Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
|---|---|---|
Theorem 4.2.15 Let \( T \) be a complete theory in a countable language. Suppose that \( \mathcal{M} \) and \( \mathcal{N} \) are countable homogeneous models of \( T \) and \( \mathcal{M} \) and \( \mathcal{N} \) realize the same types in \( {S}_{n}\left( T\right) \) for \( n \geq 1 \) . Then \( \mathcal{M} \cong \mat... | Proof We build an isomorphism \( f : \mathcal{M} \rightarrow \mathcal{N} \) by a back-and-forth argument. We will build \( {f}_{0} \subset {f}_{1} \subset \ldots \), a sequence of partial elementary maps with finite domain, and let \( f = \mathop{\bigcup }\limits_{{i = 0}}^{\infty }{f}_{i} \) . Let \( {a}_{0},{a}_{1},\... | Yes |
Corollary 4.2.16 Let \( T \) be a complete theory in a countable language. If \( \mathcal{M} \) and \( \mathcal{N} \) are prime models of \( T \), then \( \mathcal{M} \cong \mathcal{N} \) . | Proof By Theorem 4.2.8, \( \mathcal{M} \) and \( \mathcal{N} \) are atomic. Because the types in \( {S}_{n}\left( T\right) \) realized in an atomic model are exactly the isolated types, \( \mathcal{M} \) and \( \mathcal{N} \) realize the same types. By Lemma 4.2.14, countable atomic models are homogeneous. Thus, by The... | Yes |
Theorem 4.2.18 Let \( T \) be a complete theory in a countable language. If \( T \) is \( \omega \) -stable, then \( T \) is \( \kappa \) -stable for all infinite cardinals \( \kappa \) . | Proof Suppose that \( \mathcal{M} \vDash T, A \subseteq M,\left| A\right| = \kappa \) and \( \left| {{S}_{n}^{\mathcal{M}}\left( A\right) }\right| > \kappa \) . Because there are only \( \kappa \) formulas with parameters from \( A \), there is some \( {\mathcal{L}}_{A} \) -formula \( {\phi }_{\varnothing }\left( \bar{... | Yes |
Proposition 4.2.19 Let \( T \) be a complete theory in a countable language. If \( T \) is \( \omega \)-stable, then for all \( \mathcal{M} \vDash T \) and \( A \subseteq M \), the isolated types in \( {S}_{n}^{\mathcal{M}}\left( A\right) \) are dense. | Proof Suppose not. We can build a binary tree of formulas as in Theorem 4.2.11 i). As in Theorem 4.2.18, we can find a countable \( {A}_{0} \subseteq A \) such that all parameters come from \( {A}_{0} \). But then \( \left| {{S}_{n}^{\mathcal{M}}\left( {A}_{0}\right) }\right| = {2}^{{\aleph }_{0}} \), contradicting the... | Yes |
Lemma 4.2.21 Suppose that \( A \subseteq B \subseteq \mathcal{M} \vDash T \) and every \( \bar{b} \in {B}^{m} \) realizes an isolated type in \( {S}_{m}^{\mathcal{M}}\left( A\right) \) . Suppose that \( \bar{a} \in {M}^{n} \) realizes an isolated type in \( {S}_{n}^{\mathcal{M}}\left( B\right) \) . Then, \( \bar{a} \) ... | Proof Let \( \phi \left( {\bar{v},\bar{w}}\right) \) be an \( \mathcal{L} \) -formula and \( \bar{b} \in {B}^{m} \) such that \( \phi \left( {\bar{v},\bar{b}}\right) \) isolates \( {\operatorname{tp}}^{\mathcal{M}}\left( {\bar{a}/B}\right) \) . Let \( \theta \left( \bar{w}\right) \) be an \( {\mathcal{L}}_{A} \) -formu... | Yes |
Proposition 4.3.2 Let \( \kappa \geq {\aleph }_{0} \) . The following are equivalent:\n\ni) \( \mathcal{M} \) is \( \kappa \) -saturated.\n\nii) If \( A \subseteq M \) with \( \left| A\right| < \kappa \) and \( p \) is a (possibly incomplete) \( n \) -type over \( A \), then \( p \) is realized in \( \mathcal{M} \).\n\... | ## Proof\n\ni) \( \Rightarrow \) ii) If \( \mathcal{M} \) is \( \kappa \) -saturated and \( p \) is an incomplete \( n \) -type over \( A \) where \( \left| A\right| < \kappa \), then there is a complete type \( {p}^{ * } \in {S}_{n}^{\mathcal{M}}\left( A\right) \) with \( {p}^{ * } \supseteq p \) . Because \( {p}^{ * ... | Yes |
Proposition 4.3.3 If \( \mathcal{M} \) is \( \kappa \) -saturated, then \( \mathcal{M} \) is \( \kappa \) -homogeneous. | Proof Suppose that \( A \subseteq \mathcal{M},\left| A\right| < \kappa \), and \( f : A \rightarrow M \) is partial elementary. Let \( b \in M \smallsetminus A \) . Let\n\n\[ \Gamma = \left\{ {\phi \left( {v, f\left( \bar{a}\right) }\right) : \bar{a} \in {A}^{m}\text{ and }\mathcal{M} \vDash \phi \left( {b,\bar{a}}\rig... | Yes |
Proposition 4.3.4 If \( \mathcal{M} \vDash T \), then \( \mathcal{M} \) is \( {\aleph }_{0} \) -saturated if and only if \( \mathcal{M} \) is \( {\aleph }_{0} \) -homogeneous and \( \mathcal{M} \) realizes all types in \( {S}_{n}\left( T\right) \) . | Proof\n\n\( \left( \Rightarrow \right) \) Clear.\n\n\( \left( \Leftarrow \right) \) Let \( \bar{a} \in {M}^{m} \) and let \( p \in {S}_{n}^{\mathcal{M}}\left( \bar{a}\right) \) . Let \( q \in {S}_{n + m}\left( T\right) \) be the type \( \{ \phi \left( {\bar{v},\bar{w}}\right) : \phi \left( {\bar{v},\bar{a}}\right) \in ... | Yes |
Corollary 4.3.5 If \( \mathcal{M},\mathcal{N} \vDash T \) are countable saturated models, then \( \mathcal{M} \cong \) \( \mathcal{N} \) . | Proof Because \( \mathcal{M} \) and \( \mathcal{N} \) are \( {\aleph }_{0} \) -homogeneous and both realize all types in \( {S}_{n}\left( T\right) \) for all \( n < \omega \), by Theorem 4.2.15, \( \mathcal{M} \cong \mathcal{N} \) . | Yes |
Proposition 4.3.6 Let \( \mathcal{M} \vDash T \) . There is \( \mathcal{M} \prec \mathcal{N} \) such that \( \mathcal{N} \) is \( {\aleph }_{0} \) - homogeneous and \( \left| N\right| = \left| M\right| \) . | Proof We first argue that we can find \( \mathcal{M} \prec {\mathcal{N}}_{1} \) such that \( \left| M\right| = \left| {N}_{1}\right| \) , and if \( \bar{a},\bar{b}, c \in M \) and \( {\operatorname{tp}}^{\mathcal{M}}\left( \bar{a}\right) = {\operatorname{tp}}^{\mathcal{M}}\left( \bar{b}\right) \), then there is \( d \i... | Yes |
Theorem 4.3.7 T has a countable saturated model if and only if \( \left| {{S}_{n}\left( T\right) }\right| \leq \) \( {\aleph }_{0} \) for all \( n \) . | Proof We need only show that if \( \left| {{S}_{n}\left( T\right) }\right| \leq {\aleph }_{0} \) for all \( n \) then \( T \) has a countable saturated model. Let \( {p}_{0},{p}_{1},\ldots \) list all elements of \( \mathop{\bigcup }\limits_{{n \in \omega }}{S}_{n}\left( T\right) \) . Let \( {\mathcal{M}}_{0} \vDash T ... | Yes |
Corollary 4.3.14 Suppose that \( \kappa \geq {\aleph }_{1} \) is regular and \( {2}^{\lambda } \leq \kappa \) for \( \lambda < \kappa \) . Then, there is a saturated model of size \( \kappa \) . In particular, if \( \kappa \geq {\aleph }_{1} \) is strongly inaccessible, then there is a saturated model of size \( \kappa... | Proof Let \( \mathcal{M} \vDash T \) with \( \left| M\right| = \kappa \) . If \( \kappa = {\lambda }^{ + } \) for \( \lambda < \kappa \), then the corollary follows from Corollary 4.3.13. Thus, we may assume that \( \kappa \) is a limit cardinal. We build an elementary chain \( \left( {{\mathcal{M}}_{\lambda } : \lambd... | Yes |
