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Theorem 13.5 The Jacobi method with optimal choice of \( \left( {p, q}\right) \) converges super-linearly when the eigenvalues of \( A \) are simple, in the following sense. Let \( N = n(n - \) \( 1)/2 \) be the number of elements under the diagonal. Then there exists a number \( c > 0 \) such that\n\n\[ \begin{Vmatrix... | Proof. We first remark that if \( i \neq j \) with \( \{ i, j\} \neq \left\{ {{p}_{\ell },{q}_{\ell }}\right\} \), then\n\n\[ \left| {{a}_{ij}^{\left( \ell + 1\right) } - {a}_{ij}^{\left( \ell \right) }}\right| \leq \left| {t}_{\ell }\right| \sqrt{2}\begin{Vmatrix}{E}_{\ell }\end{Vmatrix} \]\n\n(13.5)\n\nwhere \( {t}_{... | Yes |
Theorem 13.7 One assumes that \( \operatorname{SpM} \) contains only one element \( \lambda \) of maximal modulus (that modulus is thus equal to \( \rho \left( M\right) \) ).\n\nIf \( \rho \left( M\right) = 0 \), the method stops because \( M{x}^{k} = 0 \) for some \( k < n \) .\n\nOtherwise, let \( {\mathbb{C}}^{n} = ... | Proof. The case \( \rho \left( M\right) = 0 \) is obvious because \( M \) is then nilpotent.\n\nAssume otherwise that \( \rho \left( M\right) > 0 \) . Let \( {x}^{0} = {y}^{0} + {z}^{0} \) be the decomposition of \( {x}^{0} \) with \( {y}^{0} \in E \) and \( {z}^{0} \in F \) . By assumption, \( {y}^{0} \neq 0 \) . Beca... | Yes |
a) Let \( \mathcal{L} = \{ + , < ,0\} \) and let \( T \) be the theory of ordered Abelian groups. Then, \( \forall x\left( {x \neq 0 \rightarrow x + x \neq 0}\right) \) is a logical consequence of \( T \) . | a) Suppose that \( \mathcal{M} = \left( {M,+, < ,0}\right) \) is an ordered Abelian group. Let \( a \in M \smallsetminus \{ 0\} \) . We must show that \( a + a \neq 0 \) . Because \( \left( {M, < }\right) \) is a linear order \( a < 0 \) or \( 0 < a \) . If \( a < 0 \), then \( a + a < 0 + a = a < 0 \) . Because \( \ne... | Yes |
Proposition 1.2.13 a) Let \( \mathcal{L} = \{ + , < ,0\} \) and let \( T \) be the theory of ordered Abelian groups. Then, \( \forall x\left( {x \neq 0 \rightarrow x + x \neq 0}\right) \) is a logical consequence of \( T \) . | a) Suppose that \( \mathcal{M} = \left( {M,+, < ,0}\right) \) is an ordered Abelian group. Let \( a \in M \smallsetminus \{ 0\} \) . We must show that \( a + a \neq 0 \) . Because \( \left( {M, < }\right) \) is a linear order \( a < 0 \) or \( 0 < a \) . If \( a < 0 \), then \( a + a < 0 + a = a < 0 \) . Because \( \ne... | Yes |
Proposition 1.2.13 a) Let \( \mathcal{L} = \{ + , < ,0\} \) and let \( T \) be the theory of ordered Abelian groups. Then, \( \forall x\left( {x \neq 0 \rightarrow x + x \neq 0}\right) \) is a logical consequence of \( T \) . | a) Suppose that \( \mathcal{M} = \left( {M,+, < ,0}\right) \) is an ordered Abelian group. Let \( a \in M \smallsetminus \{ 0\} \) . We must show that \( a + a \neq 0 \) . Because \( \left( {M, < }\right) \) is a linear order \( a < 0 \) or \( 0 < a \) . If \( a < 0 \), then \( a + a < 0 + a = a < 0 \) . Because \( \ne... | Yes |
Lemma 1.3.3 Let \( {\mathcal{L}}_{\mathrm{r}} \) be the language of ordered rings and \( (\mathbb{R}, + , - , \cdot \) , \( < ,0,1) \) be the ordered field of real numbers. Suppose that \( X \subseteq {\mathbb{R}}^{n} \) is \( A \) - definable. Then, the topological closure of \( X \) is also \( A \) -definable. | Proof Let \( \phi \left( {{v}_{1},\ldots ,{v}_{n},\bar{a}}\right) \) define \( X \) . Let \( \psi \left( {{v}_{1},\ldots ,{v}_{n},\bar{w}}\right) \) be the formula\n\n\[ \forall \epsilon \left\lbrack {\epsilon > 0 \rightarrow \exists {y}_{1},\ldots ,{y}_{n}\left( {\phi \left( {\bar{y},\bar{w}}\right) \land \mathop{\sum... | Yes |
Proposition 1.3.4 Let \( \mathcal{M} \) be an \( \mathcal{L} \) -structure. Suppose that \( {D}_{n} \) is a collection of subsets of \( {M}^{n} \) for all \( n \geq 1 \) and \( \mathcal{D} = \left( {{D}_{n} : n \geq 1}\right) \) is the smallest collection such that:\n\ni) \( {M}^{n} \in {D}_{n} \) ;\n\nii) for all \( n... | Proof We first show that the definable sets satisfy the closure properties i)-viii). Because \( \mathcal{D} \) is the smallest collection with these closure properties, every \( X \in {D}_{n} \) is definable.\n\ni) \( {M}^{n} \) is definable by \( {v}_{1} = {v}_{1} \) .\n\nii) The graph of \( {f}^{\mathcal{M}} \) is de... | Yes |
Proposition 1.3.5 Let \( \mathcal{M} \) be an \( \mathcal{L} \) -structure. If \( X \subset {M}^{n} \) is \( A \) -definable, then every \( \mathcal{L} \) -automorphism of \( \mathcal{M} \) that fixes \( A \) pointwise fixes \( X \) setwise (that is, if \( \sigma \) is an automorphism of \( M \) and \( \sigma \left( a\... | Proof Let \( \psi \left( {\bar{v},\bar{a}}\right) \) be the \( \mathcal{L} \) -formula defining \( X \) where \( \bar{a} \in A \) . Let \( \sigma \) be an automorphism of \( \mathcal{M} \) with \( \sigma \left( \bar{a}\right) = \bar{a} \), and let \( \bar{b} \in {M}^{n} \) .\n\nIn the proof of Theorem 1.1.10, we showed... | Yes |
Corollary 1.3.6 The set of real numbers is not definable in the field of complex numbers. | Proof If \( \mathbb{R} \) were definable, then it would be definable over a finite \( A \subset \mathbb{C} \) . Let \( r, s \in \mathbb{C} \) be algebraically independent over \( A \) with \( r \in \mathbb{R} \) and \( s \notin \mathbb{R} \) . There is an automorphism \( \sigma \) of \( \mathbb{C} \) such that \( \sigm... | Yes |
Lemma 1.3.7 If \( \left( {A, < }\right) \) is a linear order, then for all vertices \( x \) in \( {G}_{A} \) , there is a unique \( i \leq 5 \) such that \( {G}_{A} \vDash {\theta }_{i}\left( x\right) \) . | Proof Let \( x \) be a vertex in \( {G}_{A} \) . If \( x \) is part of a 3-cycle, then \( x = {x}_{i}^{a} \) for some \( a \in A \) . Recall that the valence of \( x \) is the number of edges with \( x \) as a vertex. If \( x \) has valence 3, then \( {\theta }_{1}\left( x\right) \) is the unique formula that holds. If... | Yes |
Proposition 2.1.1 If \( \mathcal{L} \) is a recursive language and \( T \) is a recursive \( \mathcal{L} \) - theory, then \( \{ \phi : T \vdash \phi \} \) is recursively enumerable; that is, there is an algorithm, that when given \( \phi \) as input will halt accepting if \( T \vdash \phi \) and not halt if \( T \nvda... | Proof There is \( {\sigma }_{0},{\sigma }_{1},{\sigma }_{2},\ldots \), a computable listing of all finite sequences of \( \mathcal{L} \) -formulas. At stage \( i \) of our algorithm, we check to see whether \( {\sigma }_{i} \) is a proof of \( \psi \) from \( T \) . This involves checking that each formula either is in... | Yes |
Corollary 2.1.3 \( T \) is consistent if and only if \( T \) is satisfiable. | Proof Suppose that \( T \) is not satisfiable. Because there are no models of \( T \), every model of \( T \) is a model of \( \left( {\phi \land \neg \phi }\right) \) . Thus, \( T \vDash \left( {\phi \land \neg \phi }\right) \) and by the Completeness Theorem \( T \vdash \left( {\phi \land \neg \phi }\right) \) . | Yes |
