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The theory of Boolean algebras which have a maximal nonzero atomless element, but infinitely many atoms, is complete and decidable. | Proof. Let \( \mathfrak{A} \) and \( \mathfrak{B} \) be two such Boolean algebras. Then we can write \( \mathfrak{A} \cong {\mathfrak{C}}_{0} \times {\mathfrak{C}}_{1} \) and \( \mathfrak{B} \cong {\mathfrak{D}}_{0} \times {\mathfrak{D}}_{1} \), where \( {\mathfrak{C}}_{0} \) and \( {\mathfrak{D}}_{0} \) are atomless, ... | Yes |
Lemma 24.2. If \( \mathbf{K} \) is a variety, then \( \mathbf{K} = \mathbf{{SK}} = \mathbf{{PK}} \) . | Proof. Say \( \mathbf{K} \) is the class of all models of the set \( \Gamma \) of identities. Obviously \( \mathbf{K} \subseteq \mathbf{{SK}} \) and \( \mathbf{K} \subseteq \mathbf{{PK}} \) . Now assume that \( f : \mathfrak{A} \rightarrowtail \mathfrak{B},\mathfrak{B} \in \mathbf{K} \), and \( \left\lbrack \left\lbrac... | Yes |
Proposition 24.7. Let \( \mathcal{L} \) be an algebraic language, and let \( \mathfrak{A} \) and \( \mathcal{L} \) be \( \mathcal{L} \) - structures.\n\n(i) If \( R \) is a congruence relation on \( \mathfrak{A} \), then \( {\left\lbrack \right\rbrack }_{R} \) is a homomorphism from \( \mathfrak{A} \) onto \( \mathfrak... | Proof. Conditions (i) and (ii) are completely straightforward. Now assume the hypothesis of (iii). Let \( g = \{ \left( {{fa},\left\lbrack a\right\rbrack }\right) : \dot{a} \in A\} \) . To see that \( g \) is a function, assume that \( {fa} = {fc} \) with \( a, c \in A \) ; we must show that \( \left\lbrack a\right\rbr... | Yes |
Proposition 24.9. If \( \mathbf{K} \) is a variety, then \( \mathbf{K} = \mathbf{{HK}} \) . | ## Proof. Obvious from 24.4. | No |
Proposition 24.11. The following conditions are equivalent:\n\n(i) \( \mathbf{K} = \mathbf{{SK}} = \mathbf{{HK}} = \mathbf{{PK}} \) .\n\n(ii) \( \mathbf{K} = \mathbf{{HSPK}} \) .\n\n(iii) \( \mathbf{K} = \mathbf{{HSPL}} \) for some \( \mathbf{L} \) . | Proof. Obviously \( \left( i\right) \Rightarrow \left( {ii}\right) \Rightarrow \left( {iii}\right) \) . Now assume (iii). Clearly \( \mathbf{{HK}} = \mathbf{K} \) and \( \mathbf{K} \subseteq \mathbf{{SK}},\mathbf{K} \subseteq \mathbf{{PK}} \) . Next,\n\n\[ \mathrm{{SK}} = \mathrm{{SHSPL}} \subseteq \mathrm{{HSSPL}} \]\... | Yes |
Lemma 24.18. Let \( \Gamma \) be a set of equations in an algebraic language. Then for any terms \( \sigma ,\tau ,\rho \) , (i) \( \Gamma { \vdash }_{\mathrm{{eq}}}\sigma = \sigma \) ; (ii) if \( \Gamma { \vdash }_{\mathrm{{eq}}}\sigma = \tau \), then \( \Gamma { \vdash }_{\mathrm{{eq}}}\tau = \sigma \) ; (iii) if \( \... | Proof. Condition \( \left( i\right) \) is obvious from 24.17(ii) and 24.17(iii). For (ii): assume that \( \Gamma { \vdash }_{\mathrm{{eq}}}\sigma = \tau \) . We also have \( \Gamma { \vdash }_{\mathrm{{eq}}}\tau \equiv \tau \) by \( \left( i\right) \), so \( \Gamma { \vdash }_{\mathrm{{eq}}}\tau \equiv \sigma \) by 24.... | Yes |
Lemma 24.20. Let \( \mathbf{K} \) be a class of \( \mathcal{L} \) -structures, where \( \mathcal{L} \) is an algebraic language. Let \( \Gamma = \{ \sigma \equiv \tau \) : the equation \( \sigma \equiv \tau \) holds in each member of \( \mathbf{K}\} \) Then \( {\mathfrak{{Fr}}}_{\mathcal{L}}/{ \equiv }_{\Gamma } \in \m... | Proof. Let \( I = {}^{2}\operatorname{Trm} \sim \{ \left( {\sigma ,\tau }\right) : \sigma = \tau \in \Gamma \} \) . Then for each \( \left( {\sigma ,\tau }\right) \in I \) there is a structure \( {\mathfrak{A}}_{\sigma \tau } \in \mathbf{K} \) such that \( \sigma = \tau \) fails to hold in \( {\mathfrak{A}}_{\sigma \ta... | Yes |
Lemma 24.21. Suppose \( \mathbf{K} = \mathbf{{SK}} = \mathbf{{UpK}} \), in a not necessarily algebraic language \( \mathcal{L} \) . If \( \mathfrak{A} \) is an \( \mathcal{L} \) -structure every finitely generated substructure of which is in \( \mathbf{K} \), then \( \mathfrak{A} \) is in \( \mathbf{K} \) . | Proof. Let \( I = \{ F : 0 \neq F \subseteq A, F \) finite \( \rbrack \) . For each \( F \in I \), let \( {\mathfrak{A}}_{F} \) be the substructure of \( \mathfrak{A} \) generated by \( F \) . Thus \( {\mathfrak{A}}_{F} \in \mathbf{K} \) by assumption. For each \( F \in I \) let \( {M}_{F} = \{ G \in I : F \subseteq G\... | Yes |
Theorem 24.22 (Birkhoff). K is a variety iff \( \mathbf{K} = \mathbf{{HSPK}} \) . | Proof. The trivial direction \( \Rightarrow \) is given by 24.2 and 24.9. Now assume that \( \mathbf{K} = \) HPSK. By 24.11, \( \mathbf{K} \) is closed under \( \mathbf{H},\mathbf{S} \), and \( \mathbf{P} \) ; and so it is also obviously closed under Up. Let \( \Gamma = \{ \sigma \equiv \tau : \sigma \equiv \tau \) hol... | Yes |
Theorem 24.23. Let \( \varphi \) be a sentence, in an algebraic language, which is preserved under \( \mathbf{H},\mathbf{S} \), and \( \mathbf{P} \) . That is, assume about \( \varphi \) the following three things:\n\n(i) if \( \mathfrak{A} \vDash \varphi \) and \( \mathfrak{A} \rightarrow \mathfrak{B} \), then \( \mat... | Proof. Let \( \mathbf{K} \) be the class of all models of \( \varphi \) . The conditions \( \left( i\right) \) -(iii) imply that \( \mathbf{K} = \) HSPK. Hence by 24.22 there is a set \( \Gamma \) of identities such that \( \mathbf{K} \) is the class of all models of \( \Gamma \) . Thus \( \Gamma \vDash \varphi \), so ... | Yes |
Let \( \mathbf{K} \) be a class of \( {\mathcal{L}}^{\prime } \) -structures and let \( \mathcal{L} \) be a reduct of \( {\mathcal{L}}^{\prime } \) . If \( \mathbf{K} \) can be characterized by first-order sentences, then \( \mathbf{S}\left( {\mathbf{K} \upharpoonright \mathcal{L}}\right) \) is a universal class. | Proof. By 25.3 and 18.29, \( \mathbf{S}\left( {\mathbf{K} \upharpoonright \mathcal{L}}\right) \) is closed under \( \mathbf{S} \) and \( \mathbf{{Up}} \) . Hence 25.5 is immediate from 25.2. | Yes |
Proposition 25.7. With notation as in 25.6, suppose that for each \( i \in I \), an \( \mathcal{L} \) -structure \( {\mathfrak{C}}_{i} \) is a model of the \( \mathcal{L} \) -closure conditions for \( \mathbf{U} \) . Then\n\n\[{\mathrm{P}}_{i \in I}\left( {{\mathfrak{C}}_{i} \upharpoonright \mathbf{U}}\right) /F \cong ... | Proof For any \( x \in {\mathrm{P}}_{i \in I}\left( {{\mathfrak{C}}_{i} \upharpoonright \mathbf{U}}\right) \), let \( {fx} = \left\lbrack x\right\rbrack \), the equivalence class of \( x \) under \( F \) in the structure \( {\mathrm{P}}_{i \in I}{\mathfrak{C}}_{i}/F \) . It is easy to see that \( f \) induces the desir... | No |
Corollary 25.8. Assume that \( \mathcal{L},\mathbf{U} \), and \( {\mathcal{L}}^{\prime } \) are as in 25.6, and that \( \Gamma \) is a set of sentences of \( {\mathcal{L}}^{\prime } \) including the set of \( \mathcal{L} \) -closure conditions for \( \mathbf{U} \) . Then \( \mathbf{S}\left( {\left( {\operatorname{Mod}\... | Proof By 25.2 and 25.3 it suffices to show that \( \left( {\left( {\operatorname{Mod}\Gamma }\right) \upharpoonright \mathcal{L}}\right) \upharpoonright \mathbf{U} \) is closed under ultraproducts. Suppose that \( {\mathfrak{A}}_{i} \in \left( {\left( {\operatorname{Mod}\Gamma }\right) \upharpoonright \mathcal{L}}\righ... | Yes |
