# Errata

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: represenatation


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: represenations


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


p. 133, l. 8: charaterization


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: funtcions


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]: charaterization


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