Theorem 2.1.4 (Compactness Theorem) \( T \) is satisfiable if and only if every finite subset of \( T \) is satisfiable. | Proof Clearly, if \( T \) is satisfiable, then every subset of \( T \) is satisfiable. On the other hand, if \( T \) is not satisfiable, then \( T \) is inconsistent. Let \( \sigma \) be a proof of a contradiction from \( T \) . Because \( \sigma \) is finite, only finitely many assumptions from \( T \) are used in the... | Yes |
Lemma 2.1.6 Suppose \( T \) is a maximal and finitely satisfiable \( \mathcal{L} \) -theory. If \( \Delta \subseteq T \) is finite and \( \Delta \vDash \psi \), then \( \psi \in T \) . | Proof If \( \psi \notin T \), then, because \( T \) is maximal, \( \neg \psi \in T \) . But then \( \Delta \cup \{ \neg \psi \} \) is a finite unsatisfiable subset of \( T \), a contradiction. | Yes |
Lemma 2.1.9 Suppose that \( T \) is a finitely satisfiable \( \mathcal{L} \) -theory and \( \phi \) is an \( \mathcal{L} \) -sentence, then, either \( T \cup \{ \phi \} \) or \( T \cup \{ \neg \phi \} \) is finitely satisfiable. | Proof Suppose that \( T \cup \{ \phi \} \) is not finitely satisfiable. Then, there is a finite \( \Delta \subseteq T \) such that \( \Delta \vDash \neg \phi \) . We claim that \( T \cup \{ \neg \phi \} \) is finitely satisfiable. Let \( \sum \) be a finite subset of \( T \) . Because \( \Delta \cup \sum \) is satisfia... | Yes |
Corollary 2.1.10 If \( T \) is a finitely satisfiable \( \mathcal{L} \) -theory, then there is a maximal finitely satisfiable \( \mathcal{L} \) -theory \( {T}^{\prime } \supseteq T \) . | Proof Let \( I \) be the set of all finitely satisfiable \( \mathcal{L} \) -theories containing \( T \) . We partially order \( I \) by inclusion. If \( C \subseteq I \) is a chain, let \( {T}_{C} = \bigcup \{ \sum : \sum \in C\} \) . If \( \Delta \) is a finite subset of \( {T}_{C} \), then there is \( \sum \in C \) s... | Yes |
Theorem 2.1.11 If \( T \) is a finitely satisfiable \( \mathcal{L} \) -theory and \( \kappa \) is an infinite cardinal with \( \kappa \geq \left| \mathcal{L}\right| \), then there is a model of \( T \) of cardinality at most \( \kappa \) . | Proof By Lemma 2.1.8, we can find \( {\mathcal{L}}^{ * } \supseteq \mathcal{L} \) and \( {T}^{ * } \supseteq T \) a finitely satisfiable \( {\mathcal{L}}^{ * } \) -theory such that any \( {\mathcal{L}}^{ * } \) -theory extending \( {T}^{ * } \) has the witness property and the cardinality of \( {\mathcal{L}}^{ * } \) i... | Yes |
Proposition 2.1.12 Let \( \mathcal{L} \) be a language containing \( \{ \cdot, e\} \), the language of groups, let \( T \) be an \( \mathcal{L} \) -theory extending the theory of groups, and let \( \phi \left( v\right) \) be an \( \mathcal{L} \) -formula. Suppose that for all \( n \) there is \( {G}_{n} \vDash T \) and... | Proof Let \( {\mathcal{L}}^{ * } = \mathcal{L} \cup \{ c\} \), where \( c \) is a new constant symbol. Let \( {T}^{ * } \) be the \( \mathcal{L} \) -theory\n\n\[ T \cup \{ \phi \left( c\right) \} \cup \left\{ {\underset{n - \text{ times }}{\underbrace{c \cdot c\cdots c}} \neq e : n = 1,2,\ldots }\right\} .\n\nIf \( G \... | Yes |
Proposition 2.1.13 Let \( \mathcal{L} = \{ \cdot , + , < ,0,1\} \) and let \( \operatorname{Th}\left( \mathbb{N}\right) \) be the full \( \mathcal{L} \) - theory of the natural numbers. There is \( \mathcal{M} \vDash \operatorname{Th}\left( \mathbb{N}\right) \) and \( a \in M \) such that a is larger than every natural... | Proof Let \( {\mathcal{L}}^{ * } = \mathcal{L} \cup \{ c\} \) where \( c \) is a new constant symbol and let\n\n\[ T = \operatorname{Th}\left( \mathbb{N}\right) \cup \left\{ {\underset{n - \text{ times }}{\underbrace{1 + 1 + \ldots + 1}} < c : \text{ for }n = 1,2,\ldots }\right\} .\n\nIf \( \Delta \) is a finite subset... | Yes |
Lemma 2.1.14 If \( T \vDash \phi \), then \( \Delta \vDash T \) for some finite \( \Delta \subseteq T \) . | Proof Suppose not. Let \( \Delta \subseteq T \) be finite. Because \( \Delta \mathrel{\text{\vDash \not{} }} \phi ,\Delta \cup \{ \neg \phi \} \) is satisfiable. Thus, \( T \cup \{ \neg \phi \} \) is finitely satisfiable and, by the Compactness Theorem, \( T \mathrel{\text{\vDash \not{} }} \phi \) . | No |
Proposition 2.2.2 Let \( T \) be an \( \mathcal{L} \) -theory with infinite models. If \( \kappa \) is an infinite cardinal and \( \kappa \geq \left| \mathcal{L}\right| \), then there is a model of \( T \) of cardinality \( \kappa \) . | Proof Let \( {\mathcal{L}}^{ * } = \mathcal{L} \cup \left\{ {{c}_{\alpha } : \alpha < \kappa }\right\} \), where each \( {c}_{\alpha } \) is a new constant symbol, and let \( {T}^{ * } \) be the \( {\mathcal{L}}^{ * } \) -theory \( T \cup \left\{ {{c}_{\alpha } \neq {c}_{\beta } : \alpha ,\beta < \kappa ,\alpha \neq \b... | Yes |
Proposition 2.2.4 The theory of torsion-free divisible Abelian groups is \( \kappa \) -categorical for all \( \kappa > {\aleph }_{0} \) . | Proof We first argue that models of \( T \) are essentially vector spaces over the field of rational numbers \( \mathbb{Q} \) . Clearly, if \( V \) is any vector space over \( \mathbb{Q} \) , then the underlying additive group of \( V \) is a model of \( T \) . On the other hand, if \( G \vDash T, g \in G \), and \( n ... | Yes |
Proposition 2.2.5 \( {\mathrm{{ACF}}}_{p} \) is \( \kappa \) -categorical for all uncountable cardinals \( \kappa \) . | Proof Two algebraically closed fields are isomorphic if and only if they have the same characteristic and transcendence degree (see, for example [58] X §1). An algebraically closed field of transcendence degree \( \lambda \) has cardinality \( \lambda + {\aleph }_{0} \) . If \( \kappa > {\aleph }_{0} \), an algebraical... | Yes |
Theorem 2.2.6 (Vaught’s Test) Let \( T \) be a satisfiable theory with no finite models that is \( \kappa \) -categorical for some infinite cardinal \( \kappa \geq \left| \mathcal{L}\right| \) . Then \( T \) is complete. | Proof Suppose that \( T \) is not complete. Then there is a sentence \( \phi \) such that \( T \mathrel{\text{\vDash \not{} }} \phi \) and \( T \mathrel{\text{\vDash \not{} }} \neg \phi \) . Because \( T \mathrel{\text{\vDash \not{} }} \psi \) if and only if \( T \cup \{ \neg \psi \} \) is satisfiable, the theories \( ... | Yes |
Lemma 2.2.8 Let \( T \) be a recursive complete satisfiable theory in a recursive language \( \mathcal{L} \). Then \( T \) is decidable. | Proof Because \( T \) is satisfiable \( A = \{ \phi : T \vDash \phi \} \) and \( B = \{ \phi : T \vDash \neg \phi \} \) are disjoint. Because \( T \) is consistent \( A \cup B \) is the set of all \( \mathcal{L} \)-sentences. By the Completeness Theorem, \( A = \{ \phi : T \vdash \phi \} \) and \( B = \{ \phi : T \vdas... | Yes |