Let \( \Gamma \cup \{ \varphi \} \) be a collection of sentences. Then the following conditions are equivalent: (i) \( \varphi \) is preserved under substructures relative to \( \Gamma \), i.e., for all \( \mathfrak{A},\mathfrak{B} \in \operatorname{Mod}\Gamma \) , if \( \mathfrak{A} \subseteq \mathfrak{B} \) and \( \m... | Proof. It is trivial that \( \left( {ii}\right) \Rightarrow \left( i\right) \) . Now assume that \( \left( i\right) \) holds. Now by 25.3, \( \operatorname{SMod}\left( {\Gamma \cup \{ \varphi \} }\right) \) is closed under \( \mathbf{{Up}} \), so by 25.2 we may write \( \operatorname{SMod}(\Gamma \cup \) \( \{ \varphi ... | Yes |
Theorem 25.11 (Tarski). Let \( \mathcal{L} \) be a language with no operation symbols, and with only finitely many relation symbols. Let \( \mathbf{K} \) be a class of \( \mathcal{L} \) structures. Then the following conditions are equivalent:\n\n(i) \( \mathbf{K} \) is a universal class;\n\n(ii) \( \mathbf{{SK}} = \ma... | \n\( \left( i\right) \Rightarrow \left( {ii}\right) \) . Assume \( \left( i\right) \) and the hypothesis of \( \left( {ii}\right) \) ; say \( \mathbf{K} = \operatorname{Mod}\Gamma \), with \( \Gamma \) a set of universal sentences. Let \( \varphi = \forall {v}_{i0}\cdots \forall {v}_{i\left( {m - 1}\right) }\psi \) be ... | Yes |
Corollary 25.17. Let \( \Gamma \cup \{ \varphi \} \) be a set of sentences such that \( \operatorname{Mod}\Gamma \) is closed under directed unions. Then the following conditions are equivalent:\n\n(i) \( \varphi \) is preserved under directed union relative to \( \Gamma \), i.e., \( \operatorname{Mod}\left( {\Gamma \c... | The corollary is proved just like 25.10. | No |
Lemma 25.19. If \( \mathfrak{A} \) pos \( \mathfrak{B} \), then there is an elementary extension \( {\mathfrak{B}}^{\prime } \succcurlyeq \mathfrak{B} \) and a homomorphism \( f : \mathfrak{A} \rightarrow {\mathfrak{B}}^{\prime } \) such that \( {\left( \mathfrak{A}, a\right) }_{a \in A} \) pos \( {\left( {\mathfrak{B}... | Proof. Let \( \mathcal{L} \) be our initial language, \( {\mathcal{L}}^{\prime } \) an \( A \) -expansion of \( \mathcal{L} \), and \( {\mathcal{L}}^{\prime \prime } \) a \( B \) -expansion of \( \mathcal{L} \) ; say we have new constants \( {\mathbf{c}}_{a}\left( {a \in A}\right) \) in \( {\mathcal{L}}^{\prime } \) an... | Yes |
Lemma 25.20. If \( \mathfrak{A} \) pos \( \mathfrak{B} \), then there is an elementary extension \( {\mathfrak{A}}^{\prime } \succcurlyeq \mathfrak{A} \) and a one-one mapping \( g : B \rightarrow {A}^{\prime } \) such that \( {\left( {\mathfrak{A}}^{\prime }, gb\right) }_{b \in B} \) pos \( {\left( \mathfrak{B}, b\rig... | Proof Let \( {\mathcal{L}}^{\prime } \) and \( {\mathcal{L}}^{\prime \prime } \) be as in the proof of 25.19. Let \( \Gamma \) be the elementary \( {\mathcal{L}}^{\prime } \) -diagram of \( \mathfrak{A} \), and let \( \Delta \) be the set \( \{ \neg \varphi : \varphi \) is a positive sentence of \( {\mathcal{L}}^{\prim... | No |
Lemma 25.21. Assume that \( \mathfrak{B} \) is a model of \( \{ \varphi : \varphi \) is a positive sentence and \( \Gamma \vDash \varphi \} \) . Then \( \Gamma \) has a model \( \mathfrak{A} \) such that \( \mathfrak{A} \) pos \( \mathfrak{B} \) . | Proof. Consider \( \Gamma \cup \{ \neg \varphi : \varphi \) is a positive sentence and \( \Gamma \vDash \neg \varphi \} \) . | No |
Theorem 25.22 (Lyndon). For any set \( \Gamma \) of sentences the following conditions are equivalent:\n\n(i) Mod \( \Gamma \) can be characterized by a set of positive sentences;\n\n(ii) Mod \( \Gamma \) is closed under homomorphisms, i.e., if \( \mathfrak{A} \in \operatorname{Mod}\Gamma \) and \( \mathfrak{A} \righta... | Proof. The trivial direction is \( \left( i\right) \Rightarrow \left( {ii}\right) \) : assume \( \left( i\right) \) . To show \( \left( {ii}\right) \) it suffices to prove\n\n(1) if \( \varphi \) is any positive formula, \( f : \mathfrak{A} \rightarrow \mathfrak{B}, x \in {}^{\omega }A \) and \( \mathfrak{A} \vDash \va... | Yes |
Theorem 26.2. Let \( \mathbf{K} \) be a class of \( \mathcal{L} \) -structures. Each of the following conditions implies the succeeding one. If \( \mathbf{K} \) is elementarily closed, then they are all equivalent.\n\n(i) \( \mathbf{K} \) is an elementary class;\n\n(ii) \( \mathbf{K} \) is a projective class;\n\n(iii) ... | Proof. Obviously \( \left( i\right) \Rightarrow \left( {ii}\right) \Rightarrow \left( {iii}\right) \Rightarrow \left( {iv}\right) \) . Now suppose that \( \mathbf{K} \) is elementarily closed and compact. We show that \( \mathbf{K} = \) Mod \( {\Theta \rho }\mathbf{K} \), so that \( \left( i\right) \) holds. Obviously ... | Yes |
Lemma 26.5. Let \( \varphi \) be a sentence in prenex normal form with \( m \) initial quantifiers, and suppose that \( \mathfrak{A}{ \equiv }_{m}\mathfrak{B} \) . Then \( \mathfrak{A} \vDash \varphi \) iff \( \mathfrak{B} \vDash \varphi \) . | Proof. We proceed by induction on \( m \) . For \( m = 0,\varphi \) is a sentence with no variables, and hence \( \mathcal{L} \) has individual constants. Thus the assumption \( \mathfrak{A}{ \equiv }_{0}\mathfrak{B} \) means that \( {\mathfrak{A}}^{ - } \cong {\mathfrak{B}}^{ - } \) . Hence, obviously, \( \mathfrak{A}... | Yes |
Lemma 26.7. Let \( \mathcal{L} \) be a language with no operation symbols and only finitely many relation symbols. Assume that \( \mathfrak{A} \) and \( \mathfrak{B} \) are \( \mathcal{L} \) -structures, \( n \in \omega \sim 1 \) , and that for any \( \varphi \in {\Delta }_{n}^{n},\mathfrak{A} \vDash \varphi \) iff \( ... | Proof. For each \( i \leq n \) let\n\n\[ \n{I}_{i} = \left\{ {\left( {x, y}\right) : x \in {}^{i}A, y \in {}^{i}B\text{, and }\mathfrak{A} \vDash \varphi \left\lbrack x\right\rbrack \text{ iff }\mathfrak{B} \vDash \varphi \left\lbrack y\right\rbrack \text{ for all }\varphi \in {\Delta }_{n}^{n - i}}\right\} .\n\]\n\nOb... | Yes |
Lemma 26.8. Let \( n \in \omega \) . If there is an \( \left( {n + 1}\right) \) -sequence for \( \mathfrak{A} \) and \( \mathfrak{B} \), then \( \mathfrak{A} \equiv {}_{n}\mathfrak{B} \) . | Proof. Let \( \left\langle {{I}_{m} : m \leq n}\right\rangle \) be an \( \left( {n + 1}\right) \) -sequence for \( \mathfrak{A},\mathfrak{B} \) . We now prove the following statement by induction on \( i \) ; the case \( i = n \) gives the desired result.\n\n(1)\n\n\[ \text{if}x{I}_{n - i}y\text{, then}{\left( \mathfra... | Yes |
Lemma 26.10. Let \( \mathfrak{A} \) and \( \mathfrak{B} \) be \( \mathcal{L} \) -structures and let \( n \in \omega \) . If there is an \( \left( {n + 1}\right) \) -sequence for \( \mathfrak{A} \) and \( \mathfrak{B} \), then there is an \( \left( {n + 1}\right) \) -system of partial isomorphisms of \( \mathfrak{A} \) ... | Proof. Let \( \left\langle {{J}_{m} : m \in n + 1}\right\rangle \) be an \( \left( {n + 1}\right) \) -sequence for \( \mathfrak{A},\mathfrak{B} \) . For each \( m \in n + 1 \) we set\n\n\[ \n{I}_{m} = \left\{ {\left\{ {\left( {{x}_{i},{y}_{i}}\right) : i < k}\right\} : k \leq n - m\text{ and }x{J}_{k}y}\right\} .\n\]\n... | Yes |