Theorem 4.3.15 Let \( \kappa \) be a regular cardinal. If \( T \) is \( \kappa \) -stable, then there is a saturated \( \mathcal{M} \vDash T \) with \( \left| M\right| = \kappa \) . Indeed, if \( {\mathcal{M}}_{0} \vDash T \) with \( \left| {M}_{0}\right| = \kappa \) , then there is a saturated elementary extension \( ... | Proof We build an elementary chain \( \left( {{\mathcal{M}}_{\alpha } : \alpha < \kappa }\right) \) where \( \left| {\mathcal{M}}_{\alpha }\right| = \kappa \) such that:\ni) \( {\mathcal{M}}_{0} \vDash T \) with \( \left| {\mathcal{M}}_{0}\right| = \kappa \) ;\n\nii) \( {\mathcal{M}}_{\alpha } = \mathop{\bigcup }\limit... | Yes |
Lemma 4.3.17 Let \( \kappa \geq {\aleph }_{0} \) . If \( \mathcal{M} \) is \( \kappa \) -saturated, then \( \mathcal{M} \) is \( {\kappa }^{ + } \) -universal. | Proof Let \( \mathcal{N} \vDash T \) with \( \left| N\right| \leq \kappa \) . Let \( \left( {{n}_{\alpha } : \alpha < \kappa }\right) \) enumerate \( N \) . Let \( {A}_{\alpha } = \left\{ {{n}_{\beta } : \beta < \alpha }\right\} \) . We build a sequence of partial elementary maps \( {f}_{0} \subset {f}_{1} \subset \ldo... | Yes |
Theorem 4.3.18 Let \( \kappa \geq {\aleph }_{0} \) . The following are equivalent.\ni) \( \mathcal{M} \) is \( \kappa \) -saturated.\nii) \( \mathcal{M} \) is \( \kappa \) -homogeneous and \( {\kappa }^{ + } \) -universal.\nIf \( \kappa \geq {\aleph }_{1} \) i) and ii) are also equivalent to:\niii) \( \mathcal{M} \) is... | Proof By Proposition 4.3.2 and Lemma 4.3.17, i) \( \Rightarrow \) ii). Clearly, ii) \( \Rightarrow \) iii). We argue that ii) \( \Rightarrow \) i) and, if \( \kappa \) is uncountable, iii) \( \Rightarrow \) i).\n\nLet \( A \subseteq M \) with \( \left| A\right| < \kappa \), and let \( p \in {S}_{1}^{\mathcal{M}}\left( ... | Yes |
Theorem 4.3.20 If \( \mathcal{M} \) and \( \mathcal{N} \) are saturated models of \( T \) of cardinality \( \kappa \) , then \( \mathcal{M} \cong \mathcal{N} \) . | Proof By Corollary 4.3.5, we may assume that \( \kappa \geq {\aleph }_{1} \) . Let \( \left( {{m}_{\alpha } : \alpha < \kappa }\right) \) enumerate \( \mathcal{M} \) and \( \left( {{n}_{\alpha } : \alpha < \kappa }\right) \) enumerate \( \mathcal{N} \) . We build a sequence of partial embeddings \( {f}_{0} \subset \ldo... | Yes |
Lemma 4.3.21 Suppose that \( \mathcal{N} \vDash T \) is \( \kappa \) -homogeneous where \( \kappa \leq \left| N\right| \) and \( \mathcal{M} \equiv \mathcal{N} \) such that every type in \( {S}_{n}\left( T\right) \) realized in \( \mathcal{M} \) is realized in \( \mathcal{N} \) for \( n < \omega \) . If \( A \subseteq ... | Proof We prove the claim by induction on \( \left| A\right| \) . Suppose that \( \left| A\right| \) is finite. Let \( A = \left\{ {{a}_{1},\ldots ,{a}_{n}}\right\} \) . Because every type realized in \( \mathcal{M} \) is realized in \( \mathcal{N} \), there is \( \bar{b} \in {N}^{n} \) such that \( {\operatorname{tp}}^... | Yes |
Corollary 4.3.22 If \( \mathcal{M} \vDash T \) is \( \kappa \) -homogeneous and realizes all types in \( {S}_{n}\left( T\right) \) for all \( n < \omega \), then \( \mathcal{M} \) is \( \kappa \) -saturated. | Proof By Lemma 4.3.21, \( \mathcal{M} \) is \( {\kappa }^{ + } \) -universal. Thus, by Theorem 4.3.18, \( \mathcal{M} \) is saturated. | No |
Theorem 4.3.23 If \( \mathcal{M} \equiv \mathcal{N} \) are homogeneous models of \( T \) of the same cardinality realizing the same types in \( {S}_{n}\left( T\right) \) for all \( n < \omega \), then \( \mathcal{M} \cong \mathcal{N} \) . | Proof If \( \mathcal{M} \) and \( \mathcal{N} \) are countable, this is Theorem 4.2.15 so we assume that \( \kappa = \left| M\right| = \left| N\right| \) is uncountable. We build an isomorphism \( f : \mathcal{M} \rightarrow \) \( \mathcal{N} \) by a back-and-forth argument. Let \( \left( {{a}_{\alpha } : \alpha < \kap... | Yes |
Corollary 4.3.26 Let \( \mathcal{M} \) be saturated, and let \( A \subset M \) with \( \left| A\right| < \left| M\right| \) . Then, \( b \) is definable from \( A \) if and only if \( b \) is fixed by all automorphisms of \( \mathcal{M} \) that fix \( A \) pointwise. | Proof By Proposition 4.3.25, \( \{ b\} \) is \( A \) -definable if and only if every automorphism that fixes \( A \) pointwise fixes the set \( \{ b\} \) . | Yes |
Proposition 4.3.28 If \( \mathcal{L} \) is a language containing a constant symbol and \( T \) is an \( \mathcal{L} \) -theory, then \( T \) has quantifier elimination if and only if whenever \( \mathcal{M} \vDash T, A \subseteq M,\mathcal{N} \vDash T \) is \( {\left| M\right| }^{ + } \) -saturated, and \( f : A \right... | ## Proof\n\n\( \left( \Rightarrow \right) \) By quantifier elimination \( f \) is a partial elementary embedding. As in the proof of Lemma 4.3.17, we can extend \( f \) to an elementary embedding of \( \mathcal{M} \) into \( \mathcal{N} \) .\n\n\( \left( \Leftarrow \right) \) We use the quantifier elimination criterion... | Yes |
Proposition 4.3.30 Let \( k \subset K \) be differential fields of characteristic zero.\n\ni) Suppose that \( f\left( {X,{X}^{\prime },\ldots ,{X}^{\left( n\right) }}\right) \in k\{ X\} \smallsetminus 0 \) and \( a, b \in K \) such that \( f\left( a\right) = f\left( b\right) = 0, a,\ldots ,{a}^{\left( n - 1\right) } \)... | i) Certainly, \( k\left( {a,\ldots ,{a}^{\left( n\right) }}\right) \) and \( k\left( {b,\ldots ,{b}^{\left( n\right) }}\right) \) are isomorphic as fields. We need only show that the isomorphism preserves the derivation. For \( i < n \) we have \( \delta \left( {a}^{\left( i\right) }\right) = {a}^{\left( i + 1\right) }... | Yes |
Corollary 4.3.31 If \( k \) is a differential field of characteristic zero, then there is \( K \supseteq k \) with \( K \vDash \mathrm{{DCF}} \) . | Proof If \( f, g \in k\{ X\} \smallsetminus \{ 0\} \) with \( g \) of lower order than \( f \), then by Proposition 4.3.30 iii) we can find \( {k}_{1} \supset k \) with \( a \in {k}_{1} \) where \( f\left( a\right) = 0 \) and \( g\left( a\right) \neq 0 \) . Iterating this process, we build \( K \supset k \) differentia... | No |
Theorem 4.3.32 DCF has quantifier elimination. | Proof Let \( K, L \) be differential closed fields where \( L \) is \( {\left| K\right| }^{ + } \) -saturated. Let \( R \) be a differential subring of \( K \), and let \( f : R \rightarrow L \) be a differential ring embedding. We must show that \( f \) extends to an embedding of \( K \) into \( L \) . Because there i... | Yes |
Theorem 4.3.34 If \( T \) has a \( \left( {\kappa ,\lambda }\right) \) -model where \( \kappa > \lambda \geq {\aleph }_{0} \), then \( T \) has an \( \left( {{\aleph }_{1},{\aleph }_{0}}\right) \) -model. | We will prove Theorem 4.3.34 by first showing that the existence of a \( \left( {\kappa ,\lambda }\right) \) -model has interesting implications for the countable models of \( T \) . | No |
