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Lemma 14.21. If \( \varphi \left( {{x}_{1},\ldots ,{x}_{n}}\right) \) is a \( {\Delta }_{0} \) formula, then\n\n\[ \varphi \left( {{x}_{1},\ldots ,{x}_{n}}\right) \;\text{ if and only if }\;\begin{Vmatrix}{\varphi \left( {{\check{x}}_{1},\ldots ,{\check{x}}_{n}}\right) }\end{Vmatrix} = 1. \] | Proof. By induction on the complexity of \( \varphi \) . | Yes |
Lemma 14.23. For every \( x \in {V}^{B} \) , \n\n\[ \n\parallel x\text{ is an ordinal }\parallel = \mathop{\sum }\limits_{{\alpha \in \text{ Ord }}}\parallel x = \check{\alpha }\parallel . \n\] | Proof. Since \ | No |
Theorem 14.24. Every axiom of ZFC is valid in \( {V}^{B} \) . | Proof. We show that \( \parallel \sigma \parallel = 1 \) for every axiom of ZFC.\n\nExtensionality. See Lemma 14.17.\n\nPairing. Given \( a, b \in {V}^{B} \), let \( c = \{ a, b{\} }^{B} \in {V}^{B} \) be such that \( \operatorname{dom}\left( c\right) = \{ a, b\} \) and \( c\left( a\right) = c\left( b\right) = 1 \) . T... | Yes |
Lemma 14.28. Let \( G \) be an \( M \) -generic ultrafilter on \( B \) . Then for all names \( x, y \in {M}^{B} \)\n\n(i) \( {x}^{G} \in {y}^{G} \) if and only if \( \parallel x \in y\parallel \in G \), \n\n(ii) \( {x}^{G} = {y}^{G} \) if and only if \( \parallel x = y\parallel \in G \) . | Proof. We prove (i) and (ii) simultaneously, by induction on pairs \( \left( {\rho \left( x\right) ,\rho \left( y\right) }\right) \) .\n\n(i) \( \parallel x \in y\parallel \in G \leftrightarrow \exists t \in \operatorname{dom}\left( y\right) \left( {y\left( t\right) \in G\text{and}\parallel x = t\parallel \in G}\right)... | Yes |
Corollary 14.30. \( M\left\lbrack G\right\rbrack \) is a model of ZFC. | Proof. By Theorem 14.24, every axiom \( \sigma \) of ZFC is valid in \( {M}^{B} \), therefore \( \parallel \sigma \parallel = 1 \in G \) and hence \( \sigma \) is true in \( M\left\lbrack G\right\rbrack \) . | Yes |
(i) \( M \subset M\left\lbrack G\right\rbrack \), and both models have the same ordinals.\n\n(ii) \( G \in M\left\lbrack G\right\rbrack \) and if \( N \supset M \) is a transitive model of ZFC such that\n\n\( G \in N \), then \( N \supset M\left\lbrack G\right\rbrack \) . | Proof. (i) For every \( x \in M \), the \( G \) -interpretation of the canonical name \( \check{x} \) is \( {\check{x}}^{G} = x \) (proved by \( \in \) -induction). Hence \( M \subset M\left\lbrack G\right\rbrack \) . To show that every ordinal in \( M\left\lbrack G\right\rbrack \) is in \( M \) (that \( M\left\lbrack ... | No |
Theorem 14.32. There is a generic extension \( V\left\lbrack G\right\rbrack \) that satisfies \( {2}^{{\aleph }_{0}} > {\aleph }_{1} \) . | Proof. We describe the notion of forcing that produces a generic extension with the desired property. Let \( P \) be the set of all functions \( p \) such that\n\n(14.33)\n\n(i) \( \operatorname{dom}\left( p\right) \) is a finite subset of \( {\omega }_{2} \times \omega \) ,\n\n(ii) \( \operatorname{ran}\left( p\right)... | Yes |
Lemma 14.35. Let \( P \) be a set of finite functions, with values in a given countable set \( C \) . Let \( p < q \) be defined as \( p \supset q \), and assume that for all \( p, q \in \) \( P \), if \( p \cup q \) is a function then \( p \cup q \in P \) (or more generally, \( \exists r \in P\left( {r \supset p \cup ... | Proof. Let \( F \) be an uncountable subset of \( P \), and let \( W \) be the set \( \{ \operatorname{dom}\left( p\right) \) : \( p \in F\} \) . As \( C \) is countable, the set \( W \) must be uncountable. By Theorem 9.18 there exists an uncountable \( \Delta \) -system \( Z \subset W \) ; let \( S = X \cap Y \) for ... | Yes |
Theorem 14.36 (Cohen). There is a model of ZF in which the real numbers cannot be well-ordered. Thus the Axiom of Choice is independent of the axioms of \( \mathrm{{ZF}} \) . | Before we construct a model without Choice, we shall prove an easy but useful lemma on automorphisms of Boolean-valued models. Let \( B \) be a complete Boolean algebra and let \( \pi \) be an automorphism of \( B \) . We define, by induction on \( \rho \left( x\right) \) an automorphism of the Boolean-valued universe ... | No |
Let \( \varphi \left( {{x}_{1},\ldots ,{x}_{n}}\right) \) be a formula. If \( \pi \) is an automorphism of \( B \), then for all \( {x}_{1},\ldots ,{x}_{n} \in {V}^{B} \) ,\n\n\[ \begin{Vmatrix}{\varphi \left( {\pi {x}_{1},\ldots ,\pi {x}_{n}}\right) }\end{Vmatrix} = \pi \left( \begin{Vmatrix}{\varphi \left( {{x}_{1},\... | Proof. (a) If \( \varphi \) is an atomic formula,(14.37) is proved by induction (as in the definition of \( \parallel x \in y\parallel ,\parallel x = y\parallel \) ). For instance,\n\n\[ \parallel {\pi x} \in {\pi y}\parallel = \mathop{\sum }\limits_{{t \in \operatorname{dom}\left( {\pi y}\right) }}\left( {\parallel {\... | Yes |
Lemma 14.38. If \( i \neq j \), then every \( p \) forces \( {\dot{a}}_{i} \neq {\dot{a}}_{j} \) . | Proof. For every \( p \) there exists a \( q \supset p \) such that for some \( n \in \omega, q\left( {i, n}\right) = 1 \) and \( q\left( {j, n}\right) = 0 \) . | Yes |
Lemma 15.1. Let \( \lambda \) be a cardinal in \( V \) . If \( G \) is a \( V \) -generic ultrafilter on \( B \) , then\n\n\[{\left( {2}^{\lambda }\right) }^{V\left\lbrack G\right\rbrack } \leq {\left( {\left| B\right| }^{\lambda }\right) }^{V}\] | Proof. Every subset \( A \subset \lambda \) in \( V\left\lbrack G\right\rbrack \) has a name \( \dot{A} \in {V}^{B} \) ; every such \( \dot{A} \) determines a function \( \alpha \mapsto \parallel \check{\alpha } \in \dot{A}\parallel \) from \( \lambda \) into \( B \) . Different subsets correspond to different function... | Yes |