Lemma 26.12. Let \( \mathfrak{A} \) and \( \mathfrak{B} \) be \( \mathcal{L} \) -structures, and let \( m \) be a positive integer. If there is an \( \left( {m + 1}\right) \) -system of partial isomorphisms of \( \mathfrak{A} \) into \( \mathfrak{B} \), then player II has a winning strategy for the m-elementary game ov... | Proof. Assume the hypothesis of 26.12, and let \( \left\langle {{I}_{k} : k \leq m}\right\rangle \) be an \( \left( {m + 1}\right) \) - system of partial isomorphisms of \( \mathfrak{A} \) into \( \mathfrak{B} \) . We define a function satisfying \( {26.11}\left( i\right) \) as follows. Let a well-ordering of \( A, B \... | Yes |
If \( \left( {\alpha , < }\right) { \equiv }_{\mathrm{{ee}}}\left( {\beta , < }\right) \), then \( \alpha \equiv \beta \left( {\;\operatorname{mod}\;{\omega }^{\omega }}\right) \), and if \( \alpha < {\omega }^{\omega } \) or \( \beta < {\omega }^{\omega } \) then \( \alpha = \beta \) . | The idea of the proof is to show that each ordinal \( < {\omega }^{\omega } \) is elementarily definable; the theorem then easily follows. To prove this, we need to express some simple concepts concerning ordinals. Let \ | No |
Lemma 26.20. If \( \mathfrak{A} \) and \( \mathfrak{B} \) are denumerable \( \mathcal{L} \) -structures and there is an \( \omega \) - sequence for \( \mathfrak{A},\mathfrak{B} \), then \( \mathfrak{A} \cong \mathfrak{B} \) . | Proof. Let \( A = \left\{ {{a}_{i} : i \in \omega }\right\} \) and \( B = \left\{ {{b}_{i} : i \in \omega }\right\} \) . Let \( \left\langle {{I}_{m} : m \in \omega }\right\rangle \) be an \( \omega \) - sequence for \( \mathfrak{A},\mathfrak{B} \) . Now we define sequences \( x \in {}^{\omega }A, y \in {}^{\omega }B \... | Yes |
Lemma 26.23. Let \( L \) be a general logic such that \( {L}_{\mathrm{{fo}}} \subseteq L \) but \( L \nsubseteq {L}_{\mathrm{{fo}}} \) . Assume also that \( L \) satisfies the downward Löwenheim-Skolem theorem, in the sense of 26.22. Let \( \mathcal{L} \) be as in 26.22. Then there is a class \( \mathbf{K} \) of \( \ma... | Proof. By 26.22 we may assume that \( L{ \subseteq }_{\text{inf }}{L}_{\text{fo }} \) . Let \( \mathbf{K} \) be an \( L \) -elementary class which is not a finitely axiomatizable elementary class in the usual sense. Then there is a finitely axiomatizable elementary class \( \mathbf{L} \) with the same infinite members ... | Yes |
Theorem 26.24 (Lindstrom). Let \( L \) be a general logic such that \( {L}_{\mathrm{{fo}}} \subseteq L \) . Assume that \( L \) satisfies the downward Löwenheim-Skolem theorem and the countable compactness theorem; that is, assume the following conditions:\n\n(i) any L-elementary class with an infinite member has a den... | Proof. Assume the hypothesis, but suppose that \( L \subseteq {L}_{\mathrm{{fo}}} \) . By Lemma 26.23, with \( \mathcal{L} \) as in 26.22, there is an expansion \( {\mathcal{L}}^{\prime } \) of \( \mathcal{L} \) and an \( L \) -elementary class \( {\mathbf{K}}_{0} \) of \( {\mathcal{L}}^{\prime } \) -structures such th... | Yes |
Lemma 26.34. Let \( \mathfrak{n} \geq {\aleph }_{0} \) . Then there is an \( F \subseteq {}^{\mathfrak{n}}\left( {\mathfrak{n}}^{\partial }\right) \) with \( \left| F\right| = {2}^{\mathfrak{n}} \) such that \( \left( {F,0,\{ \mathfrak{n}\} }\right) \) is 2-consistent over \( \mathfrak{n} \) . | Proof. Since \( G \) is empty, we really only have to find a \ | No |
Lemma 26.36. Let \( \delta \) be a limit ordinal \( < {\mathrm{m}}^{ + } \) and suppose that \( \left( {{F}_{\beta },{G}_{\beta },{D}_{\beta }}\right) \) is \( {\mathfrak{m}}_{\beta } \) -consistent over \( \mathfrak{n} \) for each \( \beta < \delta \) . Also assume that if \( \beta < \gamma < \delta \) then \( {F}_{\b... | Proof. The conditions (i)-(iii) of 26.33 are obvious. Now assume the hypothesis of \( {26.33}\left( {iv}\right) \), but assume that the conclusion fails:\n\n(1) \( n \sim \left\{ {\beta < n : \forall \gamma < \rho \left( {{f}_{\gamma }\beta = {\alpha }_{\gamma }}\right) }\right. \) and \( \left. {\forall m < n\left( {{... | Yes |
Lemma 26.38. Let \( \mathfrak{n} \geq {\aleph }_{0} \) . Assume that \( G \subseteq \mathop{\bigcup }\limits_{{\rho < {\mathfrak{n}}^{\partial }}}{}^{\mathfrak{n}}\rho \) and that \( {\mathfrak{n}}^{\partial } + \left| G\right| \leq \) \( \mathfrak{m} \geq {\aleph }_{0} \) . Suppose that \( \left( {F,0, D}\right) \) is... | Proof. Let \( I = \{ H : H \subseteq G, H \) finite \( \} \) . For each \( H \in I \) choose \( {F}_{H} \subseteq F \) by the preceding lemma so that \( \left| {F}_{H}\right| \leq \mathrm{m} \) and \( \left( {F \sim {F}_{H}, H, D}\right) \) is \( \mathrm{m} \) -consistent over \( \mathfrak{n} \) . Then clearly \( \math... | Yes |
Lemma 26.39. Suppose that \( \left( {F, G, D}\right) \) is \( \mathfrak{m} \) -consistent over \( \mathfrak{n} \) and \( \Gamma \subseteq \mathfrak{n} \) . Then there is an \( {F}^{\prime } \subseteq F \) with \( \left| {F - {F}^{\prime }}\right| < {\mathfrak{n}}^{\partial } \) such that either \( \left( {{F}^{\prime }... | Proof. Let \( {D}_{1} = \mathcal{F}\mathcal{G}\left( {D\cup \{ \Gamma \} }\right) \) and \( {D}_{2} = \mathcal{F}\mathcal{G}\left( {D\cup \{ \mathfrak{n} \sim \Gamma \} }\right) \) . Assume that \( \left( {F, G,{D}_{1}}\right) \) is not \( \mathfrak{m} \) -consistent over \( \mathfrak{n} \) . This means that there is a... | Yes |
Lemma 26.40. Suppose that \( \left( {F, G, D}\right) \) is \( \mathfrak{m} \) -consistent over \( \mathfrak{n} \) and that \( {\Gamma }_{\alpha } \subseteq \mathfrak{n} \) for every \( \alpha < \mathfrak{m} \) . Assume that \( {\mathfrak{n}}^{\partial } \leq \mathfrak{m} \) . Then there exist \( {F}^{\prime } \subseteq... | Proof. We define \( \left\langle {{H}_{\alpha } : \alpha < \mathfrak{m}}\right\rangle \) and \( \left\langle {{E}_{\alpha } : \alpha < \mathfrak{m}}\right\rangle \) by transfinite recursion. Suppose \( {H}_{\beta } \) and \( {E}_{\beta } \) have been defined for all \( \beta < \alpha \), where \( \alpha < m \), in such... | Yes |
Lemma 26.41. Assume that \( \mathfrak{n} \geq {\aleph }_{0} \) and \( {\mathfrak{n}}^{\partial } \leq \mathfrak{m} \) . Suppose that \( \left( {F,0, D}\right) \) is \( \mathfrak{m} \) -consistent over \( \mathfrak{n} \), and \( \left| F\right| > \mathfrak{m} \) . Let \( \mathfrak{A} \) be an \( \mathcal{L} \) -structur... | Proof of 26.41. Let \( A = \left\{ {{c}_{\gamma } : \gamma < \rho }\right\} \), where \( \rho < {\mathfrak{n}}^{\partial } \) . For each \( \nu < \mu \) let\n\n\[ {B}_{v} = \left\{ {\delta < n : \mathfrak{A} \vDash \left( {\exists {v}_{0}{\varphi }_{v}}\right) \left\lbrack \left\langle {{a}_{vj\delta } : j < \omega }\r... | Yes |
