Mathematics 46, 523-536 (1973). This entry is missing in the references.

p. 182, Proposition 7.1.3: $I$ is supposed to be just a cut, but it should be also assumed to be closed under addition and

multiplication

p. 246. Lemma 9.4.3 (1). One has to assume that $I$ and $J$ are not infimum and supremum of the same gap.

p. 292, Question 17 is garbled. It should say: Suppose $M$ is countable recursively saturated and $X\in{\mathcal P}(M)\setminus{\rm Def}(M)$ is such that ${\rm Th}(M,X)\in {\rm SSy}(M)$. Is there a countable recursively saturated $N$ such that $M\prec_{end} N$, and if $Y \subseteq M$ is coded in $N$, then $(M,Y) \not\equiv (M,X)$?

## Typos

p. vi, l. 7: fragment = fragment of

p. vi, l. -17: While, = While

p. vi, l, -9: proves = and proves

p. vi, l. -2 f.b.: purpose = purpose of

p. vii, l. -9: delete ``and"

p. vii, l. -2 f: For every countable model $M$, the isomorphism type if its reducts...

p. viii, l. -14: the Chapter 7 = Chapter 7

p. 1, l. 3: delete ``a

p. 3, l. -11: is $B$ = $B$ is

p. 3, l. -4: $y\in M$ = $y\in X$

p. 8, l. -5: $\neg \Theta(y)$ = $\neg \theta(y)$

p. 14, Definition 1.9.1: ``partial inductive satisfaction classshould be in italics

p. 14, l. -9 and 7: instead of Con(Th($\frak{A},\bar{a})+\exists X\Psi(X,\bar{a})$) one should read ``Th($\frak{A},\bar{a})+\exists X\Psi(X,\bar{a})$ is consistent

p. 19, l. 14: proof the = proof of the

p. 24, l. 13: [69] . = [69].

p. 24, l. 17: Kotlarski and Kaye = Kaye and Kotlarski

p. 25, l. 1: models arithmetic = models of arithmetic

p.27, l.22: In Corollary 2.1.4, last word of first line should be “end”, not “and”.

p. 32, l. 7: the contrary = to the contrary

p. 33, l. -5: replaced = replaced by

p. 33, l. 4 f.b.: single = a single

p. 48, l. 15: Wikie = Wilkie

p. 49, l. -6: each of which = each element of which

p. 51, l. 7: $n_{i}$ should be $n$ (similarly for p. 51, l. 9)

p. 51, l. 17: unboounded = unbounded

p. 52, l. 21: infinte = infinite

p. 53, l. 2: type = types

p. 54, l. 6: do not look = do not look the same

p. 55, l. 11: the another = another

p. 57, l. 14: realizes = realize

p. 57, l. -10: Should be ``Theorem 3.1.16..."

p. 57, l. -7; Should be ``$a\in {\rm Scl}(b)$...

p. 58, l. 10: since $p(x)$, is rare = , since $p(x)$ is rare

p. 60, l. 13: gap(b)$\setminus$gap(b) = gap(b)

p. 60, all $M$'s in the lines -8,-7, and -6 should be $M_0$'s.

p. 61, l. 3 $s(x)$ should be $s'(x)$.

p. 61, l. -5 : The are = There are

p. 67, l. -18: types. = types,

p. 67, l. -17: Then = then

p. 80, l. 7: $F$ = $f$

p. 85, Exercise 3.6.25: are = There are

p. 87, Definition after 3.6.38 should be $M\prec N$, not $M\prec_{end} N$.

p. 100, l. 18: Should be $K=M_{1}(b)= M\star K$

p. 107, l. 3: Theorem 4.5.32 = Corollary 4.5.32

p. 107, l. -16: representation

p. 107, l. -3: $r$ should be $k$.

p. 108, l. 23: be list all = be a list of all

p. 109. l. 4: Should be ``The inclusion $M_{i\land j}\subseteq M_i\cap M_j$..."

p. 122, l. -15: used get = used to get

p. 124, l. 14: hold for = hold for

p. 125, l. -17: the tree = on the tree

p. 128, l. -9: representations

p. 131, l. 3: (see Exercise 4.8.2)

p. 133, l. 8: characterization

p. 135, l. 7: introduced

p. 142, l. 1: there some = there exists some

p. 145, l. 2: instead = instead of ?