Corollary 2.2.10 Let \( \phi \) be a sentence in the language of rings. The following are equivalent.\n\ni) \( \phi \) is true in the complex numbers.\n\nii) \( \phi \) is true in every algebraically closed field of characteristic zero.\n\niii) \( \phi \) is true in some algebraically closed field of characteristic zer... | Proof The equivalence of i)-iii) is just the completeness of \( {\mathrm{{ACF}}}_{0} \) and v) \( \Rightarrow \) iv) is obvious.\n\nFor ii) \( \Rightarrow \) v) suppose that \( {\mathrm{{ACF}}}_{0} \vDash \phi \) . By Lemma 2.1.14, there is a finite \( \Delta \subset {\mathrm{{ACF}}}_{0} \) such that \( \Delta \vDash \... | Yes |
Theorem 2.2.11 Every injective polynomial map from \( {\mathbb{C}}^{n} \) to \( {\mathbb{C}}^{n} \) is surjective. | Proof Remarkably, the key to the proof is the simple observation that if \( k \) is a finite field, then every injective function \( f : {k}^{n} \rightarrow {k}^{n} \) is surjective. From this observation it is easy to show that the same is true for \( {\mathbb{F}}_{p}^{\text{alg }} \), the algebraic closure of the \( ... | Yes |
Theorem 2.3.4 (Upward Löwenheim-Skolem Theorem) Let \( \mathcal{M} \) be an infinite \( \mathcal{L} \) -structure and \( \kappa \) be an infinite cardinal \( \kappa \geq \left| \mathcal{M}\right| + \left| \mathcal{L}\right| \) . Then, there is \( \mathcal{N} \) an \( \mathcal{L} \) -structure of cardinality \( \kappa \... | Proof Because \( \mathcal{M} \vDash {\operatorname{Diag}}_{\mathrm{{el}}}\left( \mathcal{M}\right) ,{\operatorname{Diag}}_{\mathrm{{el}}}\left( \mathcal{M}\right) \) is satisfiable. By Theorem 2.1.11, there is \( \mathcal{N} \vDash {\operatorname{Diag}}_{\mathrm{{el}}}\left( \mathcal{M}\right) \) of cardinality \( \kap... | Yes |
Proposition 2.3.5 (Tarski-Vaught Test) Suppose that \( \mathcal{M} \) is a substructure of \( \mathcal{N} \). Then, \( \mathcal{M} \) is an elementary substructure if and only if, for any formula \( \phi \left( {v,\bar{w}}\right) \) and \( \bar{a} \in M \), if there is \( b \in N \) such that \( \mathcal{N} \vDash \phi... | Proof If \( \mathcal{N} \) is an elementary extension of \( \mathcal{M} \), the condition is clearly true. To prove the converse, we must show that for all \( \bar{a} \in M \) and all \( \mathcal{L} \)-formulas \( \psi \left( \bar{v}\right) \)\n\n\[ \mathcal{M} \vDash \psi \left( \bar{a}\right) \Leftrightarrow \mathcal... | Yes |
Lemma 2.3.6 Let \( T \) be an \( \mathcal{L} \) -theory. There are \( {\mathcal{L}}^{ * } \supseteq \mathcal{L} \) and \( {T}^{ * } \supseteq T \) an \( {\mathcal{L}}^{ * } \) -theory such that \( {T}^{ * } \) has built-in Skolem functions, and if \( \mathcal{M} \vDash T \), then we can expand \( \mathcal{M} \) to \( {... | Proof Recall from Exercise 1.4.15 that to expand \( \mathcal{M} \) we must interpret all of the symbols in \( {\mathcal{L}}^{ * } \smallsetminus \mathcal{L} \) to make \( \mathcal{M} \) a model of \( {T}^{ * } \) . We build a sequence of languages \( \mathcal{L} = {\mathcal{L}}_{0} \subseteq {\mathcal{L}}_{1} \subseteq... | No |
Theorem 2.3.7 (Löwenheim-Skolem Theorem) Suppose that \( \mathcal{M} \) is an \( \mathcal{L} \) -structure and \( X \subseteq M \), there is an elementary submodel \( \mathcal{N} \) of \( \mathcal{M} \) such that \( X \subseteq N \) and \( \left| \mathcal{N}\right| \leq \left| X\right| + \left| \mathcal{L}\right| + {\a... | Proof By Lemma 2.3.6, we may assume that \( \operatorname{Th}\left( \mathcal{M}\right) \) has built in Skolem functions. Let \( {X}_{0} = X \) . Given \( {X}_{i} \), let \( {X}_{i + 1} = {X}_{i} \cup \left\{ {{f}^{\mathcal{M}}\left( \bar{a}\right) : f}\right. \) an \( n \) - ary function symbol, \( \left. {\bar{a} \in ... | Yes |
An \( \mathcal{L} \) -theory \( T \) has a universal axiomatization if and only if whenever \( \mathcal{M} \vDash T \) and \( \mathcal{N} \) is a substructure of \( \mathcal{M} \), then \( \mathcal{N} \vDash T \) . In other words, a theory is preserved under substructure if and only if it has a universal axiomatization... | Proof Suppose that \( \mathcal{N} \subseteq \mathcal{M} \) . We showed in Proposition 1.1.8 that if \( \phi \left( \bar{v}\right) \) is a quantifier-free formula and \( \bar{a} \in N \), then \( \mathcal{N} \vDash \phi \left( \bar{a}\right) \) if and only if \( \mathcal{M} \vDash \phi \left( \bar{a}\right) \) . Thus, i... | Yes |
Proposition 2.3.11 Suppose that \( \left( {I, < }\right) \) is a linear order and \( \left( {{\mathcal{M}}_{i} : i \in I}\right) \) is an elementary chain. Then, \( \mathcal{M} = \mathop{\bigcup }\limits_{{i \in I}}{M}_{i} \) is an elementary extension of each \( {\mathcal{M}}_{i} \) . | Proof We prove by induction on formulas that\n\n\[ \mathcal{M} \vDash \phi \left( \bar{a}\right) \Leftrightarrow {\mathcal{M}}_{i} \vDash \phi \left( \bar{a}\right) \]\n\nfor all \( i \in I \), all formulas \( \phi \left( \bar{v}\right) \), and all \( \bar{a} \in {M}_{i}^{n} \) .\n\nBecause \( {\mathcal{M}}_{i} \) is a... | Yes |
Lemma 2.4.3 \( \mathop{\lim }\limits_{{N \rightarrow \infty }}{p}_{N}\left( {\psi }_{n}\right) = 1 \) for \( n = 1,2,\ldots \) | Proof Fix \( n \) . Let \( G \) be a random graph in \( {\mathcal{G}}_{N} \) where \( N > {2n} \) . Fix \( {x}_{1},\ldots ,{x}_{n},{y}_{1},\ldots ,{y}_{n}, z \in G \) distinct. Let \( q \) be the probability that\n\n\[ \neg \left( {\mathop{\bigwedge }\limits_{{i = 1}}^{n}\left( {R\left( {{x}_{i}, z}\right) \land \neg R... | Yes |
Theorem 2.4.4 (Zero-One Law for Graphs) For any \( \mathcal{L} \)-sentence \( \phi \) either \( \mathop{\lim }\limits_{{N \rightarrow \infty }}{p}_{N}\left( \phi \right) = 0 \) or \( \mathop{\lim }\limits_{{N \rightarrow \infty }}{p}_{N}\left( \phi \right) = 1 \). Moreover, \( T \) axiomatizes \( \left\{ {\phi : \matho... | Proof If \( T \vDash \phi \), then there is \( n \) such that if \( G \) is a graph and \( G \vDash {\psi }_{n} \), then \( G \vDash \phi \). Thus, \( {p}_{N}\left( \phi \right) \geq {p}_{N}\left( {\psi }_{n}\right) \) and by Lemma 2.4.3, \( \mathop{\lim }\limits_{{N \rightarrow \infty }}{p}_{N}\left( \phi \right) = 1 ... | Yes |
Proposition 2.4.5 If \( \mathcal{M} \) and \( \mathcal{N} \) are countable, then the second player has a winning strategy in \( {G}_{\omega }\left( {\mathcal{M},\mathcal{N}}\right) \) if and only if \( \mathcal{M} \cong \mathcal{N} \) . | Proof If \( \mathcal{M} \) and \( \mathcal{N} \) are isomorphic, then player II can win by playing according to the isomorphism.\n\nSuppose that player II has a winning strategy. Let \( {m}_{0},{m}_{1},\ldots \) list \( M \) and \( {n}_{0},{n}_{1},\ldots \) list \( N \) . Consider a play of the game where the second pl... | Yes |
Lemma 2.4.7 One of the players has a winning strategy in \( {G}_{n}\left( {\mathcal{M},\mathcal{N}}\right) \) . | Proof (sketch) This follows from Zermelo's theorem that in any two-person finite length game of perfect information without ties one of the players has a winning strategy (see [10] 1.7.1). It also follows from the determinacy of closed games (see [52]). We outline the proof. Suppose that player II does not have a winni... | Yes |