Lemma 4.3.36 If \( T \) has a \( \left( {\kappa ,\lambda }\right) \) -model where \( \kappa > \lambda \geq {\aleph }_{0} \), then there is \( \left( {\mathcal{N},\mathcal{M}}\right) \) a Vaughtian pair of models of \( T \) . | Proof Let \( \mathcal{N} \) be a \( \left( {\kappa ,\lambda }\right) \) -model. Suppose that \( X = \phi \left( \mathcal{N}\right) \) has cardinality \( \lambda \) . By the Löwenheim-Skolem Theorem, there is \( \mathcal{M} \prec \mathcal{N} \) such that \( X \subseteq M \) and \( \left| M\right| = \lambda \) . Because ... | Yes |
Lemma 4.3.37 If \( \left( {\mathcal{N},\mathcal{M}}\right) \) is a Vaughtian pair for \( T \), then there is a Vaughtian pair \( \left( {{\mathcal{N}}_{0},{\mathcal{M}}_{0}}\right) \) where \( {\mathcal{N}}_{0} \) is countable. | Proof Let \( \phi \) be an \( {\mathcal{L}}_{M} \) -formula such that \( \phi \left( \mathcal{M}\right) \) is infinite and \( \phi \left( \mathcal{M}\right) = \) \( \phi \left( \mathcal{N}\right) \) . Let \( {\bar{m}}_{0} \) be the parameters from \( M \) occurring in \( \phi \) . By the Löwenheim-Skolem Theorem, there... | Yes |
Lemma 4.3.40 Suppose that \( T \) is \( \omega \) -stable, \( \mathcal{M} \vDash T \), and \( \left| M\right| \geq {\aleph }_{1} \) . There is a proper elementary extension \( \mathcal{N} \) of \( \mathcal{M} \) such that if \( \Gamma \left( \bar{w}\right) \) is a countable type over \( M \) realized in \( \mathcal{N} ... | ## Proof\n\nClaim There is an \( {\mathcal{L}}_{M} \) -formula \( \phi \left( v\right) \) such that \( \left| \left\lbrack {\phi \left( v\right) }\right\rbrack \right| \geq {\aleph }_{1} \) and for all \( \psi \left( v\right) \in {\mathcal{L}}_{M} \) either \( \left| \left\lbrack {\phi \left( v\right) \land \psi \left(... | Yes |
Theorem 4.3.41 Suppose that \( T \) is \( \omega \) -stable and there is an \( \left( {{\aleph }_{1},{\aleph }_{0}}\right) \) -model of \( T \) . If \( \kappa > {\aleph }_{1} \), then there is a \( \left( {\kappa ,{\aleph }_{0}}\right) \) -model of \( T \) . | Proof Let \( \mathcal{M} \vDash T \) with \( \left| M\right| \geq {\aleph }_{1} \) such that \( \left| {\phi \left( \mathcal{M}\right) }\right| = {\aleph }_{0} \) and let \( \mathcal{M} \prec \mathcal{N} \) be as in Lemma 4.3.40. The type \( \Gamma \left( v\right) = \{ \phi \left( v\right) \} \cup \{ v \neq m : m \in M... | Yes |
Theorem 4.4.1 The following are equivalent:\n\ni) \( T \) is \( {\aleph }_{0} \) -categorical.\n\nii) Every type in \( {S}_{n}\left( T\right) \) is isolated for \( n < \omega \) .\n\niii) \( \left| {{S}_{n}\left( T\right) }\right| < {\aleph }_{0} \) for all \( n < \omega \) .\n\niv) For each \( n < \omega \), there is ... | ## Proof\n\ni) \( \Rightarrow \) ii) If \( p \in {S}_{n}\left( T\right) \) is nonisolated, then there is a countable \( \mathcal{M} \vDash T \) omitting \( p \) . There is also a countable \( \mathcal{N} \vDash T \) realizing \( p \) . Clearly, \( \mathcal{M} ≆ \mathcal{N} \) so \( T \) is not \( {\aleph }_{0} \) -cate... | Yes |
Corollary 4.4.2 Suppose that \( T \) is \( {\aleph }_{0} \) -categorical. There is a function \( f \) : \( \mathbb{N} \rightarrow \mathbb{N} \) such that if \( \mathcal{M} \vDash T, A \subset M \), and \( \left| A\right| \leq n \), then \( \left| {\operatorname{acl}\left( A\right) }\right| \leq f\left( n\right) \) . | Proof By Theorem 4.4.1, \( \left| {{S}_{n + 1}\left( T\right) }\right| \) is finite. Let \( {q}_{1},\ldots ,{q}_{k} \) list all \( n + 1 \) - types. Let \( X = \left\{ {i : {q}_{i}}\right. \) contains a formula \( \phi \left( {v,\bar{w}}\right) \) such that \( \mathcal{M} \vDash \) \( \left. {\forall {v}_{0},\ldots ,{v... | Yes |
Corollary 4.4.3 If \( F \) is an infinite field, then the theory of \( F \) is not \( {\aleph }_{0} \) - categorical. | Proof By compactness, we can find an elementary extension \( K \) of \( F \) such that \( K \) contains a transcendental element \( t \) . Because \( t,{t}^{2},{t}^{3},\ldots \) are distinct, \( \operatorname{acl}\left( t\right) \) is infinite. Thus, by Corollary 4.4.2, \( \operatorname{Th}\left( F\right) \) is not \( ... | Yes |
Corollary 4.4.4 Let \( G \) be an infinite group.\ni) If \( \operatorname{Th}\left( G\right) \) is \( {\aleph }_{0} \) -categorical, then \( G \) is locally finite. Moreover, there is a number \( b \) such that if \( g \in G \), then \( {g}^{n} = 1 \) for some \( n \leq b \) (we say that \( G \) has bounded exponent).\... | ## Proof\n\ni) By Corollary 4.4.2, there is a function \( f : \mathbb{N} \rightarrow \mathbb{N} \) such that if \( \left| X\right| \leq n \) , the group generated by \( X \) has size at most \( f\left( n\right) \) . In particular, if \( g \in G \) , then \( {g}^{n} = 1 \) for some \( n \leq f\left( 1\right) \) .\n\nii)... | Yes |
Theorem 4.4.6 \( I\left( {T,{\aleph }_{0}}\right) \neq 2 \) . | Proof Suppose that \( I\left( {T,{\aleph }_{0}}\right) = 2 \) . By Corollary 4.3.8 ii), there is \( \mathcal{N} \) a prime model of \( T \) and \( \mathcal{M} \) a countable saturated model of \( T \) . Because \( T \) is not \( {\aleph }_{0} \) -categorical, by Theorem 4.4.1, there is a nonisolated type \( p \in {S}_{... | Yes |
Lemma 4.4.9 For each countable \( \mathcal{M} \vDash T \), there is an ordinal \( \gamma < {\omega }_{1} \) such that if \( \bar{a},\bar{b} \in {M}^{n} \) and \( {\operatorname{tp}}_{\gamma }^{\mathcal{M}}\left( \bar{a}\right) = {\operatorname{tp}}_{\gamma }^{\mathcal{M}}\left( \bar{b}\right) \), then \( {\operatorname... | Proof Note first that if \( {\operatorname{tp}}_{\alpha }^{\mathcal{M}}\left( \bar{a}\right) \neq {\operatorname{tp}}_{\alpha }^{\mathcal{M}}\left( \bar{b}\right) \), then \( {\operatorname{tp}}_{\beta }^{\mathcal{M}}\left( \bar{a}\right) \neq {\operatorname{tp}}_{\beta }^{\mathcal{M}}\left( \bar{b}\right) \) for all \... | Yes |
Lemma 4.4.10 Suppose that \( \mathcal{M} \) and \( \mathcal{N} \) are countable models of \( T \) such that \( \mathcal{M} \) has height \( \gamma \) and \( \mathcal{M}{ \equiv }_{{L}_{\gamma + 1}}\mathcal{N} \) . If \( \bar{a},\bar{b} \in {N}^{n} \) and \( {\operatorname{tp}}_{\gamma }^{\mathcal{N}}\left( \bar{a}\righ... | Proof Let \( p = {\operatorname{tp}}_{\gamma }^{\mathcal{N}}\left( \bar{a}\right) = {\operatorname{tp}}_{\gamma }^{\mathcal{N}}\left( \bar{b}\right) \) and let \( \psi \left( \bar{v}\right) \) be an \( {\mathcal{L}}_{\gamma + 1} \) -formula. Let \( \Theta \) be the \( {\mathcal{L}}_{\gamma + 1} \) -formula \[ \forall \... | Yes |
Lemma 4.4.11 If \( \mathcal{M} \) and \( \mathcal{N} \) are countable models of \( T \) such that \( \mathcal{M} \) has height \( \gamma \) and \( \mathcal{M}{ \equiv }_{{L}_{\gamma + 1}}\mathcal{N} \), then \( \mathcal{M} \cong \mathcal{N} \) . | Proof Let \( {a}_{0},{a}_{1},\ldots \) list \( M \) and let \( {b}_{0},{b}_{1},\ldots \) list \( N \) . We build a sequence of finite partial embeddings \( {f}_{0} \subseteq {f}_{1} \subseteq \ldots \) such that if \( \bar{a} \) is the domain of \( {f}_{n} \) , then \( {\operatorname{tp}}_{\gamma }^{\mathcal{M}}\left( ... | Yes |
