Difference between revisions of "Errata"

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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 ==
 
== Corrections ==
Line 11: Line 13:
 
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. 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. 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. 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.
Line 19: Line 19:
 
p. 85, Exercise 3.6.17: Delete the hint.
 
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.98, l.-3: In the first line of the displayed formula, $x\leq x'$ should be replaced
+
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 $\alpha^{n+1}=\alpha^n\alpha$."
by $x < x'$ . The same change should be made in the second line.
+
  
p.106, paragraph starting with l.  
+
p.110/l.15: Should be: $A =\{1, 2, 3, 4\}$.
-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.18: Delete “$f (5) = 1,$”.  
 
+
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. 118, l. 12:  Near end of last line of the Theorem, delete the extraneous “) ”
  
p.128/Lemma 4.7.4: Interchange “$\alpha_1 : D_1  
+
p.128/Lemma 4.7.4: Interchange “$\alpha_1 : D_1\rightarrow {\rm Eq}(A_1 )$” and “$\alpha_2 : D_2\rightarrow {\rm Eq}(A_2 )$”.  
\rightarrow {\rm Eq}(A_1 )$” 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.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, 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. 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. 177, l. -12: $f(t_{1})$ should be $f(t_{2})$
+
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. 179, l. -12: [132] should be [130] 
+
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. 179, l. -8: should [166] be J. Schmerl, Peano models with many generic classes.  
+
p. 209, Theorem 8.4.7: In the assumptions, $a$ is definable without parameters in $(M,I)$.
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
+
p. 246. Lemma 9.4.3 (1). One has to assume that $I$ and $J$ are not  infimum and supremum of the same gap.
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 $(M,Y) \not\equiv (M,X)$?
 
+
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 $(M,Y) \not\equiv
+
(M,X)$?
+
  
  
Line 71: Line 60:
 
p. vi, l. 7: fragment = fragment of
 
p. vi, l. 7: fragment = fragment of
  
p. vi, l. -17: While, = While  
+
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 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'' 
+
 
+
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. vi, l, -9: proves = and proves
  
 +
p. vi, l. -2 f.b.: purpose = purpose of
  
p. 107, l. -16: represenatation 
+
p. vii, l. -9: delete ``and"
  
 +
p. vii, l. -2 f: For every countable model $M$, the isomorphism type if its reducts...
  
p. 107, l. -3: $r$ should be $k$.
+
p. viii, l. -14: the Chapter 7 = Chapter 7
  
 +
p. 1, l. 3: delete ``a''
  
p. 108, l. 23: be list all = be a list of all
+
p. 3, l. -11: is $B$ = $B$ is
  
 +
p. 3, l. -4: $y\in M$ = $y\in X$
  
p. 109. l. 4: Should be ``The inclusion $M_{i\land j}\subseteq M_i\cap M_j$..."
+
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. 122, l. -15: used get = used to get  
+
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. 124, l. 14: hold for = hold for
+
p. 24, l. 13: [69] . = [69].
  
 +
p. 24, l. 17: Kotlarski and Kaye = Kaye and Kotlarski 
  
p. 125, l. -17: the tree = on the tree
+
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. 128, l. -9: represenations
+
p. 32, l. 7: the contrary = to the contrary
  
 +
p. 33, l. -5: replaced = replaced by
  
p. 131, l. 3: (see Exercise 4.8.2)
+
p. 33, l. 4 f.b.: single = a single
  
 +
p. 48, l. 15: Wikie = Wilkie
  
p. 133, l. 8: charaterization
+
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. 135, l. 7: introduced
+
p. 51, l. 17: unboounded = unbounded 
  
 +
p. 52, l. 21: infinte  = infinite
  
p. 142, l. 1: there some = there exists some
+
p. 53, l. 2: type = types
  
 +
p. 54, l. 6: do not look = do not look the same
  
p. 145, l. 2: instead = instead of ?
+
p. 55, l. 11: the another = another
  
 +
p. 57, l. 14: realizes = realize
  
p. 151, l. 17: is not be = is not
+
p. 57, l. -10: Should be ``Theorem 3.1.16..."
  
 +
p. 57, l. -7; Should be ``$a\in {\rm Scl}(b)$...
  
p. 152, l. 17: off = of
+
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. 156, l. -18: linearly set = linearly ordered set
+
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. 157, l. 17: Theorems 5.3.4 = Theorem 5.3.4
+
p. 61, l. -5 : The are = There are
  
 +
p. 67, l. -18: types. = types,
  
 +
p. 67, l. -17: Then = then
  
p. 162, l. 10,12: Theorem 6.2.6 = Lemma 6.2.6
+
p. 80, l. 7: $F$ = $f$
  
 +
p. 85, Exercise 3.6.25: are = There are
  
p. 166, l. -10: funtcions
+
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. 168, l. -6: would no = would be no
+
p. 107, l. 3: Theorem 4.5.32 = Corollary 4.5.32 
  
 +
p. 107, l. -16: representation 
  
p. 169, second line of Theorem 6.4.3: $(M,X)_{X\in{\cal G}}$
+
p. 107, l. -3: $r$ should be $k$.
  