p. 151, l. 17: is not be = is not

p. 152, l. 17: off = of

p. 156, l. -18: linearly set = linearly ordered set

p. 157, l. 17: Theorems 5.3.4 = Theorem 5.3.4

p. 162, l. 10,12: Theorem 6.2.6 = Lemma 6.2.6

p. 166, l. -10: functions

p. 168, l. -6: would no = would be no

p. 169, second line of Theorem 6.4.3: $(M,X)_{X\in{\cal G}}$

p. 171, l. 10: the proof = of the proof

p. 173. First line of Theorem 6.4.8: $|M|\leq \kappa$.

p. 177, l. -14: possibilities

p. 177, l. -5: not = are not

p. 179, l. -14: independently

p. 184, l. -4: Propositional = Proposition

p. 189, l. -11: Theorem 2.2.8 = Theorem 2.2.16

p. 191, l.-8: Should be $b\in T^K$

p. 192, l. 8: if = of

p. 194, l. -6: Theorembut = Theorem but

p. 197, l. 12: model IA = model of IA

p. 200, l. -10: which = which is

p. 201, l. -12: models = models of

p. 203, l. 10: $ a_{\frak{A}} = \frak{a}_{\frak{A}}$

p. 206, l. 10: model = models

p. 209, Fourth line of the proof of Corollary 8.4.6: $b_1,b_2=b_0,b_1$

p. 214, l. -17: than least = than the least

p. 221, l. -12: delete ``a

p. 222, l. 5: Corollary 3.2.4 = Lemma 3.2.4

p. 225, l. 7: maximal = a maximal

p. 227, l. 12: Corollary 8.1.2 = Proposition 8.1.3. Delete the statement in parenthesis.

p. 227, l. -10: Back-and-forth, = Back-and-forth

p. 228, l. 20 f.b.: Frederike = Friederike

p. 229, l. 9: proof the = proof of the

p. 229, l. -14: the question mark appears ``upside down

p. 233, l. 12: aid = and

p. 234, l. 8: Proposition 9.1.3 = Lemma 9.1.3

p. 235, l. 2: models = model

p. 239, l. 13: index if = index of

p. 250, l. -6 f: realizes = realize

p. 253, l. -7: of countable = of a countable

p. 254, l. 14: definition

p. 267, l. -12: devoted the = devoted to the

p. 268, l. 6: delete ``an

p. 276, l. -1: Use previous = Use the previous

p. 280, l. 11: theory = theory of

p. 284, l. 5: is get = is to get

Reference [36]: G\"{o}tenborg = G\"{o}teborg

Reference [43]: add ``{\it Mathematical Logic and Foundations of Set Theory} (Proc. Internat. Colloq., Jerusalem, 1968)

Reference [54]: add ``Volume 1292 of {\it Lecture Notes in Mathematics}

Reference [83]: of 619 = 619 of

Reference [85]: delete ``(1983); this reference should appear after reference [88]

Reference [109]: od = of

Reference [113]: of pa = of ${\rm PA}$

Reference [120]: add `` in Automorphisms of first-order structures, R. Kaye, D. Macpherson (eds.)

Reference [148]: {\bf CIII} = {\bf 103}

Reference [153]: characterization

Reference [164]: add ``Volume 859 of {\it Lecture Notes in Mathematics}

Reference [167]: add ``Stud. Logic Found. Math., 120

Reference [172]: add ``Lecture Notes Logic, 12

Reference [188]: Unmglichkeit = Unm\"{o}glichkeit; vollstndigen = vollst\"{a}ndigen

Reference [189]: abzhlbar = abz\"{a}hlbar; Fundam. = Fund.

Reference [199]: add ``Volume 834 of {\it Lecture Notes in Mathematics}

Reference [204]: poljak-r\"{o}dl = Poljak-R\"{o}dl

Reference [206]: the name of the journal is usually abbreviated as ``Algebra Logic Appl.

Reference [208]: complete = complete models

Reference [212]: add ``Proceedings of the International Congress of Mathematicians