Difference between revisions of "Errata"

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(Created page with "\documentclass[10pt]{article} \usepackage[centertags]{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsthm} \title{The Structure of Models of Peano Arithmetic:...")
 
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\documentclass[10pt]{article}
 
  
\usepackage[centertags]{amsmath}
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== Corrections ==
\usepackage{amsfonts}
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\usepackage{amssymb}
+
\usepackage{amsthm}
+
\title{The Structure of Models of Peano Arithmetic: Errata}
+
  
\begin{document}
 
\maketitle
 
  
\begin{itemize}
+
p. vii, l. -3: Should be: ``type $\omega+(\omega^{*}+\omega)\rho$..."
  
\section{Corrections}
+
p. 21, Exercise 1.14.5: $b\in M$ should be $c\in M$
  
\item p. vii, l. -3: Should be: ``type $\omega+(\omega^{*}+\omega)\rho$..."
+
p. 22, l. -12: [70] should be [71]
  
\item p. 21, Exercise 1.14.5: $b\in M$ should be $c\in M$
+
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.
  
\item p. 22, l. -12: [70] should be [71]
+
p. 48, l. 3: the second instance of 2.3.2 should be 2.3.4.
  
\item 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.
 
  
\item 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.
  
\item 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.
 
  
\item 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  
 
+
\item 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.
 
by $x < x'$ . The same change should be made in the second  line.
  
 
+
p.106, paragraph starting with l.  
\item 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 $\{x \in M (D) : r \leq x\}$."  
 
-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 $\{x \in M (D) : r \leq x\}$."  
  
\item 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. 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 $\alpha^{n+1}=\alpha^n\alpha$."
  
\item p.110/l.15: Should be: $A =  
+
p.110/l.15: Should be: $A =  
 
\{1, 2, 3, 4\}$.   
 
\{1, 2, 3, 4\}$.   
  
\item p.110/l.18: Delete “$f (5) = 1,$”.  
+
p.110/l.18: Delete “$f (5) = 1,$”.  
 
   
 
   
\item p. 118, l. 12:  Near end of last line of the Theorem, delete the extraneous “) ”
+
p. 118, l. 12:  Near end of last line of the Theorem, delete the extraneous “) ”
  
  \item p.128/Lemma 4.7.4: Interchange “$\alpha_1 : D_1  
+
  p.128/Lemma 4.7.4: Interchange “$\alpha_1 : D_1  
 
\rightarrow {\rm Eq}(A_1 )$” and “$\alpha_2 : D_2
 
\rightarrow {\rm Eq}(A_1 )$” and “$\alpha_2 : D_2
 
\rightarrow {\rm Eq}(A_2 )$”.  
 
\rightarrow {\rm Eq}(A_2 )$”.  
 
   
 
   
  
\item p.152/l.21: Expression at end of line should be: $F_n (x_0 , x_1 , \dots , x_{n-1} , u) = $.  
+
p.152/l.21: Expression at end of line should be: $F_n (x_0 , x_1 , \dots , x_{n-1} , u) = $.  
 
+
\item p. 159, l. -1. $f:{\mathbb Q}\rightarrow{\mathbb P}$
+
  
\item p. 159. Lemma 6.2.5. Delete the last part on the last sentence starting with ``and if..."
+
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..."
  
  
\item p. 177, l. -12: $f(t_{1})$ should be $f(t_{2})$  
+
p. 177, l. -12: $f(t_{1})$ should be $f(t_{2})$  
  
\item p. 179, l. -12: [132] should be [130]   
+
p. 179, l. -12: [132] should be [130]   
  
\item p. 179, l. -8: should [166] be J. Schmerl, Peano models with many generic classes.  
+
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.
 
Pacific Journal of  Mathematics 46, 523-536 (1973). This entry is missing in the references.
  
\item 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
+
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
 
multiplication
  
\item 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. 246. Lemma 9.4.3 (1). One has to assume that $I$ and $J$ are not  infimum and supremum of the same gap.
  