 +
p. 108, l. 23: be list all = be a list of all
  
p. 171, l. 10: the proof = of the proof
+
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. 173. First line of Theorem 6.4.8: $|M|\leq \kappa$.
+
p. 124, l. 14: hold for = hold for
  
 +
p. 125, l. -17: the tree = on the tree
  
p. 177, l. -14: possibilities
+
p. 128, l. -9: representations
  
 +
p. 131, l. 3: (see Exercise 4.8.2)
  
p. 177, l. -5: not = are not
+
p. 133, l. 8: characterization
  
 +
p. 135, l. 7: introduced
  
p. 179, l. -14: independently
+
p. 142, l. 1: there some = there exists some
  
p. 184, l. -4: Propositional = Proposition
+
p. 145, l. 2: instead = instead of ?
  
 +
p. 151, l. 17: is not be = is not
  
p. 189, l. -11: Theorem 2.2.8 = Theorem 2.2.16
+
p. 152, l. 17: off = of
  
p. 191, l.-8: Should be $b\in T^K$
+
p. 156, l. -18: linearly set = linearly ordered set
  
 +
p. 157, l. 17: Theorems 5.3.4 = Theorem 5.3.4
  
p. 192, l. 8: if = of 
+
p. 162, l. 10,12: Theorem 6.2.6 = Lemma 6.2.6
  
 +
p. 166, l. -10: functions
  
p. 194, l. -6: Theorembut = Theorem
+
p. 168, l. -6: would no = would be no
but
+
  
 +
p. 169, second line of Theorem 6.4.3: $(M,X)_{X\in{\cal G}}$
  
p. 197, l. 12: model IA = model of IA
+
p. 171, l. 10: the proof = of the proof
  
 +
p. 173. First line of Theorem 6.4.8: $|M|\leq \kappa$.
  
p. 200, l. -10: which = which is
+
p. 177, l. -14: possibilities
  
 +
p. 177, l. -5: not = are not
  
p. 201, l. -12: models = models of
+
p. 179, l. -14: independently
  
p. 203, l. 10: $ a_{\frak{A}} = \frak{a}_{\frak{A}}$ 
+
p. 184, l. -4: Propositional = Proposition
  
 +
p. 189, l. -11: Theorem 2.2.8 = Theorem 2.2.16
  
p. 206, l. 10: model = models
+
p. 191, l.-8: Should be $b\in T^K$
  
 +
p. 192, l. 8: if = of 
  
p. 209, Fourth line of the proof of Corollary 8.4.6: $b_1,b_2=b_0,b_1$
+
p. 194, l. -6: Theorembut = Theorem but
  
p. 214, l. -17: than least = than the least
+
p. 197, l. 12: model IA = model of IA
  
p. 221, l. -12: delete ``a''
+
p. 200, l. -10: which = which is
  
p. 222, l. 5: Corollary 3.2.4 = Lemma 3.2.4
+
p. 201, l. -12: models = models of
  
p. 225, l. 7: maximal = a maximal
+
p. 203, l. 10: $ a_{\frak{A}} = \frak{a}_{\frak{A}}$ 
  
p. 227, l. 12: Corollary 8.1.2 = Proposition 8.1.3. Delete the statement in parenthesis.
+
p. 206, l. 10: model = models
  
p. 227, l. -10: Back-and-forth, = Back-and-forth
+
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. 228, l. 20 f.b.: Frederike = Friederike
+
p. 221, l. -12: delete ``a''
  
p. 229, l. 9: proof the = proof of the
+
p. 222, l. 5: Corollary 3.2.4 = Lemma 3.2.4
  
 +
p. 225, l. 7: maximal = a maximal
  
p. 229, l. -14: the question mark
+
p. 227, l. 12: Corollary 8.1.2 = Proposition 8.1.3. Delete the statement in parenthesis.
appears ``upside down''
+
  
p. 233, l. 12: aid = and
+
p. 227, l. -10: Back-and-forth, = Back-and-forth
  
p. 234, l. 8: Proposition 9.1.3 =
+
p. 228, l. 20 f.b.: Frederike = Friederike
Lemma 9.1.3 
+
  