Theorem 26.43 (Keisler, Shelah). The following conditions are equivalent:\n\n(i) \( \\mathbf{K} \) is an elementary class;\n\n(ii) \( \\mathbf{{UpK}} = \\mathbf{K} \) and if \( {}^{I}\\mathfrak{A}/F \\in \\mathbf{K} \), then \( \\mathfrak{A} \\in \\mathbf{K} \) . | Proof. The implication \( \\left( i\\right) \\Rightarrow \\left( {ii}\\right) \) is obvious. Now assume \( \\left( {ii}\\right) \). By 26.2 it suffices to show that \( \\mathbf{K} \) is elementarily closed. Assume, then, that \( \\mathfrak{A}{ \\equiv }_{\\mathrm{{ee}}}\\mathfrak{B} \\in \\mathbf{K} \). By Theorem 26.4... | Yes |
Proposition 27.2. Let \( \\left( {\\Gamma ,\\mathcal{L}}\\right) \) be a theory and let \( \\Delta \\subseteq {\\mathrm{{Fmla}}}_{\\mathcal{L}} \). The following conditions are equivalent:\n\n(i) \( \\Delta \) is an \( n \) -type over \( \\Gamma \);\n\n(ii) \( \\Delta \) is maximal consistent over \( \\Gamma \);\n\n(ii... | Proof. Obviously \( \\left( i\\right) \\Rightarrow \\left( {ii}\\right) \) and \( \\left( {iii}\\right) \\Rightarrow \\left( i\\right) \). Now assume \( \\left( {ii}\\right) \). Let us expand the language \( \\mathcal{L} \) to \( {\\mathcal{L}}^{\\prime } \) by adjoining new individual constants \( {\\mathbf{c}}_{0},\\... | Yes |
Theorem 27.4 [Omitting types theorem (Henkin, Orey)]. Let \( \Gamma \) be a consistent theory in a countable language \( \mathcal{L} \). Suppose that \( N \) is a non-empty subset of \( \omega \) and that for each \( n \in N \) a set \( {\Delta }_{n} \subseteq {\operatorname{Fmla}}_{\mathcal{L}}^{n} \) is given. Then t... | ## Proof\n\n\( \left( i\right) \Rightarrow \left( {ii}\right) \). Let \( \mathfrak{A} \) be a model of \( \Gamma \) which omits each \( {\Delta }_{n} \). Set \( {\Gamma }^{\prime } = {\Theta \rho }\mathfrak{A} \). To verify (ii), suppose \( n \in N,\psi \in {\operatorname{Fmla}}_{\mathcal{L}}^{n} \), and \( {\Gamma }^{... | Yes |
Theorem 27.6 (Ehrenfeucht). Let \( \Gamma \) be a consistent theory in a countable language \( \mathcal{L} \), and suppose \( 0 \neq N \subseteq \omega \) . For each \( n \in N \) let \( {\Delta }_{n} \) be an \( n \) -type over \( \Gamma \), and assume that no set \( {\Delta }_{n} \) contains a formula \( n \) -atomic... | Proof. We shall apply 27.4. To this end, assume that \( n \in N,\psi \in {\operatorname{Fmla}}_{\mathcal{L}}^{n} \) , and \( \Gamma \cup \left\{ {\exists {v}_{0}\cdots \exists {v}_{n - 1}\psi }\right\} \) has a model. If \( \psi \notin {\Delta }_{n} \), then \( \neg \psi \in {\Delta }_{n} \) since \( {\Delta }_{n} \) i... | Yes |
Theorem 27.7 (Vaught). Let \( \Gamma \) be a consistent theory in a countable language. Then \( \Gamma \) has a countable model \( \mathfrak{A} \) such that for every \( n \in \omega \), every \( n \) -tuple of elements of \( A \) satisfies either an n-atomic formula over \( \Gamma \) or an n-atomless formula over \( \... | Proof. For each \( n \in \omega \) let\n\n\[{\Delta }_{n} = \left\{ {\varphi \in {\operatorname{Fmla}}_{\mathcal{L}}^{n} : \neg \varphi }\right. \text{is}n\text{-atomic or}n\text{-atomless over}\left. \Gamma \right\} \text{.}\]\n\nTo check the hypothesis of 27.4, assume that \( n \in \omega ,\psi \in {\operatorname{Fml... | Yes |
Theorem 27.10 (Vaught). Let \( \Gamma \) be a complete theory in a countable language \( \mathcal{L} \) . Then the following conditions are equivalent:\n\n(i) \( \Gamma \) has an elementarily prime model:\n\n(ii) for every \( n \in \omega \) there are no n-atomless formulas over \( \Gamma \) . | Proof\n\n\( \left( i\right) \Rightarrow \left( {ii}\right) \) . Let \( \Gamma \) have an elementarily prime model \( \mathfrak{A} \) . Let \( n \in \omega \), and let \( \varphi \in {\operatorname{Fmla}}_{\mathcal{L}}^{n} \), with \( \Gamma \cup \left\{ {\exists {v}_{0}\cdots \exists {v}_{n - 1}\varphi }\right\} \) con... | Yes |
Theorem 27.11 (Vaught). Any two elementarily prime models of a theory \( \Gamma \) in a countable language \( \mathcal{L} \) are isomorphic. | Proof. We shall apply 26.20 ; so we need to construct an \( \omega \) -sequence for the elementarily prime models \( \mathfrak{A} \) and \( \mathfrak{B} \) of \( \Gamma \) . For each \( m \in \omega \), let \( {I}_{m} \) consist of all pairs \( \left( {x, y}\right) \in {}^{m}A \times {}^{m}B \) such that there is an \(... | Yes |
Theorem 27.14 (Ramsey). If \( X \) is an infinite set, \( n \in \omega \sim 1 \), and \( {\mathbf{S}}_{n}X = \) \( {A}_{0} \cup {A}_{1} \), then there is an infinite \( Z \subseteq X \) and an \( i < 2 \) such that \( {\mathbf{S}}_{n}Z \subseteq {A}_{i} \) . | Proof. We proceed by induction on \( n \) . The case \( n = 1 \) is trivial. Assume the result for \( n \), and assume that \( X \) is an infinite set and \( {\mathbf{S}}_{n + 1}X = {A}_{0} \cup {A}_{1} \) . We now define three sequences \( \left\langle {{x}_{i} : i < \omega }\right\rangle ,\left\langle {{Y}_{i} : i < ... | Yes |
Proposition 27.16. Let \( \mathfrak{A} = \langle A, \leq \rangle \) be a simple ordering structure, \( A \subseteq B \) , \( \mathfrak{B} \) an \( \mathcal{L} \) -structure. The following conditions are equivalent:\n\n(i) \( \mathfrak{A} \) is homogeneous for \( \mathfrak{B} \) ;\n\n(ii) \( \forall n \in \omega \forall... | Proof. \( \left( i\right) \Rightarrow \left( {ii}\right) \) : trivial. \( \left( {ii}\right) \Rightarrow \left( i\right) \) . Assume \( \left( {ii}\right) \), and assume that \( n \in \omega \) , \( \varphi \in {\operatorname{Fmla}}_{\mathcal{L}}^{n}, x, y \in {}^{n}A \), and\n\n(1)\n\n\[ \forall i, j < n\left( {{x}_{i... | Yes |
Proposition 28.2. Let \( \mathfrak{A} \) be an \( \mathcal{L} \) -structure and \( \mathfrak{m} \) an infinite cardinal. Then the following conditions are equivalent:\n\n(i) \( \mathfrak{A} \) is \( \mathfrak{m} \) -saturated;\n\n(ii) for every \( X \subseteq A \) with \( \left| X\right| < \mathfrak{m} \), if \( {\math... | Proof. Obviously \( \left( {ii}\right) \Rightarrow \left( i\right) \) . Now assume \( \left( i\right) \) and the hypothesis of \( \left( {ii}\right) \) . We may assume that \( \Delta \) is closed under conjunction. For each \( \varphi \in \Delta \) choose \( {m}_{\varphi } \) such that \( \mathrm{{Fv}}\varphi \subseteq... | Yes |
Proposition 28.3. If \( \mathfrak{A} \) is \( \mathfrak{m} \) -saturated, then either \( \mathfrak{A} \) is finite or \( \left| A\right| \geq \mathfrak{m} \) . | Proof. Suppose \( \mathfrak{A} \) is infinite and \( \left| A\right| < \mathfrak{m} \) . Let \( {\mathcal{L}}^{\prime } \) be an \( A \) -expansion of \( \mathcal{L} \), and let \( \Delta \) be \( \left\{ {\neg \left( {{v}_{0} = {\mathbf{c}}_{a}}\right) : a \in A}\right\} \) . Clearly every finite subset of \( \Delta \... | Yes |
Proposition 28.4. A is finite iff \( \mathfrak{A} \) is \( \mathfrak{m} \) -saturated for all \( \mathfrak{m} \) . | Proof. The second condition implies the first, by 28.3. Now suppose that \( \mathfrak{A} \) is finite, \( X \subseteq A,\left| X\right| < \mathfrak{m},{\mathcal{L}}^{\prime } \) is an \( X \) -expansion of \( \mathcal{L} \), and \( \Delta \subseteq {\operatorname{Fmla}}_{\mathcal{L}}^{1} \) . Suppose that \( \Delta \) ... | Yes |