Theorem 4.4.12 If \( T \) is scattered, then \( I\left( {T,{\aleph }_{0}}\right) \leq {\aleph }_{1} \) . | Proof For each countable \( \mathcal{M} \vDash T \), let \( i\left( \mathcal{M}\right) = \left( {\gamma ,{\operatorname{tp}}_{\gamma + 1}^{\mathcal{M}}\left( \varnothing \right) }\right) \), where \( \gamma \) is the height of \( \mathcal{M} \) . Note that \( \mathcal{M}{ \equiv }_{{L}_{\alpha }}\mathcal{N} \) if and o... | Yes |
Theorem 4.4.15 Suppose that \( X \) is countable and \( Y \subseteq {2}^{X} \) is analytic. If \( \left| Y\right| > {\aleph }_{0} \), then \( \left| Y\right| = {2}^{{\aleph }_{0}} \) . | Proof See [52] 14.13. | No |
Theorem 4.4.16 Let \( T \) be a complete theory in a countable language. If \( I\left( {T,{\aleph }_{0}}\right) > {\aleph }_{1} \), then \( I\left( {T,{\aleph }_{0}}\right) = {2}^{{\aleph }_{0}} \). | Proof For any countable fragment \( F,{S}_{n}\left( {F, T}\right) \) is analytic. Thus, by Theorem 4.4.15, we either have \( \left| {{S}_{n}\left( {F, T}\right) }\right| \leq {\aleph }_{0} \) or \( \left| {{S}_{n}\left( {F, T}\right) }\right| = {2}^{{\aleph }_{0}} \). If there is any countable fragment \( F \), where \... | Yes |
Theorem 4.5.48 If \( T \) is a complete theory in a countable language, then one of the following holds:\ni) there are no cardinals \( \kappa \) such that \( T \) is \( \kappa \) -stable,\nii) \( T \) is \( \kappa \) -stable for all \( \kappa \geq {2}^{{\aleph }_{0}} \) ,\niii) \( T \) is \( \kappa \) -stable if and on... | A proof of this theorem can be found in [7], [18] or [76]. | No |
Theorem 4.5.49 Let \( \mathcal{L} \) be a countable language and \( T \) an \( \mathcal{L} \) -theory.\ni) Assume that \( V = L.{}^{3} \) If \( \kappa > \lambda \geq {\aleph }_{0} \) and \( T \) has a \( \left( {\kappa ,\lambda }\right) \) -model, then \( T \) has a \( \left( {{\mu }^{ + },\mu }\right) \) -model for al... | The first result was proved by Chang (see [22] 7.2.7) for regular \( \mu \) under the weaker assumption that the Generalized Continuum Hypothesis holds. The general case is due to Jensen who also proved the second result (see [27] §VIII). | Yes |
Theorem 5.1.1 (Ramsey’s Theorem) If \( k, n < \omega \), then \( {\aleph }_{0} \rightarrow {\left( {\aleph }_{0}\right) }_{k}^{n} \) . | Proof We prove Ramsey’s Theorem by induction on \( n \) . For \( n = 1 \), Ramsey’s Theorem asserts that if \( X \) is infinite, \( k < \omega \), and \( f : X \rightarrow k \), then \( {f}^{-1}\left( i\right) \) is infinite for some \( i < k \) . This is just the Pigeonhole Principle that if we put infinitely many ite... | Yes |
Theorem 5.1.2 (Finite Ramsey Theorem) For all \( k, n, m < \omega \), there is \( l < \omega \) such that \( l \rightarrow {\left( m\right) }_{k}^{n} \) . | Proof Suppose that there is no \( l \) such that \( l \rightarrow {\left( m\right) }_{k}^{n} \) . For each \( l < \omega \) , let \( {T}_{l} = \left\{ {f : {\left\lbrack \{ 0,\ldots, l - 1\} \right\rbrack }^{n} \rightarrow k}\right. \) : there is no \( X \subseteq \{ 0,\ldots, l - 1\} \) of size at least \( m \), homog... | Yes |
Proposition 5.1.3 \( {2}^{{\aleph }_{0}} \nrightarrow {\left( 3\right) }_{{\aleph }_{0}}^{2} \) . | Proof We define \( F : {\left\lbrack {2}^{\omega }\right\rbrack }^{2} \rightarrow \omega \) by \( F\left( {\{ f, g\} }\right) \) is the least \( n \) such that \( f\left( n\right) \neq g\left( n\right) \) . Clearly, we cannot find \( \{ f, g, h\} \) such that \( f\left( n\right) \neq g\left( n\right) \) , \( g\left( n\... | Yes |
Corollary 5.1.5 \( {\beth }_{\alpha + n}^{ + } \rightarrow {\left( {\beth }_{\alpha }^{ + }\right) }_{{\beth }_{\alpha }}^{n + 1} \) . | Proof This follows from Erdös-Rado because \( {\beth }_{\alpha + n} = {\beth }_{n}\left( {\beth }_{\alpha }\right) \) . | Yes |
Lemma 5.2.4 Suppose that \( {T}^{ * } \) is an \( {\mathcal{L}}^{ * } \) -theory with built-in Skolem functions. Let \( \mathcal{M} \vDash {T}^{ * } \) . Let \( I \subseteq M \) be an infinite sequence of order indiscernibles. Suppose that \( \tau : I \rightarrow I \) is an order-preserving permutation. Then, there is ... | Proof For each element \( a \in \mathcal{H}\left( I\right) \), there is a Skolem term \( t \) and \( {x}_{1} < {x}_{2} < \) \( \ldots < {x}_{n} \in I \) such that \( a = t\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) . Let \( \sigma \left( a\right) = t\left( {\tau \left( {x}_{1}\right) ,\ldots ,\tau \left( {x}_{n}\right) ... | Yes |
Lemma 5.2.5 Let \( {T}^{ * } \) be an \( {\mathcal{L}}^{ * } \) -theory with built-in Skolem functions. Suppose that \( X = \left( {{x}_{i} : i \in I}\right) \) is an infinite sequence of order indiscernibles in \( \mathcal{M} \vDash {T}^{ * } \) . If \( \left( {J, < }\right) \) is any infinite ordered set, we can find... | Proof Add to \( {\mathcal{L}}^{ * } \) constant symbols \( {c}_{j} \) for \( j \in J \) and let \( \Gamma = {T}^{ * } \cup \left\{ {{c}_{i} \neq }\right. \) \( \left. {{c}_{j} : i, j \in J, i \neq j}\right\} \cup \left\{ {\phi \left( {{c}_{{i}_{1}},\ldots ,{c}_{{i}_{m}} : }\right. {i}_{1} < \ldots < {i}_{m} \in J}\righ... | Yes |
Lemma 5.2.6 Suppose that \( {T}^{ * } \) is an \( {\mathcal{L}}^{ * } \) -theory with built-in Skolem functions. If \( I \) is a sequence of order indiscernibles in \( \mathcal{M} \vDash {T}^{ * } \) and \( J \) is a sequence of order indiscernibles in \( \mathcal{N} \vDash {T}^{ * } \) with \( \operatorname{tp}\left( ... | Proof If \( a = t\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) for \( t \) a term and \( {x}_{1},\ldots ,{x}_{n} \in I \) we let \( \sigma \left( a\right) = \) \( t\left( {\tau \left( {x}_{1}\right) ,\ldots ,\tau \left( {x}_{n}\right) }\right) \) . We then argue as in Lemma 5.2.4 that this map is well-defined and elementa... | Yes |
Corollary 5.2.7 Let \( T \) be an \( \mathcal{L} \) -theory with infinite models. For any \( \kappa \geq \) \( \left| \mathcal{L}\right| + {\aleph }_{0} \), there is \( \mathcal{N} \vDash T \) of cardinality \( \kappa \) with \( {2}^{\kappa } \) automorphisms. | Proof Let \( {\mathcal{L}}^{ * } \) and \( {T}^{ * } \) be as above. We can find \( \mathcal{M} \vDash {T}^{ * } \) containing an infinite sequence of order indiscernibles \( I \) . Claim There is a linear order \( \left( {X, < }\right) \) of size \( \kappa \) with \( {2}^{\kappa } \) order-preserving permutations. Let... | Yes |
Corollary 5.2.8 Suppose that \( {T}^{ * } \) is an \( {\mathcal{L}}^{ * } \) -theory with built in Skolem functions, \( \mathcal{M} \vDash {T}^{ * },\mathcal{M} \) omits \( p - \) a type over \( \varnothing \), and \( \mathcal{M} \) contains an infinite sequence of order indiscernibles \( I \) . There are arbitrarily l... | Proof Let \( \kappa \geq {\aleph }_{0} \) . By Lemma 5.2.5, we can find \( \mathcal{N} \vDash {T}^{ * } \) containing a sequence of order indiscernibles \( J \) with \( \left| J\right| \geq \kappa \) and \( \operatorname{tp}\left( I\right) = \operatorname{tp}\left( J\right) \) . Then \( \left| {\mathcal{H}\left( J\righ... | Yes |
