Difference between revisions of "Errata"

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  p. vii, l. -3: Should be: ``type $\omega+(\omega^{*}+\omega)\rho$..."  
 
  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. 21, Exercise 1.14.5: $b\in M$ should be $c\in M$
 
  
 
  p. 22, l. -12: [70] should be [71]  
 
  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. 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.
 
  
 
  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  
 
  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.  
+
  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 $\{x \in M (D) : r \leq x\}$."  
+
  
 
  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 $\alpha^{n+1}=\alpha^n\alpha$."
  
  p.110/l.15: Should be: $A =  
+
  p.110/l.15: Should be: $A =\{1, 2, 3, 4\}$.   
\{1, 2, 3, 4\}$.   
+
  
 
  p.110/l.18: Delete “$f (5) = 1,$”.  
 
  p.110/l.18: Delete “$f (5) = 1,$”.  
Line 44: Line 32:
 
  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) = $.  
  
Line 54: Line 39:
  
 
  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. 177, l. -12: $f(t_{1})$ should be $f(t_{2})$  
Line 60: Line 44:
 
  p. 179, l. -12: [132] should be [130]   
 
  p. 179, l. -12: [132] should be [130]   
  
  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.
+
  
 
  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
Line 68: Line 51:
 
  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.
  
  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 $M\prec_{end} N$, and if $Y \subseteq M$ is coded in $N$, then $(M,Y) \not\equiv (M,X)$?
$X\in{\mathcal P}(M)\setminus{\rm Def}(M)$ 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 101: Line 81:
 
  p. 14, Definition 1.9.1: ``partial inductive satisfaction class''should be in italics  
 
  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
+
  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''   
+
  
 
  p. 19, l. 14: proof the = proof of the  
 
  p. 19, l. 14: proof the = proof of the  
Line 112: Line 91:
 
  p. 25, l. 1: models arithmetic = models of arithmetic  
 
  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  
+
  p.27, l.22: In Corollary 2.1.4, last word of first line should be “end”, not “and”.   
“and”.   
+
  
 
  p. 32, l. 7: the contrary = to the contrary  
 
  p. 32, l. 7: the contrary = to the contrary  
Line 128: Line 106:
  
 
  p. 51, l. 17: unboounded = unbounded   
 
  p. 51, l. 17: unboounded = unbounded   
 
  
 
  p. 52, l. 21: infinte  = infinite
 
  p. 52, l. 21: infinte  = infinite
Line 258: Line 235:
 
  p. 229, l. 9: proof the = proof of the  
 
  p. 229, l. 9: proof the = proof of the  
  
  p. 229, l. -14: the question mark
+
  p. 229, l. -14: the question mark appears ``upside down''  
appears ``upside down''  
+
  
 
  p. 233, l. 12: aid = and   
 
  p. 233, l. 12: aid = and   
  
  p. 234, l. 8: Proposition 9.1.3 =
+
  p. 234, l. 8: Proposition 9.1.3 = Lemma 9.1.3   
Lemma 9.1.3   
+
  
 
  p. 235, l. 2: models = model  
 
  p. 235, l. 2: models = model  
Line 288: Line 263:
 
  Reference [36]: G\"{o}tenborg = G\"{o}teborg   
 
  Reference [36]: G\"{o}tenborg = G\"{o}teborg   
  
  Reference [43]: add  ``{\it
+
  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)''  
+
  
  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  
 
  Reference [83]: of 619 = 619 of  
  
  Reference [85]: delete
+
  Reference [85]: delete ``(1983)''; this reference should appear after reference [88]  
``(1983)''; this
+
reference should appear after reference [88]  
+
  
 
  Reference [109]: od = of  
 
  Reference [109]: od = of  
Line 310: Line 281:
 
  Reference [153]: characterization  
 
  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''

Revision as of 12:12, 17 January 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. 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