Theorem 28.6. Let \( \mathcal{L},{\mathcal{L}}^{\prime },{\mathcal{L}}^{\prime \prime } \) be first-order languages, \( \mathfrak{m} \) and \( \mathfrak{n} \) cardinals, and \( \Gamma \) and \( {\Gamma }^{\prime } \) theories in \( \mathcal{L} \) and \( {\mathcal{L}}^{\prime } \) respectively, subject to the following ... | Proof. Let a well-ordering of \( C \) be given. We shall apply the model existence Theorem 18.9 in its full form. Let \( S \) be the set of all \( \Theta \subseteq {\operatorname{Sent}}_{{\mathcal{L}}^{\prime \prime }} \) such that \( \Gamma \cup \Theta \) is consistent and \( \{ \mathbf{c} \in C \) : \( \mathbf{c} \) ... | Yes |
Theorem 28.7 (Vaught). Let \( \Gamma \) be a theory in a countable language \( \mathcal{L} \) . Then the following conditions are equivalent:\n\n(i) \( \Gamma \) has a countable \( {\aleph }_{0} \) -saturated model;\n\n(ii) for each \( m \in \omega \) there are only countably many \( m \) -types over \( \Gamma \) . | \( \left( i\right) \Rightarrow \left( {ii}\right) \) . Let \( \mathfrak{A} \) be a countable \( {\aleph }_{0} \) -saturated model of \( \Gamma \), and let \( m \in \omega \) . For each \( m \) -type \( \Delta \) over \( \Gamma \) let \( {f\Delta } = \left\{ {a \in {}^{m}A : a\text{ realizes }\Delta \text{in }\mathfrak{... | Yes |
If \( \Gamma \) is a consistent theory in a countable language and \( \Gamma \) has up to isomorphism only countably many countable models, then \( \Gamma \) has a countable \( {\aleph }_{0} \) -saturated model. | Proof. Let \( m \in \omega \) . Each \( m \) -type over \( \Gamma \) is realized in some countable model of \( \Gamma \), by 27.2. Hence there are only countably many \( m \) -types over \( \Gamma \) (cf. the proof \( \left( i\right) \Rightarrow \left( {ii}\right) \) in 28.7). So by 28.7, \( \Gamma \) has a countable \... | No |
Theorem 28.10. Let \( \Gamma \) be a complete theory in a countable language. If \( \Gamma \) has a countable \( {\aleph }_{0} \) -saturated model, then \( \Gamma \) has an elementarily prime model. | Proof. Suppose that \( \Gamma \) has no elementarily prime model. Then by 27.10 there is an \( n \in \omega \) for which there is an \( n \) -atomless formula \( \varphi \) over \( \Gamma \) . We shall show that there are \( {2}^{{\aleph }_{0}}n \) -types over \( \Gamma \), so that by \( {28.7\Gamma } \) has no countab... | Yes |
Theorem 28.11. Let \( \mathcal{L},{\mathcal{L}}^{\prime } \) be the first-order languages, \( \mathfrak{m} \) and \( \mathfrak{n} \) cardinals, and \( {\Gamma }^{\prime } \) a theory in \( {\mathcal{L}}^{\prime } \), subject to the following conditions:\n\n(i) \( \left| {\mathrm{{Fmla}}}_{\mathcal{L}}\right| < \mathfra... | Proof. Let \( \Gamma = {\Gamma }^{\prime } \cap {\text{Sent}}_{\mathcal{L}} \) . By extending \( {\mathcal{L}}^{\prime } \) and \( {\Gamma }^{\prime } \) we may assume that \( \left| {\mathrm{{Fmla}}}_{{\mathcal{L}}^{\prime }}\right| = \mathfrak{n} \) . Let \( {\mathcal{L}}^{\prime \prime } \) be as in 28.6. Clearly 28... | Yes |
Corollary 28.12. Suppose \( \left| {\mathrm{{Fmla}}}_{\mathcal{L}}\right| \leq \mathrm{m} \), and \( \mathfrak{A} \) is an \( \mathcal{L} \) -structure with \( {\aleph }_{0} \leq \left| A\right| \leq {2}^{\mathfrak{m}} \) . Then there is an \( {\mathfrak{m}}^{ + } \) -saturated elementary extension \( \mathfrak{B} \) o... | Proof. Let \( {\mathcal{L}}^{ * } \) be an \( A \) -expansion of \( \mathcal{L} \), with new individual constants \( {\mathbf{c}}_{a} \) for \( a \in A \), and let \( {\mathcal{L}}^{\prime } \) be an expansion of \( {\mathcal{L}}^{ * } \) by adjoining new individual constants \( {\mathbf{d}}_{\alpha },\alpha < {2}^{\ma... | Yes |
Corollary 28.13. Suppose that \( \mathfrak{m} \) is strongly inaccessible, \( \left| {\mathrm{{Fmla}}}_{\mathcal{L}}\right| < \mathfrak{m} \), and \( \mathfrak{A} \) is an \( \mathcal{L} \) -structure with \( {\aleph }_{0} \leq \left| A\right| \leq \mathfrak{m} \) . Then \( \mathfrak{A} \) has an \( \mathfrak{m} \) -sa... | The proof of 28.13 is similar to that of 28.12. Using GCH we obtain | No |
Corollary 28.14 (GCH). Every theory in \( \mathcal{L} \) with infinite models has a saturated model of each regular cardinality \( > \left| {\mathrm{{Fmla}}}_{\mathcal{L}}\right| \) . | We now want to give different proofs for these corollaries which are perhaps more natural than the above. The proofs are based on the following lemma, which is of course, weaker than 28.12:\n\nSuppose \( \left| {\mathrm{{Fmla}}}_{\mathcal{L}}\right| \leq \mathfrak{m} \), and \( \mathfrak{A} \) is an \( \mathcal{L} \) -... | No |
Proposition 28.16. Let \( \mathfrak{A} \) be an \( \mathcal{L} \) -structure. The following conditions are equivalent:\n\n(i) \( \mathfrak{A} \) is \( \mathfrak{m} \) -saturated;\n\n(ii) for all \( \alpha < \mathfrak{m} \), if \( {\mathcal{L}}^{\prime } \) is an expansion of \( \mathcal{L} \) obtained by adding new ind... | Proof. Assume \( \left( i\right) \) and the hypothesis of \( \left( {ii}\right) \) . Let \( X = \left\{ {{a}_{\xi } : \xi < \alpha }\right\} \), and let \( {\mathcal{L}}^{\prime \prime } \) be an \( X \) -expansion of \( \mathcal{L} \), obtained by adding new individual constants \( {\mathbf{d}}_{\mathbf{x}} \) for eac... | Yes |
Lemma 28.17. If \( \mathfrak{A} \) is \( \mathfrak{m} \) -saturated, then \( \mathfrak{A} \) is \( \mathfrak{m} \) -homogeneous. | Proof. Assume that \( \mathfrak{A} \) is an \( \mathfrak{m} \) -saturated \( \mathcal{L} \) -structure, \( \alpha < \mathfrak{m}, x \in {}^{\alpha }A \) , \( y \in {}^{\alpha }A,{\left( \mathfrak{A},{x}_{\xi }\right) }_{\xi < \alpha } \equiv {}_{\mathrm{{ee}}}{\left( \mathfrak{A},{y}_{\xi }\right) }_{\xi < \alpha } \),... | Yes |
Lemma 28.18. If \( \mathfrak{A} \) is \( \mathfrak{m} \) -saturated, \( \mathfrak{A}{ \equiv }_{\mathrm{{ee}}}\mathfrak{B} \), and \( b \in {}^{\mathfrak{m}}B \), then there is an \( a \in {}^{\mathfrak{m}}A \) such that \( {\left( \mathfrak{A},{a}_{\xi }\right) }_{\xi < \mathfrak{m}} \equiv {}_{\mathrm{{ee}}}{\left( \... | Proof. Assume the hypothesis. We define the sequence \( a \in {}^{\mathfrak{m}}A \) by induction so that for each \( \alpha \leq m \) the following condition holds:\n\n(1)\n\n\[{\left( \mathfrak{A},{a}_{\xi }\right) }_{\xi < \alpha }{ \equiv }_{\mathrm{{ee}}}{\left( \mathfrak{B},{b}_{\xi }\right) }_{\xi < \alpha }.\]\n... | Yes |
Lemma 28.19. If \( \mathfrak{A} \) is \( \mathfrak{m} \) -saturated, then \( \mathfrak{A} \) is \( {\mathfrak{m}}^{ + } \) -universal. | Proof. Let \( \mathfrak{A} \) be \( \mathfrak{m} \) -saturated, and assume that \( \mathfrak{A}{ \equiv }_{\mathrm{{ee}}}\mathfrak{B} \) and \( \left| B\right| \leq \mathfrak{m} \) . Let \( b : \mathrm{m} \rightarrow B \) be an onto map. Then by 28.18 there is an \( a \in {}^{\mathrm{m}}A \) such that \( {\left( \mathf... | Yes |