Theorem 5.2.9 Let \( \mathcal{L} \) be countable and \( T \) be an \( \mathcal{L} \) -theory with infinite models. For all \( \kappa \geq {\aleph }_{0} \), there is \( \mathcal{M} \vDash {T}^{ * } \) with \( \left| M\right| = \kappa \) such that if \( A \subseteq M \), then \( \mathcal{M} \) realizes at most \( \left| ... | Proof For notational simplicity we assume that \( n = 1 \) . This is no loss of generality. Let \( {\mathcal{L}}^{ * } \) and \( {T}^{ * } \) be as above. Let \( \mathcal{M} \vDash T \) be the Skolem hull of a sequence of order indiscernibles \( I \) of order type \( \left( {\kappa , < }\right) \) . Then \( \left| M\ri... | Yes |
Corollary 5.2.10 Let \( T \) be a complete theory in a countable language with infinite models, and let \( \kappa \geq {\aleph }_{1} \) . If \( T \) is \( \kappa \) -categorical, then \( T \) is \( \omega \) -stable. | Proof If \( T \) is not \( \omega \) -stable, then we can find a countable \( \mathcal{M} \vDash T \) with \( A \subseteq M \) such that \( \left| {{S}_{n}^{\mathcal{M}}\left( A\right) }\right| > {\aleph }_{0} \) . By compactness, we can find \( \mathcal{M} \prec {\mathcal{N}}_{0} \) of cardinality \( \kappa \) realizi... | Yes |
Corollary 5.2.11 Let \( T \) be a complete theory in a countable language with infinite models. If \( \kappa \geq {\aleph }_{1} \) and \( T \) is \( \kappa \) -categorical, then \( T \) has no Vaughtian pairs and hence no \( \left( {\kappa ,\lambda }\right) \) -models for \( \kappa > \lambda \geq {\aleph }_{0} \) . | Proof Because \( T \) is \( \kappa \) -categorical, \( T \) is \( \omega \) -stable. If there is a Vaughtian pair, then by Theorem 4.3.34 there is an \( \left( {{\aleph }_{1},{\aleph }_{0}}\right) \) -model, and, by Theorem 4.3.41, a \( \left( {\kappa ,{\aleph }_{0}}\right) \) -model. Because we can find a model of \( ... | Yes |
Lemma 5.2.12 For any infinite cardinal \( \kappa \), there is a dense linear order \( \left( {A, < }\right) \) with \( B \subset A \) such that \( B \) is dense in \( A \) and \( \left| B\right| \leq \kappa < \left| A\right| \) . | Proof Let \( \lambda \leq \kappa \) be least such that \( {2}^{\lambda } > \kappa \) . Let \( A \) be the set of all functions from \( \lambda \) to \( \mathbb{Q} \) . If we order \( A \) by \( f < g \) if and only if \( f\left( \alpha \right) < g\left( \alpha \right) \), where \( \alpha \) is least such that \( f\left... | Yes |
Theorem 5.2.13 Suppose that \( \mathcal{L} \) is a countable language, \( \kappa \) is an infinite cardinal, and \( T \) is a \( \kappa \) -stable \( \mathcal{L} \) -theory. If \( \mathcal{M} \vDash T \) and \( X \subseteq M \) is an infinite sequence of order indiscernibles, then \( X \) is a set of indiscernibles. | Proof Let \( \phi \left( {{v}_{1},\ldots ,{v}_{n}}\right) \) be an \( \mathcal{L} \) -formula and \( {x}_{1},\ldots ,{x}_{n} \) be an increasing sequence from \( I \) such that \( \mathcal{M} \vDash \phi \left( {{x}_{1},\ldots ,{x}_{n}}\right) \) . Let \( {S}_{n} \) be the group of all permutations of \( \{ 1,\ldots, n... | Yes |
Lemma 5.3.4 Let \( \kappa \geq {\aleph }_{1} \) be a regular cardinal.\n\ni) If \( {C}_{0} \) and \( {C}_{1} \) are closed unbounded subsets of \( \kappa \), then \( {C}_{0} \cup {C}_{1} \) and \( {C}_{0} \cap {C}_{1} \) are closed unbounded.\n\nii) If \( \alpha < \kappa \) and \( \left( {{C}_{\beta } : \beta < \alpha ... | ## Proof\n\ni) Clearly, \( {C}_{0} \cup {C}_{1} \) is unbounded. If \( X \subseteq {C}_{0} \cup {C}_{1} \), then \( X \cap {C}_{i} \) is unbounded in \( X \) for some \( i \) and the least upper bound of \( X \cap {C}_{i} \) is the least upper bound of \( X \) .\n\nIt is easy to see that \( {C}_{0} \cap {C}_{1} \) is c... | Yes |
Lemma 5.3.5 If \( \left( {{C}_{\alpha } : \alpha < \kappa }\right) \) is a sequence of closed unbounded subsets of \( \kappa \), then \( \bigtriangleup {C}_{\alpha } \) is closed unbounded. | Proof Suppose that \( X \subseteq \bigtriangleup {C}_{\alpha } \) is bounded. Let \( \beta = \sup X \) . If \( \alpha < \beta \) , then \( \{ \gamma \in X : \alpha < \gamma \} \) is a bounded subset of \( {C}_{\alpha } \) with supremum \( \beta \) . Thus, \( \beta \in {C}_{\alpha } \) . Because \( \beta \) is in \( {C}... | Yes |
Lemma 5.3.8 (Fodor’s Lemma) Suppose that \( f : \kappa \rightarrow \kappa \) is regressive on a stationary set \( S \), then, there is \( \gamma < \kappa \) such that \( S \cap {f}^{-1}\left( \gamma \right) \) is stationary. | Proof Suppose not. Then, for each \( \alpha < \kappa \), we can find a closed unbounded set \( {C}_{\alpha } \) such that \( {C}_{\alpha } \cap S \cap {f}^{-1}\left( \alpha \right) = \varnothing \) . By Lemma 5.3.5, \( \bigtriangleup {C}_{\alpha } \) is closed unbounded; thus, there is \( \beta \in S \cap \bigtriangleu... | Yes |
Corollary 5.3.9 If \( S \) is stationary and \( S = \mathop{\bigcup }\limits_{{\alpha < \lambda }}{S}_{\alpha } \) for some \( \lambda < \kappa \) , then some \( {S}_{\alpha } \) is stationary. | Proof Let \( f : S \rightarrow \lambda \) by \( f\left( \alpha \right) = \) least \( \beta < \lambda \) such that \( \alpha \in {S}_{\beta } \) . Because \( \lambda < \kappa, f\left( \alpha \right) < \alpha \) on the stationary set \( \{ \alpha \in S : \lambda < \alpha \} \) . Thus, by Fodor’s Lemma, \( f \) is constan... | No |
Lemma 5.3.10 Suppose that \( \kappa \geq {\aleph }_{1} \) is regular. There is a family \( \left( {S}_{\alpha }\right. \) : \( \alpha < \kappa \) ) of disjoint stationary subsets of \( \kappa \) . | Proof\ncase 1: \( \kappa = {\lambda }^{ + } \).\n\nFor each ordinal \( \alpha < \kappa \), let \( {f}_{\alpha } : \lambda \rightarrow \alpha \) be surjective. For \( \beta < \lambda \) and \( \gamma < \kappa \), let \( {U}_{\beta ,\gamma } = \left\{ {\alpha < \kappa : {f}_{\alpha }\left( \beta \right) = \gamma }\right\... | Yes |
Lemma 5.3.11 Let \( \mathcal{N} \vDash {T}^{ * } \) contain an infinite sequence of order indiscernibles \( J \) linearly ordered by \( < \) .\n\ni) Suppose that \( J = {J}_{0} + {J}_{1} \), where \( \left| {J}_{1}\right| \geq 2 \) . Then, no element of \( {J}_{1} \) is in \( \mathcal{H}\left( {J}_{0}\right) \) .\n\nii... | ## Proof\n\ni) Let \( a \in {J}_{1} \) . Suppose that \( \bar{c} \) is a sequence from \( {J}_{0} \) and \( t \) is a Skolem term such that \( a = t\left( \bar{c}\right) \) . By indiscernibility, \( t\left( \bar{c}\right) = b \) for all \( b \in {J}_{1} \), thus \( \left| {J}_{1}\right| \leq 1 \) .\n\nii) Suppose that ... | Yes |
Lemma 5.3.12 The set \( C = \left\{ {\alpha < \kappa : {A}_{\alpha } = {B}_{\alpha } = \alpha }\right\} \) is closed unbounded. | Proof We show that \( {C}_{A} = \left\{ {\alpha < \kappa : {A}_{\alpha } = \alpha }\right\} \) is closed unbounded. Similarly \( \left\{ {\alpha < \kappa : {B}_{\alpha } = \alpha }\right\} \) is closed unbounded, and \( C \) is closed unbounded because it is the intersection of these two sets.\n\nBecause \( {C}_{A} \) ... | Yes |