Theorem 15.3. If \( \kappa \) is a regular cardinal and if \( P \) satisfies the \( \kappa \) -chain condition then \( \kappa \) remains a regular cardinal in the generic extension by \( P \) . | Proof. The proof is exactly as the proof of Theorem 14.34. The only difference is that the set \( {A}_{\alpha } \) is not necessarily countable but has cardinality less than \( \kappa \) . | No |
Theorem 15.6. Let \( \kappa \) be an infinite cardinal and assume that \( \left( {P, < }\right) \) is \( \kappa \) -distributive. Then if \( f \in V\left\lbrack G\right\rbrack \) is a function from \( \kappa \) into \( V \), then \( f \in V \) . In particular, \( \kappa \) has no new subsets in \( V\left\lbrack G\right... | Proof. Let \( f : \kappa \rightarrow V \) and \( f \in V\left\lbrack G\right\rbrack \), let \( \dot{f} \) be a name for \( f \) . There exist some \( A \in V \) and a condition \( {p}_{0} \in G \) such that \( {p}_{0} \) forces\n\n\[ \dot{f}\text{is a function from}\check{\kappa }\text{into}\check{A}\text{.} \]\n\nFor ... | Yes |
Lemma 15.8. If \( P \) is \( \kappa \) -closed then it is \( \kappa \) -distributive. | Proof. Let \( \left\{ {{D}_{\alpha } : \alpha < \kappa }\right\} \) be a collection of open dense sets. The intersection \( D = \mathop{\bigcap }\limits_{{\alpha < \kappa }}{D}_{\alpha } \) is clearly open; to show that \( D \) is dense, let \( p \in P \) be arbitrary. By induction on \( \alpha < \kappa \), we construc... | Yes |
Lemma 15.9 (The Product Lemma). Let \( P \) and \( Q \) be two notions of forcing in \( M \) . In order that \( G \subset P \times Q \) be generic over \( M \), it is necessary and sufficient that \( G = {G}_{1} \times {G}_{2} \) where \( {G}_{1} \subset P \) is generic over \( M \) and \( {G}_{2} \subset Q \) is gener... | Proof. First let \( G \) be an \( M \) -generic filter on \( P \times Q \) . We define \( {G}_{1} \) and \( {G}_{2} \) by (15.5). Clearly, \( {G}_{1} \) and \( {G}_{2} \) are filters, and \( G \subset {G}_{1} \times {G}_{2} \) . If \( \left( {{p}_{1},{p}_{2}}\right) \in \) \( {G}_{1} \times {G}_{2} \), then there are \... | Yes |
Lemma 15.12. If \( P \) and \( Q \) are \( \lambda \) -closed then \( P \times Q \) is \( \lambda \) -closed. More generally, if each \( {P}_{i} \) is \( \lambda \) -closed and \( P \) is the \( \kappa \) -product of the \( {P}_{i} \), with \( \lambda < \kappa \) , then \( P \) is \( \lambda \) -closed. | Proof. Let \( \alpha \leq \lambda \) and let \( {p}^{\xi } = \left\langle {{p}_{i}^{\xi } : i \in I}\right\rangle ,\xi < \alpha \), be a descending \( \alpha \) -sequence of conditions in \( P \) . If we let \( s = \mathop{\bigcup }\limits_{{\xi < \alpha }}s\left( {p}^{\xi }\right) \), then \( \left| s\right| < \kappa ... | Yes |
Lemma 15.14. If \( P \) and \( Q \) both have property \( \left( \mathrm{K}\right) \) then so does \( P \times Q \) . | Proof. Let \( W \subset P \times Q \) be uncountable. If there exists a \( p \in P \) such that the set \( X = \{ q : \left( {p, q}\right) \in W\} \) is uncountable, then since \( Q \) has property \( \left( \mathrm{K}\right) \) there exists an uncountable \( Y \subset X \) of pairwise compatible elements, and \( \{ p\... | Yes |
Theorem 15.15. If for every \( i \in I,{P}_{i} \) has property (K) then \( \mathop{\prod }\limits_{{i \in I}}{P}_{i} \) has property \( \left( \mathrm{K}\right) \) . | Proof. Let \( X \) be an uncountable subset of \( P \), and let \( W = \{ s\left( p\right) : p \in X\} \) . If \( W \) is countable, then there is a finite set \( J \subset I \) such that \( s\left( p\right) = J \) for uncountably many \( p \) . By Lemma 15.14, \( \mathop{\prod }\limits_{{i \in J}}{P}_{i} \) has proper... | Yes |
Theorem 15.17. (i) If each \( {P}_{i} \) has size \( \lambda \) (infinite) then the product of the \( {P}_{i} \) satisfies the \( {\lambda }^{ + } \) -chain condition. | Proof. (i) is a special case of (ii); thus consider \( \kappa \) -products. Let \( P \) be the \( \kappa \) -product, and let \( W \) be an antichain in \( P \) . If \( p = \left\langle {{p}_{i} : i \in I}\right\rangle \) and \( q = \left\langle {q}_{i}\right. \) : \( i \in I\rangle \) are incompatible in \( P \), then... | Yes |
Theorem 15.18 (Easton). Let \( M \) be a transitive model of ZFC and assume that the Generalized Continuum Hypothesis holds in \( M \) . Let \( F \) be a function (in \( M \) ) whose arguments are regular cardinals and whose values are cardinals, such that for all regular \( \kappa \) and \( \lambda \) :\n\n(i) \( F\le... | We have to point out that the generic extension is obtained by forcing with a class of conditions. By Lemma 15.1, a notion of forcing can only increase the size of \( {2}^{\kappa } \) for \( \kappa < \left| {B\left( P\right) }\right| \) ; thus we have to use a class of conditions. We shall describe the appropriate gene... | No |
Lemma 15.19. Let \( G \times H \) be an \( M \) -generic filter on \( P \times Q \), where \( P \) is \( \lambda \) - closed and \( Q \) satisfies the \( {\lambda }^{ + } \) -chain condition. Then every function \( f : \lambda \rightarrow M \) in \( M\left\lbrack {G \times H}\right\rbrack \) is in \( M\left\lbrack H\ri... | Proof. Let \( \dot{f} \) be a name for \( f \) ; let us assume, without loss of generality, that for some \( A \), every condition forces that \( \dot{f} \) is a function from \( \lambda \) into \( A \) . For each \( \alpha < \lambda \), let \( {D}_{\alpha } \subset P \) be defined as follows:\n\n\( p \in {D}_{\alpha }... | Yes |