Lemma 2.4.8 For each \( n \) and \( l \), there is a finite list of formulas \( {\phi }_{1},\ldots ,{\phi }_{k} \) of depth at most \( n \) in free variables \( {x}_{1},\ldots ,{x}_{l} \) such that every formula of depth at most \( n \) in free variables \( {x}_{1},\ldots ,{x}_{l} \) is equivalent to some \( {\phi }_{i... | Proof We first prove this for quantifier-free formulas. Because \( \mathcal{L} \) is finite and has no constant symbols, there are only finitely many atomic \( \mathcal{L} \) -formulas in free variables \( {x}_{1},\ldots ,{x}_{l} \) . Let \( {\sigma }_{1},\ldots ,{\sigma }_{s} \) list all such formulas.\n\nIf \( \phi \... | No |
Lemma 2.4.13 Let \( \mathcal{M} \) and \( \mathcal{N} \) be \( \mathcal{L} \) -structures, \( \bar{a} \in {M}^{l} \), and \( \bar{b} \in {N}^{l} \) . Then, \( \left( {\mathcal{M},\bar{a}}\right) { \sim }_{\alpha }\left( {\mathcal{N},\bar{b}}\right) \) if and only if \( \mathcal{N} \vDash {\phi }_{\bar{a},\alpha }^{\mat... | Proof We prove this by induction on \( \alpha \) (see Appendix A). Because \( \left( {\mathcal{M},\bar{a}}\right) { \sim }_{0}\left( {\mathcal{N},\bar{b}}\right) \) if and only if they satisfy the same atomic formulas, the lemma holds for \( \alpha = 0 \) . Suppose that \( \gamma \) is a limit ordinal and the lemma is ... | Yes |
For any infinite \( \mathcal{L} \) -structure \( \mathcal{M} \), there is an ordinal \( \alpha < \) \( {\left| M\right| }^{ + } \) such that if \( \bar{a},\bar{b} \in {M}^{l} \) and \( \left( {\mathcal{M},\bar{a}}\right) { \sim }_{\alpha }\left( {\mathcal{M},\bar{b}}\right) \), then \( \left( {\mathcal{M},\bar{a}}\righ... | Proof Let \( {\Gamma }_{\alpha } = \left\{ {\left( {\bar{a},\bar{b}}\right) : \bar{a},\bar{b} \in {M}^{l}}\right. \) for some \( l = 0,1,\ldots \) and \( \left( {\mathcal{M},\bar{a}}\right) { \mathrel{\text{\sim \not{} }} }_{\alpha } \) \( \left( {\mathcal{M},\bar{b}}\right) \} \) . Clearly, \( {\Gamma }_{\alpha } \sub... | Yes |
Theorem 2.4.15 (Scott’s Isomorphism Theorem) Let \( \\mathcal{M} \) be a countable \( \\mathcal{L} \) -structure, and let \( {\\Phi }^{\\mathcal{M}} \\in {\\mathcal{L}}_{{\\omega }_{1},\\omega} \) be the Scott sentence of \( \\mathcal{M} \). Then, \( \\mathcal{N} \\cong \\mathcal{M} \) if and only if \( \\mathcal{N} \\... | Proof Because \( \\alpha \) is the Scott rank of \( \\mathcal{M},\\mathcal{M} \\vDash {\\Phi }^{\\mathcal{M}} \). An easy induction left to the exercises shows that if \( \\mathcal{N} \\cong \\mathcal{M} \), then \( \\mathcal{M} \) and \( \\mathcal{N} \) model the same \( {\\mathcal{L}}_{\\infty ,\\omega} \)-sentences.... | No |
Lemma 3.1.2 Let \( \left( {A, < }\right) \) and \( \left( {B, < }\right) \) be countable dense linear orders, \( {a}_{1},\ldots ,{a}_{n} \in A,{b}_{1},\ldots ,{b}_{n} \in B \), such that \( {a}_{1} < \ldots < {a}_{n} \) and \( {b}_{1} < \ldots < {b}_{n} \) . Then there is an isomorphism \( f : A \rightarrow B \) such t... | Proof Modify the proof of Theorem 2.4.1 starting with \( {A}_{0} = \left\{ {{a}_{1},\ldots ,{a}_{n}}\right\} \) , \( {B}_{0} = \left\{ {{b}_{1},\ldots ,{b}_{n}}\right\} \), and the partial isomorphism \( {f}_{0} : {A}_{0} \rightarrow {B}_{0} \), where \( {f}_{0}\left( {a}_{i}\right) = {b}_{i} \) . The rest of the proof... | Yes |
Lemma 3.1.5 Let \( T \) be an \( \mathcal{L} \) -theory. Suppose that for every quantifier-free \( \mathcal{L} \) -formula \( \theta \left( {\bar{v}, w}\right) \) there is a quantifier-free formula \( \psi \left( \bar{v}\right) \) such that \( T \vDash \exists {w\theta }\left( {\bar{v}, w}\right) \leftrightarrow \psi \... | Proof Let \( \phi \left( \bar{v}\right) \) be an \( \mathcal{L} \) -formula. We wish to show that \( T \vDash \forall \bar{v}(\phi \left( \bar{v}\right) \leftrightarrow \) \( \psi \left( \bar{v}\right) \) ) for some quantifier-free formula \( \phi \left( \bar{v}\right) \) . We prove this by induction on the complexity ... | Yes |
Lemma 3.1.7 Suppose \( G \) and \( H \) are nontrivial torsion free divisible \( A \) belian groups, \( G \subseteq H,\psi \left( {\bar{v}, w}\right) \) is quantifier-free, \( \bar{a} \in G, b \in H \), and \( H \vDash \phi \left( {\bar{a}, b}\right) \) . Then, there is \( c \in G \) such that \( G \vDash \phi \left( {... | Proof We first note that \( \psi \) can be put in disjunctive normal form, namely there are atomic or negated atomic formulas \( {\theta }_{i, j}\left( {\bar{v}, w}\right) \) such that:\n\n\[ \psi \left( {\bar{v}, w}\right) \leftrightarrow \mathop{\bigvee }\limits_{{i = 1}}^{n}\mathop{\bigwedge }\limits_{{j = 1}}^{m}{\... | Yes |
Theorem 3.1.9 DAG has quantifier elimination. | Proof Suppose that \( {G}_{0} \) and \( {G}_{1} \) are torsion-free divisible Abelian groups, \( G \) is a common subgroup of \( {G}_{0} \) and \( {G}_{1},\bar{g} \in G, h \in {G}_{0} \), and \( {G}_{0} \vDash \phi \left( {\bar{g}, h}\right) \) , where \( \phi \) is quantifier-free. Let \( H \) be the divisible hull of... | Yes |
Corollary 3.1.12 Suppose that \( T \) is an \( \mathcal{L} \) -theory such that\ni) \( T \) has algebraically prime models and\nii) \( \mathcal{M}{ \prec }_{s}\mathcal{N} \) whenever \( \mathcal{M} \subseteq \mathcal{N} \) are models of \( T \) .\nThen, \( T \) has quantifier elimination. | Quantifier elimination implies a significant strengthening of ii). | No |
Proposition 3.1.14 If \( T \) has quantifier elimination, then \( T \) is model-complete. | Proof Suppose that \( \mathcal{M} \subseteq \mathcal{N} \) are models of \( T \) . We must show that \( \mathcal{M} \) is an elementary submodel. Let \( \phi \left( \bar{v}\right) \) be an \( \mathcal{L} \) -formula, and let \( \bar{a} \in M \) . There is a quantifier-free formula \( \psi \left( \bar{v}\right) \) such ... | Yes |
Proposition 3.1.15 Let \( T \) be a model-complete theory. Suppose that there is \( {\mathcal{M}}_{0} \vDash T \) such that \( {\mathcal{M}}_{0} \) embeds into every model of \( T \) . Then, \( T \) is complete. | Proof If \( \mathcal{M} \vDash T \), the embedding of \( {\mathcal{M}}_{0} \) into \( \mathcal{M} \) is elementary. In particular \( {\mathcal{M}}_{0} \equiv \mathcal{M} \) . Thus, any two models of \( T \) are elementarily equivalent. | Yes |
Lemma 3.1.16 Let \( G \) be an ordered Abelian group and \( H \) be the divisible hull of \( G \) . We can order \( H \) such that \( i : G \rightarrow H \) is order-preserving, \( \left( {H,+, < }\right) \vDash \) ODAG and if \( {H}^{\prime } \vDash \) ODAG and \( j : G \rightarrow {H}^{\prime } \) is an embedding, th... | Proof We let \( \frac{g}{n} \) denote \( \left\lbrack \left( {g, n}\right) \right\rbrack \) . We can order \( H \) by \( \frac{g}{n} < \frac{h}{m} \) if and only if \( {mg} < {nh} \) . If \( g < h \), then \( \frac{g}{1} < \frac{h}{1} \) so this extends the ordering of \( G \) . If \( \frac{{g}_{1}}{{n}_{1}} < \frac{{g... | Yes |