Lemma 5.3.13 The set \( D = \left\{ {\alpha < \kappa : {\bar{d}}_{\beta } \in {B}_{\alpha }}\right. \) for all \( \left. {\beta < \alpha }\right\} \) is closed unbounded. | Proof It is easy to see that \( D \) is closed. Let \( {\alpha }_{0} < \kappa \) . Build a sequence \( {\alpha }_{0} < {\alpha }_{1} < \ldots \) such that for all \( \beta < {\alpha }_{n},{\bar{d}}_{\beta } \in {B}_{{\alpha }_{n + 1}} \) . If \( \alpha = \sup {\alpha }_{i} \), then \( \alpha \in D \) . | No |
Lemma 5.3.15 The set \( {S}^{\prime } = \{ \alpha \in S : \alpha = \sup \left( {\alpha \cap S}\right) \} \) is stationary. | Proof The set \( X = \{ \alpha < \kappa : \alpha = \sup \left( {\alpha \cap S}\right) \} \) is closed unbounded and \( {S}^{\prime } = X \cap S \) . | Yes |
Lemma 5.3.16 If \( \alpha \in S \) and \( \alpha < \delta \), then \( {\bar{d}}_{\alpha } \in {J}_{ < \delta } \) . | Proof Because \( \delta \) is a limit point of \( S \), there is \( \beta \in S \) with \( \alpha < \beta < \delta \) . Because \( \beta \in S,{\bar{d}}_{\alpha } \in {B}_{\beta } \) . By Lemma 5.3.11 i), \( {B}_{\beta } \cap J = {J}_{ < \beta } \) . Thus \( {\bar{d}}_{\alpha } \in {J}_{ < \beta } \subset {J}_{ < \delt... | Yes |
Lemma 5.3.17 Let \( a \in {I}_{\delta } \) . There is \( x \in {J}_{ < \delta } \) and \( y \in {J}_{\delta } \) such that if \( {j}_{1},\ldots ,{j}_{n} \in J \) with \( x < {j}_{1} < \ldots < {j}_{n} < y \), then \( {t}^{\mathcal{B}}\left( {\bar{c},\bar{j}}\right) < a \) . | Proof Because \( \delta \in S,{A}_{\delta } = {B}_{\delta } \) and \( a \notin {B}_{\delta } \) . Let \( a = {s}^{\mathcal{B}}\left( {{x}_{1},\ldots ,{x}_{k},{y}_{1},\ldots ,{y}_{l}}\right) \) where \( s \) is a Skolem term, \( \bar{x} \in {J}_{ < \delta } \), and \( \bar{y} \in J \smallsetminus {J}_{ < \delta } \) . N... | Yes |
Lemma 5.3.18 \( i \) ) If \( {j}_{1},\ldots ,{j}_{n} \in {J}_{\delta } \) and \( {j}_{1} < \ldots < {j}_{n} \), then \( {t}^{\mathcal{B}}\left( {\bar{c},\bar{j}}\right) > {a}_{\alpha } \) for \( \alpha \in S \) with \( \alpha < \delta \) . | i) Because \( \delta \in S \) and \( \alpha < \delta ,{\bar{d}}_{\alpha } \in {B}_{\delta } \) . Because, by Lemma 5.3.11 i), \( {B}_{\delta } \cap J = {J}_{ < \delta },{d}_{\delta ,1} \notin {J}_{ < \delta } \) . Thus \( {d}_{\alpha, n} < {d}_{\delta ,1} \) . Because \[ {a}_{\alpha } = {t}^{\mathcal{B}}\left( {\bar{c}... | Yes |
Theorem 5.4.1 (Paris-Harrington Principle) For all natural numbers \( n, k, m \), there is a number \( l \) such that if \( f : {\left\lbrack l\right\rbrack }^{n} \rightarrow k \), then there is \( Y \subseteq l \) such that \( Y \) is homogeneous for \( f,\left| Y\right| \geq m \), and if \( {y}_{0} \) is the least el... | Proof We argue as in the proof of the finite version of Ramsey's Theorem. Suppose that there is no such \( l \) . For \( l < \omega \), let \( {T}_{l} = \left\{ {f : {\left\lbrack \{ 0,\ldots, l - 1\} \right\rbrack }^{n} \rightarrow }\right. \) \( k \) : there is no \( Y \) homogeneous for \( f \) with \( \left. {\left... | Yes |
Lemma 5.4.3 For all \( c, m, n, k < \omega \), there is \( d < \omega \) such that if \( g : {\left\lbrack d\right\rbrack }^{n} \rightarrow \) \( k \), then there is a homogeneous set \( Y \subseteq \left( {c, d}\right) \) with \( \left| Y\right| \geq m + {2n},\min Y + \) \( n + 1 \) . | Proof By the Paris-Harrington Principle, there is a \( d \) such that for any partition \( h : {\left\lbrack d\right\rbrack }^{n} \rightarrow k + 1 \) there is a homogeneous set \( Z \) with \( \left| Z\right| \geq c + m + {2n} + 1,\min Z \) . Given \( g : {\left\lbrack l\right\rbrack }^{n} \rightarrow k \), we define ... | Yes |
Lemma 5.4.8 Suppose that \( \mathcal{M} \) is a model of Peano arithmetic and \( {x}_{0} < \) \( {x}_{1} < \ldots \) is a sequence of diagonal indiscernibles for all \( {\Delta }_{0} \) -formulas. Let \( N = \left\{ {y \in M : y < {x}_{i}}\right. \) for some \( \left. {i < \omega }\right\} \) . Then, \( N \) is closed ... | Proof Suppose that \( i < j < k < l \) and \( a < {x}_{i} \) . If \( a + {x}_{j} \geq {x}_{k} \), then we can find \( b \leq a \) such that \( b + {x}_{j} = {x}_{k} \) . By indiscernibility, \( b + {x}_{j} = {x}_{l} \), so \( {x}_{k} = {x}_{l} \), a contradiction. Thus \( a + {x}_{j} < {x}_{k} \) . It follows that \( N... | Yes |
Lemma 6.1.4 (Exchange Principle) Suppose that \( D \subset M \) is strongly minimal, \( A \subseteq D \), and \( a, b \in D \) . If \( a \in \operatorname{acl}\left( {A\cup \{ b\} }\right) \smallsetminus \operatorname{acl}\left( A\right) \), then \( b \in \) \( \operatorname{acl}\left( {A\cup \{ a\} }\right) \) . | Proof We write \( \operatorname{acl}\left( {A, b}\right) \) for \( \operatorname{acl}\left( {A\cup \{ b\} }\right) \). Suppose that \( a \in \operatorname{acl}\left( {A, b}\right) \smallsetminus \operatorname{acl}\left( A\right) \). Suppose that \( \mathcal{M} \vDash \phi \left( {a, b}\right) \), where \( \phi \) is a ... | Yes |
Lemma 6.1.9 Let \( A, B \subseteq D \) be independent with \( A \subseteq \operatorname{acl}\left( B\right) \). i) Suppose that \( {A}_{0} \subseteq A,{B}_{0} \subseteq B,{A}_{0} \cup {B}_{0} \) is a basis for \( \operatorname{acl}\left( B\right) \) and \( a \in A \smallsetminus {A}_{0} \). Then, there is \( b \in {B}_... | ## Proof i) Let \( C \subseteq {B}_{0} \) be of minimal cardinality such that \( a \in \operatorname{acl}\left( {{A}_{0} \cup C}\right) \). Because \( A \) is independent, \( \left| C\right| \geq 1 \). Let \( b \in C \). By exchange, \( b \in \operatorname{acl}\left( {{A}_{0} \cup }\right. \) \( \{ a\} \cup \left( {C\s... | Yes |
Theorem 6.1.11 Suppose \( T \) is a strongly minimal theory. If \( \mathcal{M},\mathcal{N} \vDash T \) , then \( \mathcal{M} \cong \mathcal{N} \) if and only if \( \dim \left( M\right) = \dim \left( N\right) \) . | Proof Let \( B \) be a transcendence basis for \( \phi \left( \mathcal{M}\right) \) and \( C \) be a transcendence basis for \( \phi \left( \mathcal{N}\right) \) . Because \( \left| B\right| = \left| C\right| \), we can find a bijection \( f : B \rightarrow C \) . By Corollary 6.1.7, \( f \) is elementary. Let \( I = \... | Yes |
Corollary 6.1.12 If \( T \) is a strongly minimal theory, then \( T \) is \( \kappa \) -categorical for \( \kappa \geq {\aleph }_{1} \) and \( I\left( {T,{\aleph }_{0}}\right) \leq {\aleph }_{0} \) . | Proof If \( \mathcal{M} \) has cardinality \( \kappa \geq {\aleph }_{1} \), then any transcendence basis for \( \mathcal{M} \) has cardinality \( \kappa \), whereas if \( \left| M\right| = {\aleph }_{0} \), then \( \dim \left( M\right) \leq {\aleph }_{0} \) . | Yes |