Let \( \lambda \) be an uncountable cardinal. Let \( P \) be the set of all finite sequences \( \langle p\left( 0\right) ,\ldots, p\left( {n - 1}\right) \rangle \) of ordinals less than \( \lambda ;p \) is stronger than \( q \) if \( p \supset q \) . Let \( G \) be a generic filter on \( P \) and let \( f = \bigcup G;f... | As \( \left| P\right| = \lambda, P \) satisfies the \( {\lambda }^{ + } -chain condition and so all cardinals greater than \( \lambda \) are preserved (as are all cofinalities greater than \( \lambda \) ). | No |
Lemma 15.21. Let \( \kappa \) be a regular cardinal and let \( \lambda > \kappa \) be a cardinal. There is a notion of forcing \( \left( {P, < }\right) \) that collapses \( \lambda \) onto \( \kappa \), i.e., \( \lambda \) has cardinality \( \kappa \) in the generic extension. Moreover,\n\n(i) every cardinal \( \alpha ... | Proof. Let \( P \) be the set of all functions \( p \) such that:\n\n(15.18)\n\n(i) \( \operatorname{dom}\left( p\right) \subset \kappa \) and \( \left| {\operatorname{dom}\left( p\right) }\right| < \kappa \) ,\n\n(ii) \( \operatorname{ran}\left( p\right) \subset \lambda \) ,\n\nand let \( p < q \) if and only if \( p ... | Yes |
Theorem 15.22 (Lévy). Let \( \kappa \) be a regular cardinal and let \( \lambda > \kappa \) be an inaccessible cardinal. There is a notion of forcing \( \left( {P, < }\right) \) such that:\n\n(i) every \( \alpha \) such that \( \kappa \leq \alpha < \lambda \) has cardinality \( \kappa \) in \( V\left\lbrack G\right\rbr... | Proof. For each \( \alpha < \lambda \), let \( {P}_{\alpha } \) be the set of all functions \( {p}_{\alpha } \) such that \( \operatorname{dom}\left( {p}_{\alpha }\right) \subset \kappa ,\left| {\operatorname{dom}\left( {p}_{\alpha }\right) }\right| < \kappa \), and \( \operatorname{ran}\left( {p}_{\alpha }\right) \sub... | Yes |
Lemma 15.24. If \( A \) is a maximal antichain in a normal tree \( T \) and if \( A \) is bounded in \( T \) (in particular, if the height of \( T \) is a successor ordinal), then \( A \) is maximal in every extension of \( T \) . | Proof. Let \( {T}^{\prime } \) be an extension of \( T \) . Let \( \alpha < \operatorname{height}\left( T\right) \) be such that each \( a \in A \) is at level \( \leq \alpha \) . If \( {t}^{\prime } \in {T}^{\prime } - T \), then there exists \( t \in T \) at level \( \alpha \) such that \( t \subset {t}^{\prime } \) ... | Yes |
Lemma 15.25. Let \( \alpha \) be a countable limit ordinal, let \( T \in P \) be a normal \( \alpha \) -tree and let \( A \) be a maximal antichain in \( T \) . Then there exists an extension \( {T}^{\prime } \in P \) of \( T \) of height \( \alpha + 1 \) such that \( A \) is a maximal antichain in \( {T}^{\prime } \) ... | Proof. For each \( t \in T \) there exists \( a \in A \) such that either \( t \subset a \) or \( a \subset t \) . In either case, there exists a branch \( b = {b}_{t} \) of length \( \alpha \) in \( T \) such that \( t \in b \) and \( a \in b \) . Let \( {T}^{\prime } \) be the extension of \( T \) obtained by extendi... | Yes |
If \( A \) is a maximal antichain in \( T \), then the set\n\n\[ C = \left\{ {\alpha : A \cap {T}_{\alpha }\text{ is a maximal antichain in }{T}_{\alpha }}\right\} \]\n\nis closed unbounded. | Proof. It is easy to see that \( C \) is closed. To show that \( C \) is unbounded, let \( {\alpha }_{0} < {\omega }_{1} \) be arbitrary. Since \( {T}_{{\alpha }_{0}} \) is countable, there exists a countable ordinal \( {\alpha }_{1} > {\alpha }_{0} \) such that every \( t \in {T}_{{\alpha }_{0}} \) is compatible with ... | Yes |
Lemma 15.28. If \( T \) is a normal Suslin tree, then \( {P}_{T} \) is \( {\aleph }_{0} \) -distributive. | Proof. Let \( {D}_{n}, n = 0,1,2\ldots \), be open dense subsets of \( {P}_{T} \) . We shall prove that \( \mathop{\bigcap }\limits_{{n = 0}}^{\infty }{D}_{n} \) is dense in \( {P}_{T} \) . First we claim that if \( D \subset {P}_{T} \) is open dense, then there is an \( \alpha < {\omega }_{1} \) such that \( D \) cont... | Yes |
Lemma 15.30. (i) In the random real extension \( V\left\lbrack G\right\rbrack \), every \( f : \omega \rightarrow \omega \) is dominated by some \( g \in V \) . | Proof. (i) Forcing conditions are Borel sets of positive measure, and we freely confuse them with their equivalence classes in \( \mathcal{B}/{I}_{\mu } \) . Let \( p \Vdash \dot{f} : \omega \rightarrow \omega \) ; we shall find a \( q < p \) and some \( g : \omega \rightarrow \omega \) such that \( q \) forces that \(... | Yes |
Let \( \kappa \) be an infinite cardinal and let \( I = \kappa \times \omega \) . Let \( \Omega = \) \( \{ 0,1{\} }^{I} \) . Let \( T \) be the set of all finite \( 0 - 1 \) functions with \( \operatorname{dom}\left( t\right) \subset I \) . Let \( \mathcal{S} \) be the \( \sigma \) -algebra generated by the sets \( {S}... | If \( G \) is a generic ultrafilter on \( B \) then \( f = \bigcup \left\{ {t : {S}_{t} \in G}\right\} \) is a 0-1 function on \( I \), and for each \( \alpha < \kappa \), we define \( {f}_{\alpha }\left( n\right) = f\left( {\alpha, n}\right) \), for all \( n < \omega \) . The \( {f}_{\alpha } \) , \( \alpha < \kappa \... | Yes |
Theorem 15.34 (Sacks). When forcing with perfect trees, the generic filter is minimal over the ground model. | The proof uses the technique of fusion. Let \( p \) be a perfect tree. A node \( s \in p \) is a splitting node if both \( {s}^{ \frown }0 \in p \) and \( {s}^{ \frown }1 \in p \) ; a splitting node \( s \) is an \( n \) th splitting node if there are exactly \( n \) splitting nodes \( t \) such that \( t \subset s \) ... | No |