\item p. 292, Question 17 is garbled. It should say:  Suppose $M$  is  countable recursively saturated  and
+
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 $M\prec_{end} N$,
 
$X\in{\mathcal P}(M)\setminus{\rm Def}(M)$ 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
 
and if $Y \subseteq M$ is coded in $N$, then $(M,Y) \not\equiv
Line 79: Line 66:
  
  
\section{Typos}
+
== Typos ==
\item p. vi, l. 7: fragment = fragment of
+
  
\item p. vi, l. -17: While, = While
 
  
\item p. vi, l, -9: proves = and proves
+
p. vi, l. 7: fragment = fragment of
  
\item p. vi, l. -2 f.b.: purpose = purpose of
+
p. vi, l. -17: While, = While
  
\item p. vii, l. -9: delete ``and"
+
p. vi, l, -9: proves = and proves
  
 +
p. vi, l. -2 f.b.: purpose = purpose of
  
\item p. vii, l. -2 f: For every countable model $M$, the isomorphism type if its reducts...
+
p. vii, l. -9: delete ``and"
  
\item p. viii, l. -14: the Chapter 7 = Chapter 7
 
  
\item p. 1, l. 3: delete ``a''
+
p. vii, l. -2 f: For every countable model $M$, the isomorphism type if its reducts...
  
\item p. 3, l. -11: is $B$ = $B$ is
+
p. viii, l. -14: the Chapter 7 = Chapter 7
  
\item p. 3, l. -4: $y\in M$ = $y\in X$
+
p. 1, l. 3: delete ``a''
  
\item p. 8, l. -5: $\neg \Theta(y)$ = $\neg \theta(y)$  
+
p. 3, l. -11: is $B$ = $B$ is
  
\item p. 14, Definition 1.9.1: ``partial inductive satisfaction class''should be in italics
+
p. 3, l. -4: $y\in M$ = $y\in X$
  
\item p. 14, l. -9 and 7: instead of Con(Th($\frak{A},\bar{a})+\exists X\Psi(X,\bar{a})$) one should
+
p. 8, l. -5: $\neg \Theta(y)$ = $\neg \theta(y)$
 +
 
 +
p. 14, Definition 1.9.1: ``partial inductive satisfaction class''should 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''   
 
read ``Th($\frak{A},\bar{a})+\exists X\Psi(X,\bar{a})$ is consistent''   
  
\item p. 19, l. 14: proof the = proof of the  
+
p. 19, l. 14: proof the = proof of the  
  
  
  
\item p. 24, l. 13: [69] . = [69].  
+
p. 24, l. 13: [69] . = [69].  
  
\item p. 24, l. 17: Kotlarski and Kaye = Kaye and Kotlarski   
+
p. 24, l. 17: Kotlarski and Kaye = Kaye and Kotlarski   
  
\item p. 25, l. 1: models arithmetic = models of arithmetic  
+
p. 25, l. 1: models arithmetic = models of arithmetic  
  
\item p.27, l.22: In Corollary 2.1.4, last word of first line should be “end”, not  
+
p.27, l.22: In Corollary 2.1.4, last word of first line should be “end”, not  
 
“and”.   
 
“and”.   
  
  
\item p. 32, l. 7: the contrary = to the contrary  
+
p. 32, l. 7: the contrary = to the contrary  
  
\item p. 33, l. -5: replaced = replaced by  
+
p. 33, l. -5: replaced = replaced by  
  
\item p. 33, l. 4 f.b.: single = a single  
+
p. 33, l. 4 f.b.: single = a single  
  
 +
p. 48, l. 15: Wikie = Wilkie
  
\item p. 48, l. 15: Wikie = Wilkie
+
p. 49, l. -6: each of which = each element of which
  
\item 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)
  
\item p. 51, l. 7$n_{i}$  should be $n$ (similarly for p. 51, l. 9)
+
p. 51, l. 17: unboounded = unbounded  
  
\item p. 51, l. 17: unboounded = unbounded 
 
  
\item p. 52, l. 21: infinte  = infinite
+
p. 52, l. 21: infinte  = infinite
  
\item p. 53, l. 2: type = types  
+
p. 53, l. 2: type = types  
  
\item p. 54, l. 6: do not look = do not look the same  
+
p. 54, l. 6: do not look = do not look the same  
  
\item p. 55, l. 11: the another = another  
+
p. 55, l. 11: the another = another  
  
\item p. 57, l. 14: realizes = realize  
+
p. 57, l. 14: realizes = realize  
  
\item p. 57, l. -10: Should be ``Theorem 3.1.16..."
 