Lemma 6.1.13 Let \( T \) be \( \omega \) -stable.\n\ni) If \( \mathcal{M} \vDash T \), then there is a minimal formula in \( \mathcal{M} \).\n\nii) If \( \mathcal{M} \vDash T \) is \( {\aleph }_{0} \) -saturated and \( \phi \left( {\bar{v},\bar{a}}\right) \) is a minimal formula in \( \mathcal{M} \), then \( \phi \left... | ## Proof\n\ni) Suppose not. We build a tree of formulas \( \left( {{\phi }_{\sigma } : \sigma \in {2}^{ < \omega }}\right) \) such that:\n\nif \( \sigma \subset \tau \), then \( {\phi }_{\tau } \vDash {\phi }_{\sigma } \) ;\n\n\( {\phi }_{\sigma, i} \vDash \neg {\phi }_{\sigma ,1 - i} \)\n\n\( {\phi }_{\sigma }\left( \... | Yes |
Lemma 6.1.14 Suppose that \( T \) is an \( \mathcal{L} \) -theory with no Vaughtian pairs. Let \( \mathcal{M} \vDash T \), and let \( \phi \left( {{v}_{1},\ldots ,{v}_{k},{w}_{1},\ldots ,{w}_{m}}\right) \) be a formula with parameters from \( M \) . There is a number \( n \) such that if \( \bar{a} \in M \) and \( \lef... | Proof Suppose not. Then, for each \( n \in \mathbb{N} \), we can find \( {\bar{a}}_{n} \) in \( M \) such that \( \phi \left( {\mathcal{M},{\bar{a}}_{n}}\right) \) is a finite set of size at least \( n \) . Consider the language \( {\mathcal{L}}^{ * } = \) \( \mathcal{L} \cup \{ U\} \) for pairs of models of \( T \) us... | Yes |
Corollary 6.1.15 If \( T \) has no Vaughtian pairs, then any minimal formula is strongly minimal. | Proof Let \( \phi \left( \bar{v}\right) \) be a minimal formula over \( \mathcal{M} \vDash T \) (possibly with parameters). Suppose, for purposes of contradiction, that there is an elementary extension \( \mathcal{N} \) of \( \mathcal{M},\bar{b} \in N \), and an \( \mathcal{L} \) -formula \( \psi \left( {\bar{v},\bar{w... | Yes |
Lemma 6.1.17 If \( T \) has no Vaughtian pairs, \( \mathcal{M} \vDash T \), and \( X \subseteq {M}^{n} \) is infinite and definable, then no proper elementary submodel of \( \mathcal{M} \) contains \( X \) . If, in addition, \( T \) is \( \omega \) -stable, then \( \mathcal{M} \) is prime over \( X \) . | Proof Let \( \phi \left( \bar{v}\right) \) define \( X \) . If \( \mathcal{N} \) is a proper elementary submodel of \( \mathcal{M} \) containing \( X \), then \( X = \phi \left( \mathcal{M}\right) = \phi \left( \mathcal{N}\right) \) and \( \left( {\mathcal{M},\mathcal{N}}\right) \) is a Vaughtian pair.\n\nIf \( T \) is... | Yes |
Theorem 6.1.18 Let \( T \) be a complete theory in a countable language with infinite models, and let \( \kappa \) be an uncountable cardinal. \( T \) is \( \kappa \) -categorical if and only if \( T \) is \( \omega \) -stable and has no Vaughtian pairs. | ## Proof\n\n\( \\left( \\Rightarrow \\right) \) If \( T \) is \( \\kappa \) -categorical, then, by Corollaries 5.2.10 and 5.2.11, \( T \) is \( \\omega \) -stable and has no Vaughtian pairs.\n\n\( \\left( \\Leftarrow \\right) \) Suppose that \( T \) is \( \\omega \) -stable and has no Vaughtian pairs. Because \( T \) i... | Yes |
Lemma 6.1.20 Suppose that \( T \) is an \( \omega \) -stable \( \mathcal{L} \) -theory and \( \phi \left( v\right) \) is a strongly minimal \( \mathcal{L} \) -formula (with no additional parameters). Suppose that \( \mathcal{M} \vDash T \) and \( \dim \left( {\phi \left( \mathcal{M}\right) }\right) = n < {\aleph }_{0} ... | Proof Let \( {\mathcal{M}}^{ * } \vDash T \) be an \( \omega \) -saturated elementary extension of \( \mathcal{M} \) . It is easy to see that \( \dim \left( {\phi \left( {\mathcal{M}}^{ * }\right) }\right) \geq {\aleph }_{0} \) . Let \( {a}_{1},\ldots ,{a}_{m} \) be independent elements of \( \phi \left( {\mathcal{M}}^... | Yes |
Theorem 6.1.23 If \( T \) is superstable but not \( {\aleph }_{0} \) -categorical, then \( I\left( {T,{\aleph }_{0}}\right) \geq \) \( {\aleph }_{0} \) . | The proofs of these results use more detailed results from stability theory than we will develop here. The reader can find the proofs in [7] or [18]. | No |
Corollary 6.2.4 Suppose that \( \mathcal{M} \) is an \( \mathcal{L} \) -structure, \( \phi \) is an \( {\mathcal{L}}_{M} \) -formula, and \( {\mathcal{N}}_{0} \) and \( {\mathcal{N}}_{1} \) are \( {\aleph }_{0} \) -saturated elementary extensions of \( \mathcal{M} \) . Then \( {\operatorname{RM}}^{{\mathcal{N}}_{0}}\le... | Proof By Exercise 2.5.11 there is \( {\mathcal{N}}_{2} \), a common elementary extension of \( {\mathcal{N}}_{0} \) and \( {\mathcal{N}}_{1} \) . Let \( {\mathcal{N}}_{3} \) be an \( {\aleph }_{0} \) -saturated elementary extension of \( {\mathcal{N}}_{2} \) . By Lemma 6.2.3, \( {\operatorname{RM}}^{{\mathcal{N}}_{0}}\... | No |
Lemma 6.2.7 Let \( \mathcal{M} \) be an \( \mathcal{L} \) -structure and let \( X \) and \( Y \) be definable subsets of \( {M}^{n} \) .\ni) If \( X \subseteq Y \), then \( \operatorname{RM}\left( X\right) \leq \operatorname{RM}\left( Y\right) \).\nii) \( \operatorname{RM}\left( {X \cup Y}\right) \) is the maximum of \... | Proof We leave the proofs of i) and ii) as exercises.\n\niii) Let \( X = \phi \left( \mathcal{M}\right) \) . Because \( X \) is nonempty, \( \operatorname{RM}\left( \phi \right) \geq 0 \) . Because \( \phi \left( \mathcal{M}\right) \) is finite if and only if \( \phi \left( \mathcal{N}\right) \) is finite for any \( \m... | No |
Corollary 6.2.10 A formula \( \phi \) is strongly minimal if and only if \( \mathrm{{RM}}\left( \phi \right) = \) \( {\deg }_{\mathrm{M}}\left( \phi \right) = 1 \) . | Proof If \( \phi \) is strongly minimal, then, because \( \phi \left( \mathbb{M}\right) \) is infinite, \( \operatorname{RM}\left( \phi \right) \geq \) 1. Because \( \phi \left( \mathbb{M}\right) \) cannot be partitioned into two definable infinite sets, \( \operatorname{RM}\left( \phi \right) = 1 \) and \( {\deg }_{\m... | Yes |
Proposition 6.2.11 If \( T \) is totally transcendental, then no formula has the order property. | Proof Suppose, for purposes of contradiction, that \( \phi \left( {\bar{v},\bar{w}}\right) \) has the order property. By compactness and saturation, we can find \( \left( {{\bar{a}}_{q},{\bar{b}}_{q} : q \in \mathbb{Q}}\right) \) such that \( \mathbb{M} \vDash \phi \left( {{\bar{a}}_{q},{\bar{b}}_{r}}\right) \) if and ... | Yes |
Lemma 6.2.13 If \( p, q \in {S}_{n}\left( A\right) ,\operatorname{RM}\left( p\right) ,\operatorname{RM}\left( q\right) < \infty \), and \( p \neq q \), then \( {\phi }_{p} \neq {\phi }_{q} \) | Proof There is a formula \( \psi \) such that \( \psi \in p \) and \( \psi \notin q \) . Because \( {\phi }_{p} \land \psi \in p \) , \( \operatorname{RM}\left( {{\phi }_{p} \land \psi }\right) \leq \operatorname{RM}\left( {\phi }_{p}\right) \leq \operatorname{RM}\left( p\right) \) . Because \( \operatorname{RM}\left( ... | Yes |
Theorem 6.2.14 If \( T \) is \( \omega \) -stable, then \( T \) is totally transcendental. Conversely, if \( \mathcal{L} \) is countable and \( T \) is totally transcendental, then \( T \) is \( \omega \) -stable. | ## Proof\n\n\( \left( \Rightarrow \right) \) Suppose, for purposes of contradiction, that \( \phi \left( {{v}_{1},\ldots ,{v}_{n}}\right) \) is an \( {\mathcal{L}}_{{\mathbb{M}}^{ - }} \) formula such that \( \operatorname{RM}\left( \phi \right) = \infty \) . Let \( \beta = \sup \{ \operatorname{RM}\left( \psi \right) ... | Yes |