Lemma 28.20. Let \( \mathcal{L} \) be a first-order language and assume that \( \left| {\mathrm{{Fmla}}}_{\mathcal{L}}\right| < \mathfrak{m} \) . If \( \mathfrak{A} \) is \( \mathfrak{m} \) -universal and \( \mathfrak{m} \) -homogeneous, then \( \mathfrak{A} \) is \( \mathfrak{m} \) -saturated. | Proof. Suppose that \( \mathfrak{A} \) is \( m \) -universal and \( m \) -homogeneous, and assume that the hypothesis of \( {28.16}\left( {ii}\right) \) holds. Expand \( {\mathcal{L}}^{\prime } \) to \( {\mathcal{L}}^{\prime \prime } \) by adjoining a new individual constant \( \mathbf{d} \) . Now consider the set\n\n\... | Yes |
Theorem 28.22. If \( \mathfrak{A} \) and \( \mathfrak{B} \) are elementarily equivalent saturated structures of the same power, then \( \mathfrak{A} \cong \mathfrak{B} \) . | Proof. Let \( \left| A\right| = \left| B\right| = \mathfrak{m} \), and let \( a : \mathfrak{m} \gg A \) and \( b : \mathfrak{m} \gg B \) . We use the by now very familiar back-and-forth method to show that \( \mathfrak{A} \cong \mathfrak{B} \) ; otherwise the proof is similar to that for 28.18. We define sequences \( x... | Yes |
Theorem 28.23. If \( \mathfrak{A} \) is an \( {\mathfrak{m}}^{ + } \) -universal \( \mathcal{L} \) -structure of power \( \mathfrak{m} \geq \left| {\mathrm{{Fmla}}}_{\mathcal{L}}\right| \) , then \( \mathfrak{A} \) is isomorphic to a proper elementary substructure of itself. | Proof. Assume the hypothesis. Let \( \mathfrak{B} \) be a proper elementary extension of \( \mathfrak{A} \) . Choose \( b \in B \sim A \) . Let \( {\left( \mathfrak{C}, a, b\right) }_{a \in A} \) be an elementary substructure of \( {\left( \mathfrak{B}, a, b\right) }_{a \in A} \) of power \( \mathfrak{m} \) . Thus \( A... | Yes |
Corollary 28.24 (GCH). If \( \mathfrak{A} \) is any infinite \( \mathcal{L} \) -structure and \( \left| {\mathrm{{Fmla}}}_{\mathcal{L}}\right| \leq \mathfrak{m} \) , then there is \( \mathfrak{B} \equiv_{\mathrm{{ee}}} \mathfrak{A} \) of power \( \mathfrak{m} \) such that \( \mathfrak{B} \) is isomorphic to a proper el... | Proof. By Corollary 28.12, let \( \mathfrak{C} \) be a saturated elementary extension of \( \mathfrak{A} \) of any power \( n \geq m \) . By Theorem 28.21, \( \mathfrak{C} \) is \( {n}^{ + } \) -universal, so by Theorem 28.23, there is an isomorphism \( f \) of \( \mathfrak{C} \) onto a proper elementary substructure (... | Yes |
Proposition 28.26. Let \( \mathcal{L} \) be a language having a unary relation symbol \( \mathbf{P} \) , and suppose that \( \mathfrak{A} \) is an \( \mathfrak{m} \) -saturated \( \mathcal{L} \) -structure satisfying the following conditions:\n\n(i) \( \left| {\mathbf{P}}^{\mathfrak{A}}\right| \geq {\aleph }_{0} \) ;\n... | Proof. Suppose \( X \subseteq {\mathbf{P}}^{\mathfrak{A}} \) with \( \left| X\right| < \mathfrak{m} \), and let \( {\mathcal{L}}^{\prime } \) be an \( X \) -expansion of \( \mathcal{L} \) . Assume that \( \Delta \subseteq {\mathrm{{Fmla}}}_{\mathcal{L}}^{1} \), is such that each finite subset of \( \Delta \) can be rea... | Yes |
Theorem 28.27 (Specker) (GCH). Let \( \mathcal{L} \) be a first-order language having unary relation symbols \( \mathbf{P} \) and \( \mathbf{Q} \) . Suppose that \( {\mathcal{L}}^{\prime } \) is another first-order language, and that \( f \) and \( g \) are isomorphisms from \( {\mathcal{L}}^{\prime } \) into \( \mathc... | Proof. By Corollary 28.12 choose \( {\mathfrak{A}}^{\prime } \) to be a saturated structure elementarily equivalent to \( \mathfrak{A} \) . By 28.25 and \( {28.26},{\mathfrak{B}}^{\prime } \) and \( {\mathfrak{C}}^{\prime } \) are saturated, and \( \left| {B}^{\prime }\right| = \left| {C}^{\prime }\right| = \left| A\ri... | Yes |
Theorem 28.29. Suppose \( \mathfrak{A} \) is an \( \mathcal{L} \) -structure and \( \mathfrak{m} \) is a cardinal. Assume that \( {2}^{\mathfrak{n}} \leq \mathfrak{m} \) whenever \( \mathfrak{n} < \mathfrak{m} \), that \( \left| {\mathrm{{Fmla}}}_{\mathcal{L}}\right| < \mathfrak{m} \), and that \( {\aleph }_{\mathfrak{... | Proof. If \( \mathfrak{m} = {\mathfrak{n}}^{ + } \) for some \( \mathfrak{n} \), then the assumptions imply that \( {2}^{\mathfrak{n}} = {\mathfrak{n}}^{ + } \) , so the desired result is clear from 28.12. So assume that \( m \) is a limit cardinal. We define the desired sequence \( \left\langle {{\mathfrak{B}}_{n} : n... | Yes |
Lemma 28.31. If \( \mathfrak{A} \) is an uncountable special structure, then \( \mathfrak{A} \) is \( {\left| A\right| }^{ + } \) universal. | The proof of this lemma is, however, just a 'one-sided' version of the proof of 28.30; see the relationship between the proofs of 28.18 and 28.22. | No |
Proposition 28.32. If \( \mathfrak{A} \) is a special \( \mathcal{L} \) -structure and \( {\mathcal{L}}^{\prime } \) is a reduct of \( \mathcal{L} \) , then \( \mathfrak{A} \upharpoonright {\mathcal{L}}^{\prime } \) is special. | The analog of 28.26 also holds. Since it is not quite so clear, we sketch the proof of it. | No |
Proposition 28.33. Let \( \mathcal{L} \) be a language having a unary relation symbol \( \mathbf{P} \) , and suppose that \( \mathfrak{A} \) is a special \( \mathcal{L} \) -structure satisfying the following conditions:\n\n(i) \( \left| {\mathbf{P}}^{\mathfrak{A}}\right| \geq {\aleph }_{0} \) ;\n\n(ii) for each operati... | Proof. We take only the case in which \( m = \left| A\right| \) is a limit cardinal. Let \( \left\langle {{\mathfrak{C}}_{n} : n < m}\right\rangle \) be a sequence as indicated in 28.28 for \( \mathfrak{A} \) . For each \( n < m \) let \( {\mathfrak{D}}_{n} \) be obtained from \( {\mathfrak{C}}_{n} \) like \( \mathfrak... | Yes |
Proposition 29.8. Let \( \mathcal{L},\mathfrak{A},\mathfrak{B} \) and \( f \) be as in the first part of 29.7. Let \( \varphi \) be a formula and \( x \in {}^{\omega }A \) . Then \( \mathfrak{A} \vDash \varphi \left\lbrack x\right\rbrack \) iff \( \mathfrak{B} \vDash \varphi \left\lbrack {f \circ x}\right\rbrack \) . | From the very definition of satisfaction we obtain the following proposition: | No |
Proposition 29.10. Let \( \mathcal{L} \) be a relational first-order language with equality, \( \varphi \in {\operatorname{Sent}}_{\mathcal{L}} \), and \( \Gamma \) a set of \( \mathcal{L} \) -sentences such that equality does not appear in any sentence of \( \Gamma \) . \( \textit{Then}\;\Gamma { \vDash }_{\mathcal{L}... | Proof. First assume that \( \Gamma { \vDash }_{\mathcal{L}}\varphi \) . Let \( \left( {\mathfrak{A}, E}\right) \) be any model in the \( {\mathcal{L}}^{\mathrm{e}} \) -sense of \( \Gamma \cup \mathrm{E}\{ \varphi \} \), where \( \mathfrak{A} \) is the underlying \( \mathcal{L} \) -structure. Then \( E \) is an equivale... | Yes |
Theorem 29.13. Let \( \mathcal{L} \) be a relational first-order language without equality and with at least one relation symbol of rank \( \geq 2 \) . Then \( \{ \varphi : \vDash \varphi \} \) is undecidable. | Proof. Let \( \mathbf{R} \) be a relation symbol of \( \mathcal{L} \) of rank \( \geq 2 \) . Let \( {\mathcal{L}}^{\prime } \) be a first-order language with sole non-logical constant the binary relation symbol \( \mathbf{S} \) . With each formula \( \varphi \) of \( {\mathcal{L}}^{\prime } \) we associate the formula ... | Yes |