If \( X \) is a countable set of ordinals in \( V\left\lbrack G\right\rbrack \) then there exists a set \( A \in V \), countable in \( V \), such that \( X \subset A \) . | Let \( \dot{F} \) be a name and let \( p \in P \) be such that \( p \) forces \ | No |
Theorem 15.38. \( B \) is \( \left( {\kappa ,\lambda }\right) \) -distributive if and only if every \( f : \kappa \rightarrow \lambda \) in the generic extension by \( B \) is in the ground model. | Proof. If \( \parallel \dot{f} \) is a function from \( \kappa \) to \( \lambda \parallel = 1 \), then \( \{ \parallel \dot{f}\left( \alpha \right) = \beta \parallel : \beta < \lambda \} \) is a partition of \( B \) of size \( \leq \lambda \) . | No |
Lemma 15.40. Let \( X \) be a subset of a complete Boolean algebra \( B \) such that \( B \) is completely generated by \( X \) . Then for every generic \( G \) on \( B, V\left\lbrack G\right\rbrack = \) \( V\left\lbrack {X \cap G}\right\rbrack \) . | Proof. We want to show that \( V\left\lbrack G\right\rbrack \) is the least model such that the set \( A = \) \( X \cap G \) is in \( V\left\lbrack G\right\rbrack \) . It suffices to show that \( G \) can be defined in terms of \( A \) .\n\nSince \( B \) is generated by \( X \), every element of \( B \) can be obtained... | Yes |
Corollary 15.42. If \( G \) is generic on \( B \) and \( A \in V\left\lbrack G\right\rbrack \) is a subset of \( \kappa \), then there exists a \( \kappa \) -generated complete subalgebra \( D \) of \( B \) such that \( V\left\lbrack {D \cap G}\right\rbrack = \) \( V\left\lbrack A\right\rbrack \) for some \( A \subset ... | Proof. Let \( \dot{A} \) be a name for \( A \) . We let \( X = \left\{ {{u}_{\alpha } : \alpha < \kappa }\right\} \), where \( {u}_{\alpha } = \) \( \parallel \check{\alpha } \in A\parallel \) . Now let \( D \) be the complete subalgebra completely generated by \( X \) ; by Lemma 15.40 we have \( V\left\lbrack {X \cap ... | Yes |
Lemma 15.43. Let \( G \) be generic on \( B \) . If \( M \) is a model of ZFC such that \( V \subset M \subset V\left\lbrack G\right\rbrack \), then there exists a complete subalgebra \( D \subset B \) such that \( M = V\left\lbrack {D \cap G}\right\rbrack \) . | Proof. We show that \( M = V\left\lbrack A\right\rbrack \), where \( A \) is a set of ordinals. Then the lemma follows from Corollary 15.42. First we note that since \( M \) satisfies the Axiom of Choice, there exists for every \( X \in M \) a set of ordinals \( {A}_{X} \in M \) such that \( X \in V\left\lbrack {A}_{X}... | Yes |
Lemma 15.44. Let \( i : P \rightarrow Q \) be such that\n\n(i) if \( {p}_{1} \leq {p}_{2} \) then \( i\left( {p}_{1}\right) \leq i\left( {p}_{2}\right) \),\n\n(ii) if \( {p}_{1} \) and \( {p}_{2} \) are incompatible then \( i\left( {p}_{1}\right) \) and \( i\left( {p}_{2}\right) \) are incompatible,\n\n(iii) for every ... | Proof. If \( G \) is generic on \( Q \) then \( {i}^{-1}\left( G\right) \) is generic on \( P \) . | No |
Lemma 15.45. Let \( h : Q \rightarrow P \) be such that\n\n(i) if \( {q}_{1} \leq {q}_{2} \) then \( h\left( {q}_{1}\right) \leq h\left( {q}_{2}\right) \).\n\n(ii) for every \( q \in Q \) and every \( p \leq h\left( q\right) \) there exists a \( {q}^{\prime } \) compatible with \( q \) such that \( h\left( {q}^{\prime ... | Proof. If \( D \subset P \) is open dense then \( {h}^{-1}\left( D\right) \) is predense in \( Q \) . It follows that if \( G \) is generic on \( Q \) then \( \{ p \in P : p \geq h\left( q\right) \) for some \( q \in G\} \) is generic on \( P \) . | Yes |
Theorem 15.46 (Vopěnka). Let \( V = L\left\lbrack A\right\rbrack \) where \( A \) is a set of ordinals. Then \( V \) is a generic extension of the model HOD. There is a Boolean algebra \( B \in {HOD} \) complete in \( {HOD} \), and there is an ultrafilter \( G \subset B \), generic over \( {HOD} \), such that \( V = {H... | Proof. Let \( \kappa \) be such that \( A \subset \kappa \) . We let \( C = {OD} \cap P\left( {P\left( \kappa \right) }\right) \) be the family of all ordinal definable sets of subsets of \( \kappa \) . Let us consider the partial ordering \( \left( {C, \subset }\right) \) .\n\nFirst we claim that there is a hereditari... | Yes |
Let \( A \) be infinite, and let \( \mathcal{G} \) be the group of all permutations of \( A \) . Let \( \mathcal{F} \) be generated by \( \{ \operatorname{fix}\left( E\right) : E \subset A \) is finite \( \} \), and let \( U \) be the permutation model. In the model \( U \) the set \( A \), although infinite, has no co... | Assume that there exists an \( f \in U \) that is a one-to-one mapping of \( \omega \) into \( A \) . Let \( E \) be a finite subset of \( A \) such that \( {\pi f} = f \) for every \( \pi \in \operatorname{fix}\left( E\right) \) . Since \( E \) is finite, there exists an \( a \in A - E \) such that \( a = f\left( n\ri... | Yes |
Let \( A \) be a disjoint countable union of pairs: \( A = \mathop{\bigcup }\limits_{{n = 0}}^{\infty }{P}_{n} \) , \( {P}_{n} = \left\{ {{a}_{n},{b}_{n}}\right\} \), and let \( \mathcal{G} \) be the group of all permutations of \( A \) such that \( \pi \left( \left\{ {{a}_{n},{b}_{n}}\right\} \right) = \left\{ {{a}_{n... | Proof. Each \( {P}_{n} \) is a symmetric set since \( \pi \left( {P}_{n}\right) = {P}_{n} \) for all \( \pi \in \mathcal{G} \) . For the same reason, \( \pi \left\langle {{P}_{n} : n \in \omega }\right\rangle = \pi \left( \left\{ {\left( {n,{P}_{n}}\right) : n \in \omega }\right\} \right) = \left\langle {{P}_{n} : n \i... | Yes |