Corollary 3.1.17 ODAG is a complete decidable theory with quantifier elimination. In particular, every ordered divisible Abelian group is elementarily equivalent to \( \left( {\mathbb{Q},+, < }\right) \) . | Proof By Lemma 3.1.16, ODAG \( {}_{\forall } \) is the theory of ordered Abelian groups and ODAG has algebraically prime models. From the remarks above and Corollary 3.1.12 we see that ODAG has quantifier elimination. The ordered group of rationals embeds into every ordered divisible Abelian group; thus, by Proposition... | Yes |
Lemma 3.1.19 Let \( \left( {G,+, < ,{P}_{2},{P}_{3},\ldots }\right) \vDash T \) . There is \( H \supseteq G \) such that \( H \vDash \Pr \) and if \( {H}^{\prime } \supseteq G \) and \( {H}^{\prime } \vDash \Pr \), then there is \( h : H \rightarrow {H}^{\prime } \) such that \( h \mid G \) is the identity. | Proof We define \( H \) a subgroup of the divisible hull of \( G \) . Let \( H = \left\{ \frac{x}{n}\right. \) : \( \left. {x \in G\text{and}n = 1\text{or}{P}_{n}\left( x\right) }\right\} \) . We let \( {P}_{n}^{H} = {nH} \) .\n\nWe first show that \( H \) is a subgroup of the divisible hull of \( G \) . Suppose that \... | No |
Lemma 3.1.20 If \( G, H \vDash \Pr \) and \( G \subseteq H \), then \( G{ \prec }_{s}H \) . | Proof Let \( \bar{a} \in G \), let \( \phi \left( {v,\bar{w}}\right) \) be a quantifier-free formula, and let \( b \in H \) such that \( H \vDash \phi \left( {b,\bar{a}}\right) \) . We claim that we may assume that \( \phi \left( {v,\bar{a}}\right) \) is of the form \[ \bigwedge {m}_{i}v = {g}_{i} \land \bigwedge {P}_{... | Yes |
Corollary 3.1.21 Presburger arithmetic is a complete decidable theory with quantifier elimination in the language \( {\mathcal{L}}^{ * } \) . | Proof Because \( \mathbb{Z} \) can be embedded in any model of \( \Pr ,\Pr \) is complete by Proposition3.1.15. Because we have given a recursive set of axioms Pr is decidable by Lemma 2.2.8. | No |
Proposition 3.1.22 Suppose that \( T \) is a decidable theory with quantifier elimination. Then, there is an algorithm which when given a formula \( \phi \) as input will output a quantifier-free formula \( \psi \) such that \( T \vDash \phi \leftrightarrow \psi \) . | Proof Given input \( \phi \left( \bar{v}\right) \) we search for a quantifier-free formula \( \psi \left( \bar{v}\right) \) such that \( T \vDash \forall \bar{v}\left( {\phi \left( \bar{v}\right) \leftrightarrow \psi \left( \bar{v}\right) }\right) \) . Because \( T \) is decidable this is an effective search. Because \... | Yes |
Lemma 3.2.1 \( {\mathrm{{ACF}}}_{\forall } \) is the theory of integral domains. | Proof The axioms for integral domains are universal consequences of ACF. If \( D \) is an integral domain, then the algebraic closure of the fraction field of \( D \) is a model of ACF. Because every integral domain is a subring of an algebraically closed field, \( {\mathrm{{ACF}}}_{\forall } \) is the theory of integr... | Yes |
Theorem 3.2.2 ACF has quantifier elimination. | Proof We will apply Corollary 3.1.12. If \( D \) is an integral domain, then the algebraic closure of the fraction field of \( D \) embeds into any algebraically closed field containing \( D \) . Thus, ACF has algebraically prime models.\n\nTo prove quantifier elimination, we need only show that if \( K \) and \( F \) ... | Yes |
Corollary 3.2.3 ACF is model-complete and \( {\mathrm{{ACF}}}_{p} \) is complete where \( p = \) \( 0 \) or \( p \) is prime. | Proof Model-completeness is an immediate consequence of quantifier elimination.\n\nThe completeness of \( {\mathrm{{ACF}}}_{p} \) was proved in Proposition 2.2.5, but it also follows from quantifier elimination. Suppose that \( K, L \vDash {AC}{F}_{p} \) . Let \( \phi \) be any sentence in the language of rings. By qua... | Yes |
Lemma 3.2.4 Let \( K \) be a field.\ni) If \( X \subseteq {K}^{n} \), then \( I\left( X\right) \) is a radical ideal. | i) Suppose that \( p, q \in I\left( X\right) \) and \( f \in K\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) . If \( a \in X \), then \( p\left( a\right) + q\left( a\right) = f\left( a\right) p\left( a\right) = 0 \) . Thus, \( p + q,{fp} \in I\left( X\right) \) and \( I\left( X\right) \) is an ideal. If \( {f}^... | Yes |
Lemma 3.2.7 Let \( K \) be a field. The subsets of \( {K}^{n} \) defined by atomic formulas are exactly those of the form \( V\left( p\right) \) for some \( p \in K\left\lbrack \bar{X}\right\rbrack \) . A subset of \( {K}^{n} \) is quantifier-free definable if and only if it is a Boolean combination of Zariski closed s... | Proof If \( \phi \left( {\bar{x},\bar{y}}\right) \) is an atomic \( {\mathcal{L}}_{\mathrm{r}} \) -formula, then there is \( q\left( {\bar{X},\bar{Y}}\right) \in \mathbb{Z}\left\lbrack {\bar{X},\bar{Y}}\right\rbrack \) such that \( \phi \left( {\bar{x},\bar{y}}\right) \) is equivalent to \( q\left( {\bar{x},\bar{y}}\ri... | Yes |
Corollary 3.2.8 Let \( K \) be an algebraically closed field.\ni) \( X \subseteq {K}^{n} \) is constructible if and only if it is definable.\nii) (Chevalley's Theorem) The image of a constructible set under a polynomial map is constructible. | Proof i) By Lemma 3.2.7, the constructible sets are exactly the quantifier-free definable sets, but by quantifier elimination every definable set is quantifier-free definable.\n\nii) Let \( X \subseteq {K}^{n} \) be constructible and \( p : {K}^{n} \rightarrow {K}^{m} \) be a polynomial map. Then, the image of \( X = \... | Yes |
Corollary 3.2.9 If \( K \) is an algebraically closed field and \( X \subseteq K \) is algebraically closed, then either \( X \) or \( K \smallsetminus X \) is finite. Thus, ACF is strongly minimal. | Proof By quantifier elimination \( X \) is a finite Boolean combination of sets of the form \( V\left( p\right) \), where \( p \in K\left\lbrack X\right\rbrack \) . But \( V\left( p\right) \) is either finite or (if \( p = 0 \) ) all of \( K \) . | Yes |
Theorem 3.2.11 (Hilbert’s Nullstellensatz) Let \( K \) be an algebraically closed field. Suppose that \( I \) and \( J \) are radical ideals in \( K\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) and \( I \subset J \) . Then \( V\left( J\right) \subset V\left( I\right) \) . Thus \( X \mapsto I\left( X\right) \) ... | Proof Let \( p \in J \smallsetminus I \) . By Lemma 3.2.10, there is a prime ideal \( P \supseteq I \) such that \( p \notin P \) . We will show that there is \( x \in V\left( P\right) \subseteq V\left( I\right) \) such that \( p\left( x\right) \neq 0 \) . Thus \( V\left( I\right) \neq V\left( J\right) \) . Because \( ... | Yes |