 +
p. 229, l. 9: proof the = proof of the
  
p. 235, l. 2: models = model
+
p. 229, l. -14: the question mark appears ``upside down''
  
p. 239, l. 13: index if = index of  
+
p. 233, l. 12: aid = and  
  
p. 250, l. -6 f: realizes = realize  
+
p. 234, l. 8: Proposition 9.1.3 = Lemma 9.1.3  
  
p. 253, l. -7: of countable = of a countable 
+
p. 235, l. 2: models = model
  
 +
p. 239, l. 13: index if = index of 
  
p. 254, l. 14: definition
+
p. 250, l. -6 f: realizes = realize 
  
p. 267, l. -12: devoted the = devoted to the
+
p. 253, l. -7: of countable = of a countable 
  
p. 268, l. 6: delete ``an'' 
+
p. 254, l. 14: definition
  
 +
p. 267, l. -12: devoted the = devoted to the
  
p. 276, l. -1: Use previous = Use the previous
+
p. 268, l. 6: delete ``an'' 
  
p. 280, l. 11: theory = theory of
+
p. 276, l. -1: Use previous = Use the previous
  
p. 284, l. 5: is get = is to get
+
p. 280, l. 11: theory = theory of
  
Reference [36]: G\"{o}tenborg = G\"{o}teborg 
+
p. 284, l. 5: is get = is to get
  
Reference [43]: add  ``{\it
+
Reference [36]: G\"{o}tenborg = G\"{o}teborg 
Mathematical Logic and Foundations of Set Theory} (Proc. Internat. Colloq., Jerusalem, 1968)''
+
  
Reference [54]: add  ``Volume
+
Reference [43]: add  ``{\it Mathematical Logic and Foundations of Set Theory} (Proc. Internat. Colloq., Jerusalem, 1968)''  
1292 of {\it Lecture Notes in Mathematics}''  
+
  
Reference [83]: of 619 = 619 of
+
Reference [54]: add  ``Volume 1292 of {\it Lecture Notes in Mathematics}''
  
Reference [85]: delete
+
Reference [83]: of 619 = 619 of
``(1983)''; this
+
reference should appear after reference [88]
+
  
Reference [109]: od = of
+
Reference [85]: delete ``(1983)''; this reference should appear after reference [88]
  
Reference [113]: of pa = of ${\rm PA}$
+
Reference [109]: od = of  
  
Reference [120]: add `` in Automorphisms of first-order structures, R. Kaye, D. Macpherson (eds.)''
+
Reference [113]: of pa = of ${\rm PA}$
  
Reference [148]: {\bf CIII} = {\bf 103}
+
Reference [120]: add `` in Automorphisms of first-order structures, R. Kaye, D. Macpherson (eds.)''
  
Reference [153]: charaterization
+
Reference [148]: {\bf CIII} = {\bf 103}
  
 +
Reference [153]: characterization
  
Reference [164]: add  ``Volume
+
Reference [164]: add  ``Volume 859 of {\it Lecture Notes in Mathematics}''  
859 of {\it Lecture Notes in Mathematics}''  
+
  
Reference [167]: add  ``Stud.
+
Reference [167]: add  ``Stud. Logic Found. Math., 120''  
Logic Found. Math., 120''  
+
  
Reference [172]: add ``Lecture
+
Reference [172]: add ``Lecture Notes Logic, 12''  
Notes Logic, 12''  
+
  
Reference [188]: Unmglichkeit =
+
Reference [188]: Unmglichkeit = Unm\"{o}glichkeit; vollstndigen = vollst\"{a}ndigen
Unm\"{o}glichkeit; vollstndigen = vollst\"{a}ndigen
+
  
Reference [189]: abzhlbar = abz\"{a}hlbar; Fundam. = Fund.  
+
Reference [189]: abzhlbar = abz\"{a}hlbar; Fundam. = Fund.  
  
Reference [199]: add ``Volume
+
Reference [199]: add ``Volume 834 of {\it Lecture Notes in Mathematics}''  
834 of {\it Lecture Notes in
+
Mathematics}''  
+
  
Reference [204]: poljak-r\"{o}dl = Poljak-R\"{o}dl  
+
Reference [204]: poljak-r\"{o}dl = Poljak-R\"{o}dl  
  
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.''  
+
  
  Reference [208]: complete = complete models
+
Reference [208]: complete = complete models
  
Reference [212]: add ``Proceedings of the International Congress of Mathematicians''
+
Reference [212]: add ``Proceedings of the International Congress of Mathematicians''

Latest revision as of 04:05, 7 April 2013

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. 209, Theorem 8.4.7: In the assumptions, $a$ is definable without parameters in $(M,I)$.

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