  
\item p. 57, l. -7; Should be ``$a\in {\rm Scl}(b)$...
+
p. 57, l. -10: Should be ``Theorem 3.1.16..."
  
\item p. 58, l. 10: since $p(x)$, is rare = , since $p(x)$ is rare
 
  
\item p. 60, l. 13: gap(b)$\setminus$gap(b) = gap(b)  
+
p. 57, l. -7; Should be ``$a\in {\rm Scl}(b)$...
  
\item p. 60, all $M$'s in the lines -8,-7, and -6 should be $M_0$'s.
 
  
\item p. 61, l. 3 $s(x)$ should be $s'(x)$.
+
p. 58, l. 10: since $p(x)$, is rare = , since $p(x)$ is rare
  
  
\item p. 61, l. -5 : The are = There are
+
p. 60, l. 13: gap(b)$\setminus$gap(b) = gap(b)
  
\item p. 67, l. -18: types. = types,
 
  
\item p. 67, l. -17: Then = then
+
p. 60, all $M$'s in the lines -8,-7, and -6 should be $M_0$'s.
  
\item p. 80, l. 7: $F$ = $f$
 
  
 +
p. 61, l. 3 $s(x)$ should be $s'(x)$.
  
\item p. 85, Exercise 3.6.25: are = There are
 
  
\item p. 87, Definition after 3.6.38  should be $M\prec N$, not $M\prec_{end} N$.
 
  
\item p. 100, l. 18: Should be $K=M_{1}(b)= M\star K$
+
p. 61, l. -5 : The are = There are
  
\item p. 107, l. 3: Theorem 4.5.32 = Corollary 4.5.32 
+
p. 67, l. -18: types. = types,
  
\item p. 107, l. -16: represenatation 
 
  
\item p. 107, l. -3: $r$ should be $k$.
+
p. 67, l. -17: Then = then
  
\item p. 108, l. 23: be list all = be a list of all
 
  
\item p. 109. l. 4: Should be ``The inclusion $M_{i\land j}\subseteq M_i\cap M_j$..."
+
p. 80, l. 7: $F$ = $f$  
  
\item p. 122, l. -15: used get = used to get 
 
  
\item p. 124, l. 14: hold for = hold for
+
p. 85, Exercise 3.6.25: are = There are
  
\item p. 125, l. -17: the tree = on the tree
+
p. 87, Definition after 3.6.38  should be $M\prec N$, not $M\prec_{end} N$.
  
\item p. 128, l. -9: represenations
+
p. 100, l. 18: Should be $K=M_{1}(b)= M\star K$
  
\item p. 131, l. 3: (see Exercise 4.8.2)
 
  
\item p. 133, l. 8: charaterization
+
p. 107, l. 3: Theorem 4.5.32 = Corollary 4.5.32 
  
\item p. 135, l. 7: introduced
 
  
\item p. 142, l. 1: there some = there exists some
+
p. 107, l. -16: represenatation 
  
\item p. 145, l. 2: instead = instead of ?
 
  
\item p. 151, l. 17: is not be = is not
+
p. 107, l. -3: $r$ should be $k$.
  
\item p. 152, l. 17: off = of
 
  
\item p. 156, l. -18: linearly set = linearly ordered set
+
p. 108, l. 23: be list all = be a list of all
  
\item p. 157, l. 17: Theorems 5.3.4 = Theorem 5.3.4
 
  
 +
p. 109. l. 4: Should be ``The inclusion $M_{i\land j}\subseteq M_i\cap M_j$..."
  