Proposition 3.2.14 If \( X \subseteq {K}^{n} \) is constructible and \( f : X \rightarrow K \) is definable, then there are constructible sets \( {X}_{1},\ldots ,{X}_{m} \) and quasirational functions \( {\rho }_{1},\ldots ,{\rho }_{m} \) such that \( \bigcup {X}_{i} = X \) and \( f\left| {{X}_{i} = {\rho }_{i}}\right|... | Proof Let \( \Gamma \left( {{v}_{1},\ldots ,{v}_{n}}\right) = \{ f\left( \bar{x}\right) \neq \rho \left( \bar{x}\right) : \rho \) a quasirational function \( \} \) \( \cup \{ \bar{v} \in X\} \cup \mathrm{{ACF}} \cup \operatorname{Diag}\left( K\right) . \n\nClaim \( \Gamma \) is not satisfiable.\n\nSuppose that \( \Gamm... | Yes |
Proposition 3.2.15 Let \( K \vDash \mathrm{{ACF}} \) and \( A \subseteq K \) . Then, \( a \in \operatorname{acl}\left( A\right) \) if and only if \( a \) is algebraic over the subfield of \( K \) generated by \( A \) . | Proof Let \( k \) be the field generated by \( A \) . If \( a \) is algebraic over \( k \), there are polynomials \( {q}_{0}\left( {{X}_{1},\ldots ,{X}_{n}}\right) ,\ldots ,{q}_{m}\left( {{X}_{1},\ldots ,{X}_{n}}\right) \in \mathbb{Z}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) and \( {b}_{1},\ldots ,{b}_{n} ... | Yes |
Lemma 3.2.16 There is a definable function \( f : {k}^{nm} \rightarrow {k}^{l} \) for some \( l \in \omega \) such that \( \bar{c}E\bar{d} \) if and only if \( f\left( \bar{c}\right) = f\left( \bar{d}\right) \) . | Proof Suppose that \( \bar{c} = \left( {{\bar{c}}_{1},\ldots ,{\bar{c}}_{m}}\right) \) where \( {\bar{c}}_{i} = \left( {{c}_{i,1},\ldots ,{c}_{i, n}}\right) \) . Let \( {q}_{i}^{\bar{c}} \) be the polynomial\n\n\[ Y - \mathop{\sum }\limits_{{j = 1}}^{n}{c}_{i, j}{X}_{j} \]\n\nin \( k\left\lbrack {{X}_{1},\ldots ,{X}_{n... | Yes |
Lemma 3.2.18 Suppose that \( \bar{c} \) is algebraic over \( \bar{a}/E,\bar{d},\bar{b} \) and \( \bar{b} \) is algebraic over \( \bar{a}/E,\bar{d} \), then \( \bar{c} \) is algebraic over \( \bar{a}/E,\bar{d} \) . | Proof Exercise 3.4.16. | No |
Lemma 3.2.19 Suppose that \( K \) is an algebraically closed field and \( E \) is a definable equivalence relation on \( {K}^{n} \) . Let \( \psi \left( {\bar{x},\bar{y},\bar{d}}\right) \) define \( E \) . If \( \bar{a} \in {K}^{n} \) , then there is \( \bar{c} \in {K}^{n} \) algebraic over \( \bar{a}/E,\bar{d} \) such... | Proof Let \( \psi \left( {\bar{x},\bar{y},\bar{d}}\right) \) define \( E \) . Let \( 0 \leq m \leq n \) be maximal such that there are \( {c}_{1},\ldots ,{c}_{m} \) algebraic over \( \bar{a}/E,\bar{d} \) such that\n\n\[ K \vDash \exists {v}_{m + 1}\ldots \exists {v}_{n}\psi \left( {\bar{c},\bar{v},\bar{a},\bar{d}}\righ... | Yes |
Theorem 3.2.20 Suppose that \( K \) is an algebraically closed field, \( A \subseteq K \) , and \( E \) is an \( A \) -definable equivalence relation on \( {K}^{n} \) . Then for some \( l \) there is an A-definable function \( f : {K}^{n} \rightarrow {K}^{l} \) such that \( \bar{x}E\bar{y} \) if and only if \( f\left( ... | Proof For notational simplicity, we will assume that \( E \) is defined over \( \varnothing \) .\n\nFor each formula \( \phi \left( {\bar{x},\bar{y}}\right) \) and \( k > 0 \), let \( {\Theta }_{\phi, k}\left( \bar{y}\right) \) be the conjunction of\n\ni) \( \forall \bar{x}\left( {\phi \left( {\bar{x},\bar{y}}\right) \... | Yes |
Theorem 3.3.5 Let \( F \) be a formally real field. The following are equivalent.\ni) \( F \) is real closed.\nii) \( F\left( i\right) \) is algebraically closed (where \( {i}^{2} = - 1 \) ).\niii) For any \( a \in F \), either a or \( - a \) is a square and every polynomial of odd degree has a root. | Proof We can axiomatize real closed fields by:\ni) axioms for fields\nii) for each \( n \geq 1 \), the axiom\n\[ \forall {x}_{1}\ldots \forall {x}_{n}{x}_{1}^{2} + \ldots + {x}_{n}^{2} + 1 \neq 0 \]\niii) \( \forall x\exists y\left( {{y}^{2} = x \vee {y}^{2} + x = 0}\right) \)\niv) for each \( n \geq 0 \), the axiom\n\... | No |
Corollary 3.3.6 The class of real closed fields is an elementary class of \( {\mathcal{L}}_{\mathrm{r}} \) -structures. | Proof We can axiomatize real closed fields by:\n\ni) axioms for fields\n\nii) for each \( n \geq 1 \), the axiom\n\n\[ \forall {x}_{1}\ldots \forall {x}_{n}{x}_{1}^{2} + \ldots + {x}_{n}^{2} + 1 \neq 0 \]\n\niii) \( \forall x\exists y\left( {{y}^{2} = x \vee {y}^{2} + x = 0}\right) \)\n\niv) for each \( n \geq 0 \), th... | Yes |
Proposition 3.3.8 If \( F \) is a real closed field and \( X \subseteq {F}^{n} \) is definable by an \( {\mathcal{L}}_{\text{or }} \) -formula, then \( X \) is definable by an \( {\mathcal{L}}_{\mathrm{r}} \) -formula. | Proof Replace all instances of \( {t}_{i} < {t}_{j} \) by \( \exists v\left( {v \neq 0 \land {v}^{2} + {t}_{i} = {t}_{j}}\right) \), where \( {t}_{i} \) and \( {t}_{j} \) are terms occurring in the definition of \( X \) (see Exercise 1.4.15). | No |
Lemma 3.3.11 If \( \left( {F, < }\right) \) is an ordered field, \( 0 < x \in F \), and \( x \) is not a square in \( F \), then we can extend the ordering of \( F \) to \( F\left( \sqrt{x}\right) \) . | Proof We can extend the ordering to \( F\left( \sqrt{x}\right) \) by \( 0 < a + b\sqrt{x} \) if and only if\ni) \( b = 0 \) and \( a > 0 \), or\n\nii) \( b > 0 \) and \( \left( {a > 0}\right. \) or \( \left. {x > \frac{{a}^{2}}{{b}^{2}}}\right) \), or\n\niii) \( b < 0 \) and \( \left( {a < 0}\right. \) and \( \left. {x... | Yes |
Corollary 3.3.14 RCF has algebraically prime models. | Proof Let \( \left( {D, < }\right) \) be an ordered domain, and let \( \left( {R, < }\right) \) be the real closure of the fraction field compatible with the ordering of \( D \) . Let \( \left( {F, < }\right) \) be any real closed field extension of \( \left( {D, < }\right) \) . Let \( K = \{ \alpha \in F : \alpha \) i... | Yes |
Theorem 3.3.15 The theory RCF admits elimination of quantifiers in \( {\mathcal{L}}_{\text{or }} \) . | Proof Because RCF has algebraically prime models, by Corollary 3.1.12, we need only show that \( F{ \prec }_{s}K \) when \( F, K \vDash \operatorname{RCF} \) and \( F \subseteq K \) . Let \( \phi \left( {v,\bar{w}}\right) \) be a quantifier-free formula and let \( \bar{a} \in F, b \in K \) be such that \( K \vDash \phi... | Yes |
Corollary 3.3.16 RCF is complete, model complete, and decidable. Thus RCF is the theory of \( \left( {\mathbb{R},+,\cdot , < }\right) \) and RCF is decidable. | Proof By quantifier elimination, RCF is model complete.\n\nEvery real closed field has characteristic zero; thus, the rational numbers are embedded in every real closed field. Therefore, \( {\mathbb{R}}_{\text{alg }} \), the field of real algebraic numbers (i.e., the real closure of the rational numbers) is a subfield ... | Yes |
Corollary 3.3.19 If \( F \vDash {RCF} \) and \( A \subseteq {F}^{n} \) is semialgebraic, then the closure (in the Euclidean topology) of \( F \) is semialgebraic. | Proof We repeat the main idea of Lemma 1.3.3. Let \( d \) be the definable function\n\n\[ d\left( {{x}_{1},\ldots ,{x}_{n},{y}_{1},\ldots ,{y}_{n}}\right) = z\text{ if and only if }z \geq 0 \land {z}^{2} = \mathop{\sum }\limits_{{i = 1}}^{n}{\left( {x}_{i} - {y}_{i}\right) }^{2}. \]\n\nThe closure of \( A \) is\n\n\[ \... | Yes |