We claim that in \( N, A \) has no countable subset. Thus assume that some \( f \in N \) is a one-to-one function from \( \omega \) into \( A \) . Let \( \dot{f} \in {HS} \) and let \( {p}_{0} \in G \) be such that\n\n\[ {p}_{0} \vDash \dot{f}\text{maps}\check{\omega }\text{one-to-one into}\dot{A}\text{.} \] | The contradiction is obtained as in Lemma 14.39. We let \( E \) be a support of \( \dot{f} \), i.e., a finite subset of \( \omega \) such that \( \operatorname{sym}\left( \dot{f}\right) \supset \operatorname{fix}\left( E\right) \) . We pick \( i \in \omega \) such that \( i \notin E \), and find \( p \leq {p}_{0} \) an... | Yes |
Theorem 15.53 (Jech-Sochor). Let \( U \) be a permutation model, let \( A \) be its set of atoms, and let \( \alpha \) be an ordinal. There exist a symmetric model \( N \) of \( \mathrm{{ZF}} \) and an embedding \( x \mapsto \widetilde{x} \) of \( U \) into \( N \) such that \[ {\left( {P}_{\alpha }\left( A\right) \rig... | Proof. We work in the theory ZFA, plus the Axiom of Choice. We denote \( A \) the set of all atoms, and let \( M \) be the kernel, \( M = {P}^{\infty }\left( \varnothing \right) \) . We consider a group \( \mathcal{G} \) of permutations of \( A \), and a filter \( \mathcal{F} \) on \( \mathcal{G} \), and let \( U \) be... | Yes |
For all \( x \) and \( y, x \in y \) if and only if \( \widetilde{x} \in \widetilde{y} \), and \( x = y \) if and only if \( \widetilde{x} = \widetilde{y} \). | First we note that \( \begin{Vmatrix}{{\dot{x}}_{a,\xi } = {\dot{x}}_{{a}^{\prime },{\xi }^{\prime }}}\end{Vmatrix} = 0 \) whenever \( \left( {a,\xi }\right) \neq \left( {{a}^{\prime },{\xi }^{\prime }}\right) \), and that \( \begin{Vmatrix}{{\dot{x}}_{a,\xi } = \check{z}}\end{Vmatrix} = 0 \) for all \( z \in M \) . Co... | Yes |
Lemma 15.55. For all \( x, x \in U \) if and only if \( \dot{\widetilde{x}} \in {HS} \) . | Proof. It suffices to show that \( x \) is symmetric if and only if \( \dot{\widetilde{x}} \) is symmetric. If \( \sigma \in \mathcal{G} \) and \( \pi \in \bar{\sigma } \), then \( \pi \dot{\widetilde{x}} \) is the canonical name for \( \left( {\sigma x}\right) \), and so \( {\operatorname{sym}}_{\bar{G}}\left( \dot{\w... | Yes |
Lemma 15.56. For all \( x, x \in U \) if and only if \( \widetilde{x} \in N \) . | Proof. By Lemma 15.55, it suffices to show that if \( \widetilde{x} \in N \), then \( x \in U \) . Assume otherwise, and let \( x \) be of least rank such that \( \widetilde{x} \in N \) and \( x \notin U \) . Thus \( x \subset U \), and since \( \widetilde{x} \in N \), there exist a name \( \dot{z} \in {HS} \) and some... | Yes |
Let \( M \) be a transitive model of ZFC. There is a model \( N \supset M \) such that \( {\left( {\aleph }_{1}\right) }^{N} = {\left( {\aleph }_{\omega }\right) }^{M} \) ; hence \( {\aleph }_{1} \) is singular in \( N \) . | First we construct a generic extension \( M\left\lbrack G\right\rbrack \) by adjoining collapsing maps \( {f}_{n} : \omega \rightarrow {\omega }_{n} \), for all \( n \in \omega \) : We let \( \left( {P, \supset }\right) \) consist of finite functions with domain \( \subset \omega \times \omega \), such that \( p\left( ... | Yes |
Lemma 15.58. If \( \operatorname{sym}\left( \dot{x}\right) \supset {H}_{n} \) and \( p \Vdash \varphi \left( \dot{x}\right) \), then \( p \upharpoonright n \Vdash \varphi \left( \dot{x}\right) \), where \( p \upharpoonright n \) is the restriction of \( p \) to \( \{ \left( {k, i}\right) : k \leq n\} \) . | Proof. Let us assume that \( p \upharpoonright n \) does not force \( \varphi \left( \dot{x}\right) \) and let \( q \supset p \upharpoonright n \) be such that \( q \Vdash \neg \varphi \left( \dot{x}\right) \) . It is easy to find some \( \pi \in {H}_{n} \) such that \( {\pi p} \) and \( q \) are compatible; since \( {... | Yes |
Let \( M \) be a transitive model of ZFC. There is a model \( N \supset M \) such that in \( N \), there is no nonprincipal ultrafilter on \( \omega \) . | The model \( N \) is obtained by adjoining to \( M \) infinitely many generic reals \( {a}_{n}, n < \omega \), without putting in \( N \) the set \( \left\{ {{a}_{n} : n \in \omega }\right\} \) (unlike in Example 15.52 where \( \left. {\left\{ {{a}_{n} : n \in \omega }\right\} \text{is in}N}\right) \) . First we constr... | Yes |
Theorem 16.2. (i) Let \( G \) be a \( V \) -generic filter on \( P \), let \( Q = {\dot{Q}}^{G} \), and let \( H \) be a \( V\left\lbrack G\right\rbrack \) -generic filter on \( Q \) . Then\n\n\[ G * H = \left\{ {\left( {p,\dot{q}}\right) \in P * \dot{Q} : p \in G\text{ and }{\dot{q}}^{G} \in H}\right\} \]\n\nis a \( V... | Proof. (i) Let us prove that if \( D \in V \) is a dense subset of \( P * Q \) then \( D \cap \left( {G * H}\right) \) is nonempty. In \( V\left\lbrack G\right\rbrack \), let\n\n\[ {D}_{1} = \left\{ {{\dot{q}}^{G} : \exists p \in G\text{ such that }\left( {p,\dot{q}}\right) \in D}\right\} .\n\nThe set \( {D}_{1} \) is ... | Yes |
Lemma 16.3. \( D \) is a complete Boolean algebra, and \( B \) embeds in \( D \) as a complete subalgebra. | Proof. If \( X \subset D \), let \( \dot{X} \in {V}^{B} \) be such that \( \operatorname{dom}\left( \dot{X}\right) = X \) and \( \dot{X}\left( \dot{c}\right) = 1 \) for all \( \dot{c} \in X \) . Since \( \dot{C} \) is a complete Boolean algebra in \( {V}^{B} \) and \( {V}^{B} \) is full, there exists a \( \dot{c} \) su... | Yes |