Theorem 29.16. Let \( \mathfrak{A} \) be an \( \mathcal{L} \) -structure, \( \mathcal{L} \) a relational language without equality, and let \( A \subseteq B \) . Then there is an \( \mathcal{L} \) -structure \( \mathfrak{B} \) with universe \( B \) such that \( \mathfrak{A} \preccurlyeq \mathfrak{B} \) . | Proof. Fix \( a \in A \) and define \( f : B \rightarrow A \) by setting \( {fx} = x \) for \( x \in A \) and \( {fx} = \) \( a \) for \( x \in B \sim A \) . For each \( m \) -ary relational symbol \( \mathbf{R} \), let \( {\mathbf{R}}^{\mathfrak{B}} = \left\{ {x \in {}^{m}B : f \circ }\right. \) \( x \in {\mathbf{R}}^... | Yes |
Theorem 29.17. Let \( \mathcal{L} \) be a relational language without equality, and let \( \varphi \) and \( \psi \) be sentences of \( \mathcal{L} \) such that \( \vDash \varphi \rightarrow \psi \) while neither \( \vDash \neg \varphi \) nor \( \vDash \psi \) . Then there is a sentence \( \chi \) such that \( \vDash \... | Proof. First note:\n\n(1)\n\nthere is a nonlogical constant common to \( \varphi \) and \( \psi \) .\n\nFor, suppose not. Let \( \mathfrak{A} \) be a model of \( \varphi \) and \( \mathfrak{B} \) a model of \( \neg \psi \) . By taking elementary extensions, by 29.16 we may assume that \( A = B \) . Let \( C = A \), and... | Yes |
Theorem 29.20. Let \( \left( {\mathcal{L},\tau, O}\right) \) be a descriptive triple, \( \Gamma \) a set of sentences in \( \left( {\mathcal{L},\tau ,\mathbf{O}}\right) ,\sigma \) a term of \( \left( {\mathcal{L},\tau ,\mathbf{O}}\right) \) and \( \varphi \) a formula of \( \left( {\mathcal{L},\tau ,\mathbf{O}}\right) ... | Proof. We proceed by simultaneous induction on \( \sigma \) and \( \varphi \) . If \( \sigma \) is \( {v}_{i} \), then for \( \psi \) in \( \left( i\right) \) we may obviously take the formula \( {v}_{i} = {v}_{k} \) . If \( \sigma \) is \( \mathbf{O}{\rho }_{0}\cdots {\rho }_{m - 1} \) and \( {v}_{k} \) does not occur... | Yes |
Theorem 29.21. For a descriptive triple \( \left( {\mathcal{L},\tau ,\mathrm{O}}\right) \) the set \( {\mathcal{g}}^{+ * }\{ \varphi : \varphi \) a sentence of \( \left( {\mathcal{L},\tau ,\mathbf{O}}\right) \) and \( \vDash \varphi \} \) is recursively enumerable. | This gives rise to a possibility of giving logical axioms for \( \tau \), and this has been done, leading to a notion \( \Gamma { \vdash }_{\mathrm{D}}\varphi \) . The completeness theorem can thus be proved for this notion. | No |
Proposition 29.24. Let the notation be as above. Suppose that \( \sigma \) is a term of \( \left( {\mathcal{L},\mathbf{\varepsilon },\mathbf{O}}\right) ,\varphi \) is a formula of \( \left( {\mathcal{L},\mathbf{\varepsilon },\mathbf{O}}\right) \), and \( x \in {}^{\omega }A \) . Then \( {\sigma }^{\mathfrak{A}f}x = {\s... | Proof. The simultaneous inductive proof on \( \sigma \) and \( \varphi \) is straightforward. | No |
Theorem 29.25. The set \( {\mathcal{g}}^{+ * }\{ \varphi : \varphi \) is a sentence \( \left( {\mathcal{L},\mathbf{\varepsilon },\mathbf{O}}\right) \) and \( \vDash \varphi \} \) is recursively enumerable. | Proof. We claim that \( \vDash \varphi \) iff \( \Gamma \vDash {\varphi }^{ * } \), using the notation of 29.23, where \( \Gamma \) is the following set of sentences of \( {\mathcal{L}}^{\prime } \) :\n\nthe Skolem set of \( {\mathcal{L}}^{\prime } \) over \( \mathcal{L} \) ;\n\n\( {\chi }_{ij\varphi \psi } \) for \( i... | Yes |
Theorem 30.1. The set \( \\left\\{ {{g}^{ + }\\psi : \\psi }\\right. \) is a sentence of \( {\\mathcal{L}}_{\\text{nos }} \) in weak second-order logic and \( {\\mathrm{Q}}^{\\mathrm{w}} \\vDash \\psi \\} \) is not elementarily definable in \( \\left( {\\omega ,+,\\cdot, s,0}\\right) \), in weak second-order logic. | Now \( {\\mathrm{Q}}^{\\mathrm{w}} \\vDash \\psi \) iff \( \\vDash \\bigwedge {\\mathrm{Q}}^{\\mathrm{w}} \\rightarrow \\psi \) . Hence \( \\left\\{ {{\\mathcal{G}}^{ + }\\psi : \\vDash \\psi }\\right\\} \) is not weak second order elementarily definable in \( \\left( {\\omega ,+,\\cdot ,0, s}\\right) \) either. Thus b... | Yes |
Lemma 30.2. Let \( \mathfrak{A} \) and \( \mathfrak{B} \) be \( \mathcal{L} \) -structures, with \( \mathfrak{A} \subseteq \mathfrak{B} \) . Then the following two conditions are equivalent:\n\n(i) \( \mathfrak{A} \preccurlyeq \mathfrak{B} \) in the weak second-order sense;\n\n(ii) for every weak second-order formula \... | The proof is just like for 19.16. | No |
Theorem 30.3 (Tarski). Let \( \mathfrak{B} \) be an \( \mathcal{L} \) -structure, let \( \mathfrak{m} \) be a cardinal such that \( \left| {\mathrm{{Fmla}}}_{\mathcal{L}}\right| \leq \mathfrak{m} \leq \left| B\right| \), and let \( C \) be a subset of \( B \) of power \( \leq \mathfrak{m} \) . Then there is an elementa... | Proof. The proof is very similar to that for 19.17, but for completeness we shall sketch it. Let \( \mathcal{C} \) be a choice function for nonempty subsets of \( B \cup \mathbf{S}B \) . We now define a sequence \( \left\langle {{D}_{m} : m \in \omega }\right\rangle \) by recursion. Let \( {D}_{0} \) be any subset of \... | Yes |
Theorem 30.4. The set \( \left\{ {{\mathcal{g}}^{ + }\psi : \psi }\right. \) is a sentence of \( {\mathcal{L}}_{\text{nos }} \) in monadic second-order logic and \( {\mathrm{Q}}^{\mathrm{m}} \vDash \psi \} \) is not elementarily definable in \( \left( {\omega ,+,\cdot ,\delta ,0}\right) \), in monadic second-order logi... | Thus again \( \left\{ {{g}^{ + }\psi : \vDash \psi }\right\} \) is not r.e., and there is no reasonable proof theory for this logic. | Yes |
Theorem 30.9. For any sentence \( \varphi \) of \( \mathcal{L},{ \vDash }_{\mathrm{f}}\varphi \) iff \( \Gamma { \vDash }_{{\mathcal{L}}^{n}}{\varphi }^{ * } \) . | Proof. The direction \( \Leftarrow \) is given by the preceding two lemmas. For \( \Rightarrow \) , assume that \( { \vDash }_{\mathrm{f}}\varphi \), and let \( \mathfrak{B} \) be any \( {\mathcal{L}}^{\prime \prime } \) -structure which is a model of \( \Gamma \) . Set \( A = {\mathbf{T}}^{o\mathfrak{B}} \) . We now d... | Yes |
Theorem 30.10 ( \( \omega \) -completeness theorem). With the above notation, let \( \Theta \) be a theory in \( {\mathcal{L}}^{\prime } \) which includes \( \Gamma \), has a model, and is \( \omega \) -complete. Then (C) has an \( \omega \) -model, i.e., it has a model which is an \( \omega \) -structure. | Proof. From the above remarks we see that it suffices to find a model of \( \Theta \) which omits \( \Delta \) . We apply 27.4 with \( N = \{ 1\} \) . Suppose that \( \psi \in {\operatorname{Fmla}}_{\mathcal{L}}^{1} \) , and \( \Theta \cup \left\{ {\exists {v}_{0}\psi }\right\} \) has a model. To obtain a contradiction... | Yes |
Corollary 30.14. In the \( \mathfrak{m} \) -interpretation, any \( \mathbf{Q} \) -language is \( \left( {{\aleph }_{0},\pi \mathfrak{m}}\right) \) -compact. | Thus in the \( {\left( \exp {\aleph }_{0}\right) }^{ + } \) -interpretation, any \( \mathbf{Q} \) -language is \( \left( {{\aleph }_{0},{\aleph }_{1}}\right) \) -compact: any countable set of \( \mathbf{Q} \) -sentences such that every finite subset has a model, also has a model. Since \( \pi {\aleph }_{1} = {\aleph }_... | No |