Corollary 3.3.20 Let \( F \) be a real closed field. If \( X \subseteq {F}^{n} \) is closed and bounded, and \( f \) is a continuous semialgebraic function, then \( f\left( X\right) \) is closed and bounded. | Proof If \( F = \mathbb{R} \), then \( X \) is closed and bounded if and only if \( X \) is compact. Because the continuous image of a compact set is compact, the continuous image of a closed and bounded set is closed and bounded.\n\nIn general, there are \( \bar{a},\bar{b} \in F \) and formulas \( \phi \) and \( \psi ... | Yes |
Theorem 3.3.22 (Hilbert’s 17th Problem) If \( f \) is a positive semidefinite rational function over a real closed field \( F \), then \( f \) is a sum of squares of rational functions. | Proof Suppose that \( f\left( {{X}_{1},\ldots ,{X}_{n}}\right) \) is a positive semidefinite rational function over \( F \) that is not a sum of squares. By Theorem 3.3.3, there is an ordering of \( F\left( \bar{X}\right) \) so that \( f \) is negative. Let \( R \) be the real closure of \( F\left( \bar{X}\right) \) ex... | Yes |
Corollary 3.3.23 RCF is an o-minimal theory. | Proof Let \( R \vDash \) RCF. We need to show that every definable subset of \( R \) is a finite union of points and intervals with endpoints in \( R \cup \{ \pm \infty \} \) . By quantifier elimination, very definable subset of \( R \) is a finite Boolean combination of sets of the form\n\n\[ \n\{ x \in R : p\left( x\... | Yes |
Lemma 3.3.24 If \( f : \mathbb{R} \rightarrow \mathbb{R} \) is semialgebraic, then for any open interval \( U \subseteq \mathbb{R} \) there is a point \( x \in U \) such that \( f \) is continuous at \( x \) . | ## Proof\ncase 1: There is an open set \( V \subseteq U \) such that \( f \) has finite range on \( V \) .\n\nPick an element \( b \) in the range of \( f \) such that \( \{ x \in V : f\left( x\right) = b\} \) is infinite. By o-minimality, there is an open set \( {V}_{0} \subseteq V \) such that \( f \) is constantly \... | Yes |
Corollary 3.3.25 Let \( F \) be a real closed field and \( f : F \rightarrow F \) is a semi-algebraic function. Then, we can partition \( F \) into \( {I}_{1} \cup \ldots \cup {I}_{m} \cup X \), where \( X \) is finite and the \( {I}_{j} \) are pairwise disjoint open intervals with endpoints in \( F \cup \{ \pm \infty ... | Proof Let\n\n\[ D = \{ x : F \vDash \exists \epsilon > 0\;\forall \delta > 0\;\exists y\;\left| {x - y}\right| < \delta \land \left| {f\left( x\right) - f\left( y\right) }\right| > \epsilon \} \]\n\nbe the set of points where \( f \) is discontinuous. Because \( D \) is definable, by o-minimality \( D \) is either fini... | Yes |
Corollary 3.3.26 Let \( F \) be a real closed field and \( X \subseteq {F}^{n + m} \) be semial-gebraic. There is a semialgebraic function \( f : {F}^{n} \rightarrow {F}^{m} \) such that for all \( \bar{x} \in {F}^{n} \), if there is a \( \bar{y} \in {F}^{m} \) such that \( \left( {\bar{x},\bar{y}}\right) \in X \), the... | Proof We proceed by induction on \( m \) . For each \( m \) we will show that our claim holds for all \( n \) and for all definable \( X \subseteq {F}^{n + m} \) .\n\nAssume that \( m = 1 \) . For \( \bar{a} \in {F}^{n} \), let \( {X}_{\bar{a}} = \{ y : \left( {\bar{a}, y}\right) \in X\} \) . By o-minimality, \( {X}_{\... | Yes |
Corollary 3.3.27 (Curve Selection) Let \( F \) be a real closed field. Let \( X \subseteq {F}^{n} \) be semialgebraic and \( \bar{a} \) be a point in the closure of \( X \). There is a continuous semialgebraic function \( f : \left( {0, r}\right) \rightarrow {F}^{n} \) such that for all \( \epsilon \in \left( {0, r}\ri... | Proof Let\n\n\[ \nX = \left\{ {\left( {\epsilon ,\bar{y}}\right) : \mathop{\sum }\limits_{{i = 1}}^{n}{\left( {y}_{i} - {a}_{i}\right) }^{2} < \epsilon }\right\} .\n\] \n\nFor all \( \epsilon > 0 \), there is \( \bar{y} \in {F}^{n} \) with \( \left( {\epsilon ,\bar{y}}\right) \in X \). By Corollary 3.3.26, there is a d... | Yes |
Corollary 3.3.28 Let \( F \) be a real closed field. Let \( E \subseteq {F}^{n} \times {F}^{n} \) be a definable equivalence relation. There is a definable \( X \subset {F}^{n} \) such that for all \( a \in {F}^{n} \) there is a unique \( b \in X \) such that \( {aEb} \) . We call \( X \) a definable transversal of \( ... | Proof Let \( f : {F}^{n} \rightarrow {F}^{n} \) be a definable invariant Skolem function. Then, \( {aEf}\left( a\right) \) for all \( a \in {F}^{n} \) and if \( {aEb} \), then \( f\left( a\right) = f\left( b\right) \) . Let \( X \) be the range of \( f \) . | Yes |
Lemma 3.3.30 (Uniform Bounding) Let \( X \subseteq {F}^{n + 1} \) be semialgebraic. There is a natural number \( N \) such that if \( \bar{a} \in {F}^{n} \) and \( {X}_{\bar{a}} = \{ y : \left( {\bar{a}, y}\right) \in X\} \) is finite, then \( \left| {X}_{\bar{a}}\right| < N \) . | Proof First, note that \( {X}_{\bar{a}} \) is infinite if and only if there is no interval \( \left( {c, d}\right) \) such that \( \left( {c, d}\right) \subseteq {X}_{\bar{a}} \) . Thus \( \left\{ {\left( {\bar{a}, b}\right) \in X : {X}_{\bar{a}}}\right. \) is finite \( \} \) is definable. Without loss of generality, w... | Yes |
Proposition 4.1.3 Let \( \mathcal{M} \) be an \( \mathcal{L} \) -structure, \( A \subseteq M \), and \( p \) an n-type over \( A \) . There is \( \mathcal{N} \) an elementary extension of \( \mathcal{M} \) such that \( p \) is realized in \( \mathcal{N} \) . | Proof Let \( \Gamma = p \cup {\operatorname{Diag}}_{\mathrm{{el}}}\left( \mathcal{M}\right) \) . We claim that \( \Gamma \) is satisfiable.\n\nSuppose that \( \Delta \) is a finite subset of \( \Gamma \) . Without loss of generality, \( \Delta \) is the single formula\n\n\[ \phi \left( {{v}_{1},\ldots ,{v}_{n},{a}_{1},... | Yes |
Corollary 4.1.4 \( p \in {S}_{n}^{\mathcal{M}}\left( A\right) \) if and only if there is an elementary extension \( \mathcal{N} \) of \( \mathcal{M} \) and \( \bar{a} \in {N}^{n} \) such that \( p = {\operatorname{tp}}^{\mathcal{N}}\left( {\bar{a}/A}\right) \) . | Proof If \( \bar{a} \in {N}^{n} \), then \( {\operatorname{tp}}^{\mathcal{N}}\left( {\bar{a}/A}\right) \in {S}_{n}^{\mathcal{N}}\left( A\right) = {S}_{n}^{\mathcal{M}}\left( A\right) \) . On the other hand if \( p \in {S}_{n}^{\mathcal{M}}\left( A\right) \), then, by Proposition 4.1.3, there is an elementary extension ... | Yes |