\item p. 162, l. 10,12: Theorem 6.2.6 = Lemma 6.2.6
 
  
\item p. 166, l. -10: funtcions
+
p. 122, l. -15: used get = used to get 
  
\item p. 168, l. -6: would no = would be no
 
  
\item p. 169, second line of Theorem 6.4.3: $(M,X)_{X\in{\cal G}}$
+
p. 124, l. 14: hold for = hold for
  
\item p. 171, l. 10: the proof = of the proof
 
  
\item p. 173. First line of Theorem 6.4.8: $|M|\leq \kappa$.
+
p. 125, l. -17: the tree = on the tree
  
  
\item p. 177, l. -14: possibilities
+
p. 128, l. -9: represenations
  
\item p. 177, l. -5: not = are not
 
  
\item p. 179, l. -14: independently
+
p. 131, l. 3: (see Exercise 4.8.2)
  
\item p. 184, l. -4: Propositional = Proposition
 
  
\item p. 189, l. -11: Theorem 2.2.8 = Theorem 2.2.16
+
p. 133, l. 8: charaterization
  
\item p. 191, l.-8: Should be $b\in T^K$
 
  
\item p. 192, l. 8: if = of 
+
p. 135, l. 7: introduced
  
\item p. 194, l. -6: Theorembut = Theorem
+
 
 +
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  
 
but  
  
\item p. 197, l. 12: model IA = model of IA
 
  
\item p. 200, l. -10: which = which is  
+
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
  
\item p. 201, l. -12: models = models of
 
  
\item p. 203, l. 10: $ a_{\frak{A}} = \frak{a}_{\frak{A}}$
+
p. 209, Fourth line of the proof of Corollary 8.4.6: $b_1,b_2=b_0,b_1$
  
\item p. 206, l. 10: model = models
+
p. 214, l. -17: than least = than the least
  
\item p. 209, Fourth line of the proof of Corollary 8.4.6: $b_1,b_2=b_0,b_1$
+
p. 221, l. -12: delete ``a''
  
\item p. 214, l. -17: than least = than the least
+
p. 222, l. 5: Corollary 3.2.4 = Lemma 3.2.4
  
\item p. 221, l. -12: delete ``a''
+
p. 225, l. 7: maximal = a maximal
  
\item p. 222, l. 5: Corollary 3.2.4 = Lemma 3.2.4
+
p. 227, l. 12: Corollary 8.1.2 = Proposition 8.1.3. Delete the statement in parenthesis.
  
\item p. 225, l. 7: maximal = a maximal
+
p. 227, l. -10: Back-and-forth, = Back-and-forth
  
\item p. 227, l. 12: Corollary 8.1.2 = Proposition 8.1.3. Delete the statement in parenthesis.
 
  
\item p. 227, l. -10: Back-and-forth, = Back-and-forth
+
p. 228, l. 20 f.b.: Frederike = Friederike
  
\item p. 228, l. 20 f.b.: Frederike = Friederike
+
p. 229, l. 9: proof the = proof of the
  
\item p. 229, l. 9: proof the = proof of the
 
  
\item p. 229, l. -14: the question mark
+
p. 229, l. -14: the question mark
 
appears ``upside down''  
 
appears ``upside down''  
  
\item p. 233, l. 12: aid = and   
+
p. 233, l. 12: aid = and   
  
\item p. 234, l. 8: Proposition 9.1.3 =
+
p. 234, l. 8: Proposition 9.1.3 =
 
Lemma 9.1.3   
 
Lemma 9.1.3   
  
\item p. 235, l. 2: models = model
 
  
\item p. 239, l. 13: index if = index of 
+
p. 235, l. 2: models = model
  
\item p. 250, l. -6 f: realizes = realize  
+
p. 239, l. 13: index if = index of  
  
\item p. 253, l. -7: of countable = of a countable  
+
p. 250, l. -6 f: realizes = realize  
  
\item p. 254, l. 14: definiton
+
p. 253, l. -7: of countable = of a countable 
  
\item p. 267, l. -12: devoted the = devoted to the
 
  
\item p. 268, l. 6: delete ``an'' 
+
p. 254, l. 14: definition
  
\item p. 276, l. -1: Use previous = Use the previous
+
p. 267, l. -12: devoted the = devoted to the  
  