Theorem 31.3. Let \( \mathcal{L} \) have only countably many nonlogical constants. If \( S \) is a consistency family for \( \left( {\mathcal{L},{\mathcal{L}}^{\prime }}\right) \) and \( \Gamma \in S \), then \( \Gamma \) has a model. | Proof. Let \( \Delta \) be the closure of \( \Gamma \) under the following operations: passage to subformulas; passage from \( \varphi \) to \( \operatorname{Sub}{f}_{\mathbf{c}}^{\alpha }\varphi \), where \( \mathbf{c} \in C \) ; passage from \( \neg \varphi \) to \( \varphi \rightarrow \) ; formation of \( \mathbf{c}... | Yes |
Theorem 31.4. Let \( \varphi \) and \( \psi \) be sentences in an \( {\mathrm{L}}_{{\omega }_{1}\omega } \) -language such that \( { \vDash }_{\varphi } \rightarrow \psi \) . Then there is a sentence \( \chi \) such that \( { \vDash }_{\varphi } \rightarrow \chi ,{ \vDash }_{\chi } \rightarrow \psi \), and any nonlogic... | The proof is almost identical with that of 22.1 , and we omit it. | No |
Lemma 31.6. Any strongly compact cardinal is a strong limit cardinal. | Proof. Suppose that \( m \) is not a strong limit cardinal. We shall show that it is not strongly compact. Clearly there is a cardinal \( n < m \) such that \( m \leq {2}^{n} \) . Let \( \mathcal{L} \) be a first-order language with unary relation symbols \( {\mathbf{P}}_{\xi \varepsilon } \) for each \( \xi < n \) and... | Yes |
Lemma 31.7. Any weakly compact cardinal is regular. | Proof. Assume that \( m \) is singular, say \( m = \mathop{\bigcup }\limits_{{\alpha < n}}{p}_{\alpha } \), where \( n < m \) and each \( {\mathfrak{p}}_{\alpha } < \mathfrak{m} \) . Let \( \mathcal{L} \) be a first-order language with two nonlogical constants, a binary relation symbol \( < \) and a unary relation symb... | Yes |
Theorem 31.11. Every weakly compact cardinal is weakly inaccessible. | Proof. By 31.7 it suffices to show that any successor cardinal \( {m}^{ + } > {\aleph }_{0} \) is not weakly compact. Let \( \Gamma \) be the following set of sentences:\n\n(1) \( < \) is a well-ordering;\n\n(2) \( \exists {v}_{0}{\varphi }_{\alpha } \) for each \( \alpha < {\mathrm{m}}^{ + } \) ;\n\n(3) \( \exists {v}... | No |
Theorem 31.14 (Ulam). Every measurable cardinal is strongly inaccessible. | Proof. Let \( m \) be measurable, and suppose \( n < m \) and \( {p}_{\alpha } < m \) for each \( \alpha < n \) ; we shall show that \( {\Pi }_{\alpha < n}{\mathfrak{p}}_{\alpha } < m \) (this is one of the equivalent conditions for \( m \) to be strongly inaccessible). Let \( \mu \) be a measure on \( S\left( m\right)... | Yes |
Theorem 31.17. Assume that \( \mathfrak{m} \) is strongly inaccessible. Then the following conditions are equivalent;\n\n(i) \( \mathfrak{m} \) is weakly compact;\n\n(ii) \( \mathfrak{m} \) has the tree property;\n\n(iii) if \( \mathfrak{A} \) is an \( \mathfrak{m} \) -complete, \( \mathfrak{m} \) -distributive Boolean... | Proof. We shall show \( \left( i\right) \Rightarrow \left( {ii}\right) \Rightarrow \left( {iii}\right) \Rightarrow \left( {iv}\right) \Rightarrow \left( v\right) \Rightarrow \left( {vi}\right) \Rightarrow \left( {ii}\right) \Rightarrow \left( i\right) \) .\n\n\( \left( i\right) \Rightarrow \left( {ii}\right) \) . Assum... | Yes |
Theorem 1.2. An open set \( B \subset {\mathbb{C}}^{n} \) is a Reinhardt domain if and only if there exists an open set \( W \subset V \) with \( B = {\tau }^{-1}\left( W\right) \) . | 1. Let \( B = {\tau }^{-1}\left( W\right), W \subset V \) open. For \( \mathfrak{z} \in B,\tau \left( \mathfrak{z}\right) \in W \) ; therefore \( {\tau }^{-1}\tau \left( 3\right) \subset {\tau }^{-1}\left( W\right) = B \) .\n\n2. Let \( B \) be a Reinhardt domain. Then \( B = {\tau }^{-1}\tau \left( B\right) \) and it ... | Yes |
Theorem 1.3. Every complete Reinhardt domain is proper. | Proof. Let \( G \) be a complete Reinhardt domain. There exists a point \( {\mathfrak{z}}_{1} \in G \cap \) \( {\mathbb{C}}^{n} \), and by definition \( 0 \in {P}_{31} \subset G \) . It remains to show that \( G \) is connected.\n\na. Let \( {\mathfrak{z}}_{1} \in G \) be a point in a general position (i.e., \( {\mathf... | No |
Theorem 1.4. Let \( \mathfrak{P}\left( \mathfrak{z}\right) = \mathop{\sum }\limits_{{v = 0}}^{\infty }{a}_{v}{\mathfrak{z}}^{v} \) be a formal power series in \( {\mathbb{C}}^{n} \) . Then the region of convergence \( B = B\left( {\mathfrak{P}\left( \mathfrak{z}\right) }\right) \) is a complete Reinhardt domain. \( \ma... | ## Proof\n\n1. Let \( {\mathfrak{z}}_{1} \in B \) . Then \( {U}_{\varepsilon }^{\prime }\left( {\mathfrak{z}}_{1}\right) = \left\{ {{\mathfrak{z \in C}}^{n} : \left| {\mathfrak{z} - {\mathfrak{z}}_{1}}\right| < \varepsilon }\right\} = {U}_{\varepsilon }\left( {z}_{1}^{\left( 1\right) }\right) \times \cdots \times \) \(... | Yes |
Theorem 2.1. Let \( B \subset {\mathbb{C}}^{n} \) be a region and \( f : B \rightarrow \mathbb{C} \) complex differentiable at \( {\mathfrak{z}}_{0} \in B \) . Then the values of the functions \( {\Delta }_{1},\ldots ,{\Delta }_{n} \) at \( {\mathfrak{z}}_{0} \) are uniquely determined. | Proof. \( {E}_{v} \mathrel{\text{:=}} \left\{ {3 \in {\mathbb{C}}^{n} : {z}_{\lambda } = {z}_{\lambda }^{\left( 0\right) }\text{for}\lambda \neq v}\right\} \) is a complex one-dimensional plane. Let \( {B}_{v} \mathrel{\text{:=}} \left\{ {\zeta \in \mathbb{C} : \left( {{z}_{1}^{\left( 0\right) },\ldots ,{z}_{v - 1}^{\l... | Yes |
Theorem 2.2. Let \( B \subset {\mathbb{C}}^{n} \) be a region and \( f \) complex differentiable at \( {\mathfrak{z}}_{0} \in B \) . Then \( f \) is continuous at \( {\mathfrak{z}}_{0} \) . | Proof. We have \( f\left( 3\right) = f\left( {3}_{0}\right) + \mathop{\sum }\limits_{{v = 1}}^{n}\left( {{z}_{v} - {z}_{v}^{\left( 0\right) }}\right) {\Delta }_{v}\left( 3\right) \) ; the right side of this equation is clearly continuous at \( {\mathfrak{z}}_{0} \) . | No |
Theorem 2.3. Let \( B \subset {\mathbb{C}}^{n} \) be a region, \( f \) holomorphic in \( B \) . Then \( f \) is complex differentiable in \( B \) . | Proof. Let \( {\mathfrak{z}}_{0} \in B \) . Then there is a neighborhood \( U = U\left( {\mathfrak{z}}_{0}\right) \) and a power series \( \mathop{\sum }\limits_{{v = 0}}^{\infty }{a}_{v}{\left( 3 - {3}_{0}\right) }^{v} \) which in \( U \) converges uniformly to \( f\left( 3\right) \) . Without loss of generality let \... | Yes |
Theorem 3.1. Let \( B \subset {\mathbb{C}}^{n} \) be a region, \( P \) a polycylinder with \( \bar{P} \subset B \) and \( T \) the \( n \) -dimensional torus belonging to \( P \) . If \( f \) is complex differentiable in \( B \) then \( f \mid P = \operatorname{ch}\left( {f \mid T}\right) \) . | Proof. This theorem is a generalization of the familar 1-dimensional Cauchy integral formula.\n\nThe function \( {f}_{n}^{ \star } \) with \( {f}_{n}^{ \star }\left( {z}_{n}\right) \mathrel{\text{:=}} f\left( {{\xi }_{1},\ldots ,{\xi }_{n - 1},{z}_{n}}\right) \) is complex differentiable for fixed \( \left( {{\xi }_{1}... | Yes |
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