Lemma 4.1.6 Let \( \mathcal{M},\mathcal{N}, B \) be as above and let \( f : B \rightarrow N \) be partial elementary. If \( b \in M \), there is an elementary extension \( {\mathcal{N}}_{1} \) of \( \mathcal{N} \) and \( g \) : \( B \cup \{ b\} \rightarrow {\mathcal{N}}_{1} \) a partial elementary map extending \( f \)... | Proof Let \( \Gamma = \left\{ {\phi \left( {v, f\left( {a}_{1}\right) ,\ldots, f\left( {a}_{n}\right) }\right) : \mathcal{M} \vDash \phi \left( {b,{a}_{1},\ldots ,{a}_{n}}\right) ,{a}_{1},\ldots ,{a}_{n} \in }\right. \n\n\( B\} \cup {\operatorname{Diag}}_{\mathrm{{el}}}\left( \mathcal{N}\right) \) .\n\nSuppose that we ... | Yes |
Corollary 4.1.7 If \( \mathcal{M} \) and \( \mathcal{N} \) are \( \mathcal{L} \) -structures, \( B \subseteq M \) and \( f : B \rightarrow N \) is a partial elementary map, then there is \( {\mathcal{N}}^{\prime } \) an elementary extension of \( \mathcal{N} \) and \( g : \mathcal{M} \rightarrow {\mathcal{N}}^{\prime }... | Proof Let \( \kappa = \left| M\right| \), and let \( \left\{ {{a}_{\alpha } : \alpha < \kappa }\right\} \) be an enumeration of \( M \) . Let \( {\mathcal{N}}_{0} = \mathcal{N},{B}_{0} = B \), and \( {g}_{0} = f \) . Let \( {B}_{\alpha } = B \cup \left\{ {{a}_{\beta } : \beta < \alpha }\right\} \) . We inductively buil... | Yes |
Proposition 4.1.11 Let \( p \in {S}_{n}^{\mathcal{M}}\left( A\right) \) . The following are equivalent.\n\ni) \( p \) is isolated.\n\nii) \( \{ p\} = \left\lbrack {\phi \left( \bar{v}\right) }\right\rbrack \) for some \( {\mathcal{L}}_{A} \) -formula \( \phi \left( \bar{v}\right) \) . We say that \( \phi \left( \bar{v}... | Proof\n\ni) \( \Rightarrow \) ii) If \( X \) is open, then\n\n\[X = \mathop{\bigcup }\limits_{{i \in I}}\left\lbrack {\phi }_{i}\right\rbrack\]\n\nfor some collection of formulas \( \left( {{\phi }_{i} : i \in I}\right) \) . If \( \{ p\} \) is open, then \( \{ p\} = \left\lbrack \phi \right\rbrack \) for some formula \... | Yes |
Proposition 4.1.16 The map \( p \mapsto {I}_{p} \) is a continuous bijection from \( {S}_{n}^{K}\left( k\right) \) to \( \operatorname{Spec}\left( {k\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack }\right) \) . | Proof We have shown that the map is one-to-one so we need only show that it is continuous. Suppose that \( {f}_{1},\ldots ,{f}_{m} \in k\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) . Then, the inverse image of \( \left\{ {P \in \operatorname{Spec}\left( {k\left\lbrack \overline{X}\right\rbrack }\right) : {f}_... | Yes |
Corollary 4.1.17 The Zariski topology on \( \operatorname{Spec}\left( {k\left\lbrack \bar{X}\right\rbrack }\right) \) is compact. | Proof This is clear because \( {S}_{n}^{K}\left( k\right) \) is compact and \( p \mapsto {I}_{p} \) is continuous. | Yes |
Corollary 4.1.18 Suppose that \( K \vDash \mathrm{{ACF}} \) and \( k \) is a subfield of \( K \) . Then \( \left| {{S}_{n}^{K}\left( k\right) }\right| = \left| k\right| + {\aleph }_{0} \) | Proof By Hilbert’s Basis Theorem, all ideals in \( k\left\lbrack \bar{X}\right\rbrack \) are finitely generated. Thus, there are only \( \left| k\right| + {\aleph }_{0} \) prime ideals. | Yes |
Proposition 4.2.2 If \( \phi \left( \bar{v}\right) \) isolates \( p \), then \( p \) is realized in any model of \( T \cup \{ \exists \bar{v}\phi \left( \bar{v}\right) \} \) . In particular, if \( T \) is complete, then every isolated type is realized. | Proof If \( \mathcal{M} \vDash T \) and \( \mathcal{M} \vDash \phi \left( \bar{a}\right) \), then \( \bar{a} \) realizes \( p \) . If \( T \) is complete and \( T \cup \{ \phi \left( \bar{v}\right) \} \) is satisfiable, then \( T \vDash \exists \bar{v}\phi \left( \bar{v}\right) \) . | No |
Theorem 4.2.4 Let \( \mathcal{L} \) be a countable language, and let \( T \) be an \( \mathcal{L} \) -theory. Let \( X \) be a countable collection of nonisolated types over \( \varnothing \) . There is a countable \( \mathcal{M} \vDash T \) that omits all of the types \( p \in X \) . | Proof (Sketch) Let \( {p}_{0},{p}_{1},\ldots \) list \( X \) . Let \( C \) be as in the proof of Theorem 4.2.3, and let \( {\bar{d}}_{0},{\bar{d}}_{1}\ldots \) list all finite sequences from \( C \) . Fix \( \pi : \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N} \) , a bijection.\n\nWe do a Henkin-style argument as ... | Yes |
If \( \mathcal{M} \) is a countable model of \( {PA} \), then there is \( \mathcal{M} \prec \mathcal{N} \) such that \( \mathcal{N} \) is a proper end extension of \( \mathcal{M} \) . | Consider the language \( {\mathcal{L}}^{ * } \) where we have constant symbols for all elements of \( M \) and a new constant symbol \( c \) . Let \( T = {\operatorname{Diag}}_{\mathrm{{el}}}\left( \mathcal{M}\right) \cup \{ c > m \) : \( m \in M\} \), and for \( a \in M \smallsetminus \mathbb{N} \) let \( {p}_{a} \) b... | Yes |
Theorem 4.2.8 Let \( \mathcal{L} \) be a countable language and let \( T \) be a complete \( \mathcal{L} \) -theory with infinite models. Then, \( \mathcal{M} \vDash T \) is prime if and only if it is countable and atomic. | ## Proof\n\n\( \left( \Rightarrow \right) \) We have argued that prime models are atomic. Because \( \mathcal{L} \) is countable, \( T \) has a countable model. Thus, the prime model must be countable.\n\n\( \left( \Leftarrow \right) \) Let \( \mathcal{M} \) be countable and atomic. Let \( \mathcal{N} \vDash T \) . We ... | Yes |
Lemma 4.2.9 Suppose that \( \left( {\bar{a},\bar{b}}\right) \in {M}^{m + n} \) realizes an isolated type in \( {S}_{m + n}\left( T\right) \) . Then \( \bar{a} \) realizes an isolated type in \( {S}_{m}\left( T\right) \) . Indeed if \( A \subseteq M \) and \( \left( {\bar{a},\bar{b}}\right) \in {M}^{m + n} \) realizes a... | Proof Let \( \phi \left( {\bar{v},\bar{w}}\right) \) isolate \( {\operatorname{tp}}^{\mathcal{M}}\left( {\bar{a},\bar{b}/A}\right) \) . We claim that \( \exists {w\phi }\left( {\bar{v},\bar{w}}\right) \) isolates \( {\operatorname{tp}}^{\mathcal{M}}\left( {\bar{a}/A}\right) \) . Let \( \psi \left( \bar{v}\right) \) be ... | Yes |
Proposition 4.2.13 Suppose that \( \mathcal{M} \) is homogeneous, \( A \subset M,\left| A\right| < \left| M\right| \) , and \( f : A \rightarrow M \) is a partial elementary map. Then, there is an automorphism \( \sigma \) of \( \mathcal{M} \) with \( \sigma \supseteq f \) . | Proof Let \( \left| M\right| = \kappa \), and let \( \left( {{a}_{\alpha } : \alpha < \kappa }\right) \) be an enumeration of \( M \) . We build a sequence of partial elementary maps \( \left( {{f}_{\alpha } : \alpha < \kappa }\right) \) extending \( f \) with \( {f}_{\alpha } \subseteq {f}_{\beta } \) for \( \alpha < ... | Yes |
Lemma 4.2.14 If \( \mathcal{M} \) is atomic, then \( \mathcal{M} \) is \( {\aleph }_{0} \) -homogeneous. In particular, countable atomic models are homogeneous. | Proof Suppose that \( \bar{a} \mapsto \bar{b} \) is elementary and \( c \in M \) . Let \( \phi \left( {\bar{v}, w}\right) \) isolate \( {\operatorname{tp}}^{\mathcal{M}}\left( {\bar{a}, c}\right) \) . Because \( \mathcal{M} \vDash \exists {w\phi }\left( {\bar{a}, w}\right) \) and \( \bar{a} \mapsto \bar{b} \) is elemen... | Yes |
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