Theorem 16.4. Let \( \kappa \) be a regular uncountable cardinal. If \( P \) satisfies the \( \kappa \) -chain condition and if in \( {V}^{P},\dot{Q} \) satisfies the \( \kappa \) -chain condition, then \( P * \dot{Q} \) satisfies the \( \kappa \) -chain condition. | Proof. Assume that \( \left( {{p}_{\alpha },{\dot{q}}_{\alpha }}\right) ,\alpha < \kappa \), are mutually incompatible in \( P * \dot{Q} \) . Let \( \dot{Z} \in {V}^{P} \) be the canonical name for the set \( \left\{ {\alpha : {p}_{\alpha } \in G}\right\} \) (where \( G \) is a generic filter on \( P \) ), i.e., \( \pa... | Yes |
Lemma 16.5. If \( P * \dot{Q} \) satisfies the \( \kappa \) -chain condition then \( { \Vdash }_{P}\dot{Q} \) satisfies the \( \kappa \) -chain condition. | Of course \( P \) satisfies the \( \kappa \) -c.c. because \( B\left( P\right) \) is a complete subalgebra of \( B\left( {P * \dot{Q}}\right) \) . Proof. Let \( D = B * \dot{C} \) and assume that \( D \) satisfies the \( \kappa \) -chain condition. Let \( \dot{W} \in {V}^{B} \) and \( {b}_{0} \in {B}^{ + } \) be such t... | Yes |
Lemma 16.7. If \( P \) is \( \kappa \) -closed and \( { \Vdash }_{P}\dot{Q} \) is \( \kappa \) -closed, then \( P * \dot{Q} \) is \( \kappa \) -closed. | Proof. Let \( \lambda \leq \kappa \) and let \( \left( {{p}_{1},{\dot{q}}_{1}}\right) \geq \left( {{p}_{2},{\dot{q}}_{2}}\right) \geq \ldots \geq \left( {{p}_{\alpha },{\dot{q}}_{\alpha }}\right) \geq \ldots \left( {\alpha < \lambda }\right) \) be a descending sequence in \( P * \dot{Q} \) . Then \( {\left\{ {p}_{\alph... | Yes |
Theorem 16.9. Let \( \kappa \) be a regular uncountable cardinal. Let \( {P}_{\alpha } \) be the iteration with finite support of \( \left\langle {{\dot{Q}}_{\beta } : \beta < \alpha }\right\rangle \), such that for each \( \beta < \alpha \) , \( { \Vdash }_{\beta }{\dot{Q}}_{\beta } \) satisfies the \( \kappa \) -chai... | Proof. By induction on \( \alpha \) . If \( \alpha = \beta + 1 \) then \( {P}_{\alpha } = {P}_{\beta } * {\dot{Q}}_{\beta } \) and the assertion follows from Theorem 16.4. Thus let \( \alpha \) be a limit ordinal. For each \( p \in {P}_{\alpha } \) , let \( s\left( p\right) \) denote the support of \( p \) .\n\nLet \( ... | Yes |
Lemma 16.12. Martin's Axiom is equivalent to its restriction to partial orders of cardinality \( < \mathfrak{c} \): (16.5) If \( \left( {P, < }\right) \) is a partially ordered set that satisfies the countable chain condition and \( \left| P\right| < {2}^{{\aleph }_{0}} \), and if \( \mathcal{D} \) is a collection of a... | Proof. Let \( P \) be a c.c.c. partially ordered set and let us assume that (16.5) holds. Let \( \mathcal{D} \) be a family of fewer than \( \mathfrak{c} \) dense subsets of \( P \). For each \( D \in \) \( \mathcal{D} \), we let \( {W}_{D} \) be a maximal incompatible subset of \( D \). Since each \( {W}_{D} \) is cou... | Yes |
Lemma 16.14. If \( \lambda < \kappa \) and \( X \subset \lambda \) is in \( V\left\lbrack G\right\rbrack \) then \( X \in V\left\lbrack {G}_{\alpha }\right\rbrack \) for some \( \alpha < \kappa \) . | Proof. Let \( \dot{X} \) be a name for \( X \) . Every Boolean value \( \parallel \xi \in \dot{X}\parallel \) (where \( \xi < \lambda \) ) is determined by a countable antichain in \( P \) and hence \( \dot{X} \) is determined by at most \( \lambda \) conditions in \( P \) . Every condition has finite support which in ... | Yes |
Lemma 16.15. Let \( \left( {Q, < }\right) \in V\left\lbrack G\right\rbrack \) and \( \mathcal{D} \in G \) be such that \( \left( {Q, < }\right) \) is a c.c.c. partial order, \( \left| Q\right| < \kappa \) and \( \left| \mathcal{D}\right| < \kappa \) . There exists in \( V\left\lbrack G\right\rbrack \) a \( \mathcal{D} ... | Proof of Lemma 16.15. By Lemma 16.14, both \( \left( {Q, < }\right) \) and \( \mathcal{D} \) are in \( V\left\lbrack {G}_{\beta }\right\rbrack \) , for some \( \beta < \kappa \) . Let \( \dot{Q} \) be a name for \( Q \) in \( {V}^{{P}_{\beta }} \) . We may assume that \( Q \) has a greatest element, and let \( \gamma \... | Yes |
Theorem 16.16. If \( {\mathrm{{MA}}}_{{\aleph }_{1}} \) holds, then there is no Suslin tree. | Proof. Let us assume that \( T \) is a normal Suslin tree and let \( {P}_{T} \) be the partially ordered set obtained from \( T \) by reversing the order. \( {P}_{T} \) satisfies the countable chain condition. For each \( \alpha < {\omega }_{1} \), let \( {D}_{\alpha } \) be the union of all levels above \( \alpha \) :... | Yes |
Lemma 16.18. If \( T \) is an Aronszajn tree and \( W \) is an uncountable collection of finite pairwise disjoint subsets of \( T \), then there exist \( S,{S}^{\prime } \in W \) such that any \( x \in S \) is incomparable with any \( y \in {S}^{\prime } \) . | Proof. Since uncountably many elements of \( W \) have the same size, we may as well assume that there exists a natural number \( n \) such that \( \left| S\right| = n \) for all \( S \in W \) ; furthermore let us consider a fixed enumeration \( \left\{ {{z}_{1},\ldots ,{z}_{n}}\right\} \) of each set \( S \in W \) . L... | Yes |
Lemma 16.19. \( \left( {P, < }\right) \) satisfies the countable chain condition. | Proof. Let \( W \) be an uncountable subset of \( P \) . Note that the set \( \{ \operatorname{dom}\left( p\right) \) : \( p \in W\} \) is uncountable (there are only countably many functions from a finite set into \( \omega \) ). By \( \Delta \) -Lemma, there is an uncountable \( {W}_{1} \subset W \), and a finite set... | Yes |
Theorem 16.20 (Martin-Solovay). Martin's Axiom implies that \( \mathfrak{c} \) is regular, and \( {2}^{\kappa } = \mathfrak{c} \) for all infinite cardinals \( \kappa < \mathfrak{c} \) . | Proof. Assuming MA, we prove that \( {2}^{\kappa } = {2}^{{\aleph }_{0}} \) for every \( \kappa < {2}^{{\aleph }_{0}} \) . Regularity of \( \mathfrak{c} \) follows, as \( \operatorname{cf}{2}^{{\aleph }_{0}} = \operatorname{cf}{2}^{\kappa } > \kappa \) for all \( \kappa < {2}^{{\aleph }_{0}} \) . Let \( \kappa < {2}^{{... | Yes |