Corollary 6.2.18 Suppose that \( T \) is \( \omega \) -stable, \( \mathcal{M} \vDash T, X \subseteq {\mathbb{M}}^{n}, Y \subseteq {\mathbb{M}}^{m} \) are definable, and \( f : X \rightarrow Y \) is a definable finite-to-one function from \( X \) onto \( Y \) . Then \( \operatorname{RM}\left( X\right) = \operatorname{RM... | Proof Let \( A \subset \mathbb{M} \) such that \( X, Y \) and \( f \) are definable over \( A \) . Suppose that \( f\left( \bar{a}\right) = \bar{b} \) . Then, \( \bar{b} \) is definable over \( A,\bar{a} \) and, because \( f \) is finite-to-one, \( \bar{a} \) is algebraic over \( A,\bar{b} \) . By Lemma 6.2.17\n\n\[ \o... | Yes |
Lemma 6.2.20 Let \( T \) be strongly minimal. Suppose that \( C \subseteq {\mathbb{M}}^{m + n} \) is definable. Let \( {C}_{\bar{a}} = \left\{ {\bar{x} \in {\mathbb{M}}^{n} : \left( {\bar{a},\bar{x}}\right) \in C}\right\} \) for \( \bar{a} \in {\mathbb{M}}^{m} \) . The set \( {Y}_{n, k} = \) \( \left\{ {\bar{a} \in {\m... | Proof We prove this by induction on \( n \) .\n\nSuppose that \( n = 1 \) . We first note that there is a number \( N \) such that \( \left| {C}_{\bar{a}}\right| < N \) or \( \left| {\mathbb{M} \smallsetminus {C}_{\bar{a}}}\right| < N \) for all \( \bar{a} \in {\mathbb{M}}^{m} \) because otherwise the type\n\n\[ \left\... | Yes |
Theorem 6.2.22 If \( K \) is an algebraically closed field and \( V \subseteq {K}^{n} \) is an irreducible variety, then the Krull dimension of \( V \) is equal to the transcendence degree of the function field \( K\left( V\right) \) over \( K \) . | Proof See [6] §11. | No |
Corollary 6.2.23 If \( K \) is an algebraically closed field and \( V \subseteq {K}^{n} \) is an irreducible variety, then \( \operatorname{RM}\left( V\right) \) is equal to the Krull dimension of \( V \) . | Proof We prove this by induction on the Krull dimension of \( V \) . If \( V \) has Krull dimension 0, then \( V \) is a point and \( \operatorname{RM}\left( V\right) = 0 \) .\n\nSuppose that \( V \) has Krull dimension \( k > 0 \) . Let \( \phi \left( \bar{v}\right) \) be the \( {\mathcal{L}}_{K} \) -formula defining ... | Yes |
Lemma 6.2.26 Suppose that \( V \subseteq {K}^{n} \) is an irreducible closed set, \( X \subseteq {K}^{n} \) is constructible, and \( \operatorname{RM}\left( X\right) = \operatorname{RM}\left( V\right) \) . There is an open \( O \subseteq {K}^{n} \) such that \( O \cap V \subseteq X \) and \( O \cap V \neq \varnothing \... | Proof By quantifier elimination, \( X = \mathop{\bigcup }\limits_{{i = 1}}^{n}{F}_{i} \cap {O}_{i} \) where \( {F}_{i} \subseteq F \) is Zariski closed, \( {O}_{i} \subseteq {K}^{n} \) is Zariski open, and \( {F}_{i} \cap {O}_{i} \) is nonempty. Because \( \operatorname{RM}\left( X\right) = \operatorname{RM}\left( V\ri... | Yes |
Theorem 6.3.2 (Existence of nonforking extensions) Suppose that \( p \in {S}_{n}\left( A\right) \) and \( A \subseteq B \) .\n\ni) There is \( q \in {S}_{n}\left( B\right) \) a nonforking extension of \( p \) .\n\nii) There are at most \( {\deg }_{\mathrm{M}}\left( p\right) \) nonforking extensions of \( p \) in \( {S}... | Proof Let \( \phi \left( \bar{v}\right) \in p \) be of minimal Morley rank and degree with \( \operatorname{RM}\left( \phi \right) = \) \( \alpha \) .\n\ni) Let \( \mathcal{M} \) be an \( {\aleph }_{0} \) -saturated model containing \( B \) . Let \( \psi \left( \bar{v}\right) \) be an \( {\mathcal{L}}_{M} \) - formula ... | No |
Lemma 6.3.4 Suppose that \( \mathcal{M} \) is \( {\aleph }_{0} \) -saturated, \( \phi \left( \bar{v}\right) \) is an \( {\mathcal{L}}_{M} \) -formula with \( \operatorname{RM}\left( \phi \right) = \alpha \), and \( \psi \left( \bar{v}\right) \) is an \( {\mathcal{L}}_{\mathbb{M}} \) -formula with \( \operatorname{RM}\l... | Proof We prove this by induction on \( \alpha \) . If \( \alpha = 0 \), this is clear because \( \phi \left( \mathbb{M}\right) \) is finite and \( \phi \left( \mathcal{M}\right) = \phi \left( \mathbb{M}\right) \) . Suppose that \( \alpha > 0 \) . If \( {\deg }_{\mathrm{M}}\left( \phi \right) = d > 1 \) , then we can fi... | Yes |
Corollary 6.3.6 If \( p \in {S}_{n}\left( A\right) \), then \( p \) is definable over \( {A}_{0} \) for some finite \( {A}_{0} \subseteq A \) . | Proof Let \( \phi \left( \bar{v}\right) \in p \) be of minimal Morley rank and degree. Let \( {A}_{0} \subseteq A \) such that \( \phi \) is an \( {\mathcal{L}}_{{A}_{0}} \) -formula, and let \( \operatorname{RM}\left( \phi \right) = \alpha \) . For any formula \( \psi \left( {\bar{v},\bar{w}}\right) \) and \( \bar{a} ... | Yes |
Corollary 6.3.7 Suppose that \( A \subseteq M \) and \( X \subseteq {M}^{n} \) is \( A \) -definable. Then, any \( Y \subseteq {X}^{m} \) is \( A \cup X \) -definable. | Proof Let \( \psi \left( {\bar{v},\bar{b}}\right) \) define \( Y \) . Then \( Y = \left\{ {\bar{c} \in {X}^{n} : \psi \left( {\bar{c},\bar{b}}\right) \in \operatorname{tp}\left( {\bar{b}/X}\right) }\right\} \) . Because \( \operatorname{tp}\left( {\bar{b}/X}\right) \) is definable over \( X, Y \) is definable over \( A... | Yes |
Proposition 6.3.8 Suppose that \( p \in {S}_{n}\left( A\right) \) and \( {\deg }_{\mathrm{M}}\left( p\right) = 1 \) . Then, \( p \) is definable over \( A \) . If \( B \supseteq A \), let\n\n\[ \n{p}_{B} = \left\{ {\psi \left( {\bar{v},\bar{b}}\right) : \mathbb{M} \vDash {d}_{p}\psi \left( \bar{b}\right) ,\bar{b} \in B... | Proof Suppose that \( p \in {S}_{n}\left( A\right) ,\operatorname{RM}\left( p\right) = \alpha \), and \( {\deg }_{\mathrm{M}}\left( p\right) = 1 \) . Let \( \phi \left( \bar{v}\right) \) be an \( {\mathcal{L}}_{A} \) -formula with \( \operatorname{RM}\left( \phi \right) = \alpha \) and \( {\deg }_{\mathrm{M}}\left( p\r... | Yes |
Theorem 6.3.9 Suppose that \( A \subseteq B, p \in {S}_{n}\left( A\right), q \in {S}_{n}\left( B\right) \), and \( q \) does not fork over \( A \) . There is \( \alpha \in {\operatorname{acl}}^{\mathrm{{eq}}}\left( A\right) \) such that \( q \) is definable over \( A \cup \{ \alpha \} \) . In other words, there is an A... | Proof We choose \( \mathcal{M} \), an \( {\aleph }_{0} \) -saturated model containing \( B \), and \( {q}^{ * } \in \) \( {S}_{n}\left( M\right) \), a nonforking extension of \( q \) . The argument above shows that \( {q}^{ * } \) , and hence \( q \), is definable over \( {\operatorname{acl}}^{\mathrm{{eq}}}\left( A\ri... | Yes |
Corollary 6.3.10 If \( \mathcal{M} \vDash T, p \in {S}_{n}\left( \mathcal{M}\right), M \subseteq B \) and \( q \in {S}_{n}\left( B\right) \) is a nonforking extension of \( p \), then \( q \) is definable over \( \mathcal{M} \) . | Proof If \( E \) is a definable equivalence relation with finitely many classes, then, because \( \mathcal{M} \prec \mathbb{M} \), every \( \bar{a} \in \mathbb{M} \) is equivalent to an element of \( M \) . | No |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.