\item p. 280, l. 11: theory = theory of
+
p. 268, l. 6: delete ``an'' 
  
\item p. 284, l. 5: is get = is to get
 
  
\item Reference [36]: G\"{o}tenborg = G\"{o}teborg 
+
p. 276, l. -1: Use previous = Use the previous
  
\item Reference [43]: add  ``{\it
+
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)''  
 
Mathematical Logic and Foundations of Set Theory} (Proc. Internat. Colloq., Jerusalem, 1968)''  
  
\item Reference [54]: add  ``Volume
+
Reference [54]: add  ``Volume
 
1292 of {\it Lecture Notes in Mathematics}''  
 
1292 of {\it Lecture Notes in Mathematics}''  
  
 +
Reference [83]: of 619 = 619 of
  
\item Reference [83]: of 619 = 619 of
+
Reference [85]: delete
 
+
\item Reference [85]: delete
+
 
``(1983)''; this
 
``(1983)''; this
 
reference should appear after reference [88]  
 
reference should appear after reference [88]  
  
\item Reference [109]: od = of  
+
Reference [109]: od = of  
  
\item Reference [113]: of pa = of ${\rm PA}$  
+
Reference [113]: of pa = of ${\rm PA}$  
  
\item Reference [120]: add `` in Automorphisms of first-order structures, R. Kaye, D. Macpherson (eds.)''  
+
Reference [120]: add `` in Automorphisms of first-order structures, R. Kaye, D. Macpherson (eds.)''  
  
\item Reference [148]: {\bf CIII} = {\bf 103}  
+
Reference [148]: {\bf CIII} = {\bf 103}  
  
\item Reference [153]: charaterization  
+
Reference [153]: charaterization  
  
  
\item Reference [164]: add  ``Volume
+
Reference [164]: add  ``Volume
 
859 of {\it Lecture Notes in Mathematics}''  
 
859 of {\it Lecture Notes in Mathematics}''  
  
\item Reference [167]: add  ``Stud.
+
Reference [167]: add  ``Stud.
 
Logic Found. Math., 120''  
 
Logic Found. Math., 120''  
  
\item Reference [172]: add ``Lecture
+
Reference [172]: add ``Lecture
 
Notes Logic, 12''  
 
Notes Logic, 12''  
  
\item Reference [188]: Unmglichkeit =
+
Reference [188]: Unmglichkeit =
 
Unm\"{o}glichkeit; vollstndigen = vollst\"{a}ndigen
 
Unm\"{o}glichkeit; vollstndigen = vollst\"{a}ndigen
  
\item Reference [189]: abzhlbar = abz\"{a}hlbar; Fundam. = Fund.  
+
Reference [189]: abzhlbar = abz\"{a}hlbar; Fundam. = Fund.  
  
\item Reference [199]: add ``Volume
+
Reference [199]: add ``Volume
 
834 of {\it Lecture Notes in
 
834 of {\it Lecture Notes in
 
Mathematics}''  
 
Mathematics}''  
  
\item Reference [204]: poljak-r\"{o}dl = Poljak-R\"{o}dl  
+
Reference [204]: poljak-r\"{o}dl = Poljak-R\"{o}dl  
  
\item Reference [206]: the name of the journal is usually abbreviated as
+
Reference [206]: the name of the journal is usually abbreviated as
 
``Algebra Logic Appl.''  
 
``Algebra Logic Appl.''  
  
\item  Reference [208]: complete = complete models
+
  Reference [208]: complete = complete models
 
+
\item Reference [212]: add ``Proceedings of the International Congress of Mathematicians''
+
 
+
\end{itemize}
+
 
+
\end{document}
+
 
+
 
+
 
+
 
+
 
+
  
\end{itemize}
+
Reference [212]: add ``Proceedings of the International Congress of Mathematicians''
\end{document}
+

Revision as of 11:55, 17 January 2013

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