Theorem 16.21. \( {\mathrm{{MA}}}_{{\aleph }_{1}} \) implies that every partially ordered set that satisfies the countable chain condition has property \( \left( \mathrm{K}\right) \) . | Proof. Let \( P \) be a partially ordered set that satisfies the countable chain condition and let \( W = \left\{ {{w}_{\alpha } : \alpha < {\omega }_{1}}\right\} \) be an uncountable subset of \( P \) . We will use \( {\mathrm{{MA}}}_{{\aleph }_{1}} \) to find a filter \( G \) such that \( Z = G \cap W \) is uncountab... | Yes |
Corollary 16.22. \( {\mathrm{{MA}}}_{{\aleph }_{1}} \) implies that if every \( {P}_{i}, i \in I \), satisfies the countable chain condition then so does the product \( \mathop{\prod }\limits_{{i \in I}}{P}_{i} \) (with finite support). | Proof. Theorem 15.15. | No |
Theorem 16.23. Martin's Axiom implies that the intersection of fewer than \( \mathfrak{c} \) dense open sets of reals is dense. | Proof. Let \( \kappa < \mathfrak{c} \) and let \( {U}_{\alpha },\alpha < \kappa \), be dense open sets of reals. Let \( I \) be a bounded open interval. We’ll show that \( \mathop{\bigcap }\limits_{{\alpha < \kappa }}{U}_{\alpha } \cap I \neq \varnothing \) . Let \( P \) be the following notion of forcing: Conditions a... | Yes |
Corollary 16.25. MA implies that there exists a c-scale. | Proof. A scale is constructed by transfinite induction, using an enumeration of \( {\omega }^{\omega } \) of order-type \( \mathfrak{c} \) . | No |
Corollary 16.26. MA implies that \( \mathfrak{c} \) is not real-valued measurable. | Proof. Lemma 10.16. | No |
Theorem 16.27 (Booth). Martin's Axiom implies that there exists a p-point. | Proof. Let \( {\mathcal{A}}_{\alpha },\alpha < {2}^{{\aleph }_{0}} \), be an enumeration of all decreasing sequences \( {\left\{ {A}_{n}\right\} }_{n = 0}^{\infty } \) of subsets of \( \omega \) . We construct, by induction on \( \alpha < {2}^{{\aleph }_{0}} \), a chain of families \( {\mathcal{G}}_{0} \subset \ldots \... | Yes |
Lemma 16.28. Assume MA, and let \( \mathcal{G} \) be a family of subsets of \( \omega \) with the finite intersection property such that \( \left| \mathcal{G}\right| < {2}^{{\aleph }_{0}} \) . Let \( {A}_{0} \supset {A}_{1} \supset \ldots {A}_{n} \supset \ldots \) be a decreasing sequence of elements of \( \mathcal{G} ... | Proof. We may assume that that if \( X, Y \in \mathcal{G} \), then \( X \cap Y \in \mathcal{G} \) . For each \( X \in \mathcal{G} \), let \( {h}_{X} : \omega \rightarrow \omega \) be some function such that \( {h}_{X}\left( n\right) \in X \cap {A}_{n} \) . By Theorem 16.24 the family \( \left\{ {{h}_{X} : X \in \mathca... | Yes |
Theorem 16.30. Let \( \kappa \) be a regular uncountable cardinal and let \( \alpha \) be a limit ordinal. Let \( {P}_{\alpha } \) be an iteration such that for each \( \beta < \alpha ,{P}_{\beta } = {P}_{\alpha } \upharpoonright \beta \) satisfies the \( \kappa \) -chain condition. If \( {P}_{\alpha } \) is a direct l... | Proof. Exactly as the proof of Theorem 16.9. The only difference is that we apply Fodor’s Theorem not to \( C \), but to the stationary subset of \( C \) consisting of all \( \xi \) such that \( {P}_{{\alpha }_{\xi }} \) is a direct limit. | Yes |
Lemma 17.2. If \( U \) is a \( \sigma \) -complete ultrafilter, then \( \left( {\mathrm{{Ult}},{ \in }^{ * }}\right) \) is a wellfounded model. | Proof. We shall show that there is no infinite descending \( { \in }^{ * } \) -sequence in Ult if \( U \) is a \( \sigma \) -complete ultrafilter on \( S \) . Let us assume that \( {f}_{0},{f}_{1},\ldots ,{f}_{n},\ldots \) is such a descending sequence. Thus for each \( n \), the set\n\n\[ \n{X}_{n} = \left\{ {x \in S ... | Yes |
Lemma 17.4. Let \( j : V \rightarrow M \) be a nontrivial elementary embedding, let \( \kappa \) be the least ordinal moved, and let \( D \) be the ultrafilter on \( \kappa \) defined in (17.2). Let \( {j}_{D} : V \rightarrow \) Ult be the canonical embedding of \( V \) in the ultrapower \( {\operatorname{Ult}}_{D}\lef... | Proof. For each \( \left\lbrack f\right\rbrack \in \) Ult, let\n\n(17.4)\n\n\[ k\left( \left\lbrack f\right\rbrack \right) = \left( {j\left( f\right) }\right) \left( \kappa \right) \]\n\n(Here \( f \) is a function on \( \kappa \) and \( j\left( f\right) \) is a function on \( j\left( \kappa \right) \).)\n\nWe shall fi... | Yes |
Lemma 17.5. Let \( D \) be a nonprincipal \( \kappa \) -complete ultrafilter on \( \kappa \) . Then the following are equivalent:\n\n(i) \( D \) is normal.\n\n(ii) In the ultrapower \( {\operatorname{Ult}}_{D}\left( V\right) \) ,\n\n\[ \kappa = \left\lbrack d\right\rbrack \]\n\nwhere \( d \) is the diagonal function.\n... | Proof. (i) implies (ii): Every function \( f{ \in }^{ * }d \) is regressive, and hence represents an ordinal \( \gamma < \kappa \).\n\n(ii) implies (iii): If \( X \subset \kappa \), then \( X \in D \) if and only if \( d\left( \alpha \right) \in X \) for almost all \( \alpha \), that is, if and only if \( \left\lbrack ... | Yes |
Lemma 17.6. Let \( j \) be an elementary embedding of the universe and let \( \kappa \) be the least ordinal moved (i.e., \( j\left( \kappa \right) > \kappa \) ). If \( C \) is a closed unbounded subset of \( \kappa \), then \( \kappa \in j\left( C\right) \) . | Proof. Since \( j\left( \alpha \right) = \alpha \) for all \( \alpha < \kappa \), we have \( j\left( C\right) \cap \kappa = C \) . Thus \( j\left( C\right) \cap \kappa \) is unbounded in \( \kappa \) ; and because \( j\left( C\right) \) is closed (in \( j\left( V\right) \) and hence in the universe), we have \( \kappa ... | Yes |
Theorem 17.7 (Kunen). If \( j : V \rightarrow M \) is a nontrivial elementary embedding, then \( M \neq V \) . | First we prove the following lemma:\n\nLemma 17.8. Let \( \lam | No |
Lemma 17.8. Let \( \lambda \) be an infinite cardinal such that \( {2}^{\lambda } = {\lambda }^{{\aleph }_{0}} \) . There exists a function \( F : {\lambda }^{\omega } \rightarrow \lambda \) such that whenever \( A \) is a subset of \( \lambda \) of size \( \lambda \) and \( \gamma < \lambda \), there exists some \( s ... | Proof. Let \( \left\{ {\left( {{A}_{\alpha },{\gamma }_{\alpha }}\right) : \alpha < {2}^{\lambda }}\right\} \) be an enumeration of all pairs \( \left( {A,\gamma }\right) \) where \( \gamma < \lambda \) and \( A \) is a subset of \( \lambda \) of size \( \lambda \) . We define, by induction on \( \alpha \), a sequence ... | Yes |
Lemma 17.9. Let \( U \) be a nonprincipal \( \kappa \) -complete ultrafilter on \( \kappa \), let \( M = \) \( {\operatorname{Ult}}_{U}\left( V\right) \) and let \( j = {j}_{U} \) be the canonical elementary embedding of \( V \) in \( M \) .\n\n(i) \( {M}^{\kappa } \subset M \), i.e., every \( \kappa \) -sequence \( \l... | Proof. (i) Let \( \left\langle {{a}_{\xi } : \xi < \kappa }\right\rangle \) be a \( \kappa \) -sequence of elements of \( M \) . For each \( \xi < \kappa \), let \( {g}_{\xi } \) be a function that represents \( {a}_{\xi } \), and let \( h \) be a function that represents \( \kappa \):\n\n\[ \left\lbrack {g}_{\xi }\rig... | Yes |
Theorem 17.10. Every measurable cardinal \( \kappa \) is weakly compact and if \( D \) is a normal measure on \( \kappa \) then the set \( \{ \alpha < \kappa : \alpha \) is weakly compact \( \} \) is in \( D \) . | Proof. The first statement was proved in Lemma 10.18. Let \( D \) be a normal measure on \( \kappa \), and let \( {j}_{D} : V \rightarrow M \) be the canonical embedding. Since \( {P}^{M}\left( \kappa \right) = P\left( \kappa \right) \), it follows that \( \kappa \) is weakly compact in \( M \), and since \( {\left\lbr... | Yes |
Lemma 17.11. Let \( \kappa \) be a measurable cardinal. If \( {2}^{\kappa } > {\kappa }^{ + } \), then the set \( \left\{ {\alpha < \kappa : {2}^{\alpha } > {\alpha }^{ + }}\right\} \) has measure one for every normal measure on \( \kappa \) . | Proof. Let \( D \) be a normal measure on \( \kappa \), and let \( M = {\operatorname{Ult}}_{D}\left( V\right) \) . If \( {2}^{\alpha } = {\alpha }^{ + } \) for almost all \( \alpha \), then, since \( {\left\lbrack d\right\rbrack }_{D} = \kappa \), we have \( M \vDash {2}^{\kappa } = {\kappa }^{ + } \) . In other words... | Yes |
Lemma 17.12. Let \( \kappa \) be a measurable cardinal, let \( D \) be a normal measure on \( \kappa \) and let \( j : V \rightarrow M \) be the corresponding elementary embedding. Let \( \lambda > \kappa \) be a strong limit cardinal of cofinality \( \kappa \) . Then \( {2}^{\lambda } < j\left( \lambda \right) \) . | Proof. Since cf \( \lambda = \kappa \), we have \( j\left( \lambda \right) > \lambda \) . We shall show that \( {2}^{\lambda } = {\lambda }^{\kappa } \leq \) \( {\left( {\lambda }^{\kappa }\right) }^{M} \leq {\left( {\lambda }^{j\left( \kappa \right) }\right) }^{M} < j\left( \lambda \right) \) . The first equality hold... | Yes |
Every measurable cardinal is \( {\Pi }_{1}^{2} \) -indescribable. | Let \( \kappa \) be a measurable cardinal, let \( U \subset {V}_{\kappa } \) and let \( \sigma \) be a \( {\Pi }_{1}^{2} \) sentence of the (third order) language \( \{ \in, U\} \) . Let us assume that \( \left( {{V}_{\kappa }, \in, U}\right) \vDash \sigma \) .\n\nWe have \( \sigma = \forall {X\varphi }\left( X\right) ... | Yes |
Lemma 17.16. If \( \kappa \) is not inaccessible, then it is describable by a first order sentence, i.e., \( {\Pi }_{m}^{0} \) -describable for some \( m \) . | Proof. Let \( \kappa \) be a singular cardinal, and let \( f \) be a function with \( \operatorname{dom}\left( f\right) = \) \( \lambda < \kappa \) and \( \operatorname{ran}\left( f\right) \) cofinal in \( \kappa \) . Let \( {U}_{1} = f \) and \( {U}_{2} = \{ \lambda \} \), and let \( \sigma \) be the first order sente... | Yes |
Lemma 17.17. If \( \kappa \) is a weakly compact cardinal, then for every \( U \subset {V}_{\kappa } \) , the model \( \left( {{V}_{\kappa }, \in, U}\right) \) has a transitive elementary extension \( \left( {M, \in ,{U}^{\prime }}\right) \) such that \( \kappa \in M \) . | Proof. Let \( \sum \) be the set of all \( {\mathcal{L}}_{\kappa ,\kappa } \) sentences true in the model \( {\left( {V}_{\kappa }, \in, U, x\right) }_{x \in {V}_{\kappa }} \) plus the sentences\n\n\[ c\text{is an ordinal,}\]\n\n\[ c > \alpha ,\;\left( {\text{all }\alpha < \kappa }\right) .\n\nClearly \( \left| \sum \r... | Yes |
Theorem 17.18 (Hanf-Scott). A cardinal \( \kappa \) is \( {\Pi }_{1}^{1} \) -indescribable if and only if it is weakly compact. | Proof. First we show that every \( {\Pi }_{1}^{1} \) -indescribable cardinal is weakly compact. If \( \kappa \) is \( {\Pi }_{1}^{1} \) -indescribable, then by Lemma 17.16, \( \kappa \) is inaccessible, and it suffices to show that \( \kappa \) has the tree property. In fact, by the proof of Theorem 17.13(i) it suffice... | Yes |
Every weakly compact cardinal \( \kappa \) is a Mahlo cardinal, and the set of Mahlo cardinals below \( \kappa \) is stationary. | Proof. Let \( C \subset \kappa \) be a closed unbounded set. Since \( \kappa \) is inaccessible, \( \left( {{V}_{\kappa }, \in, C}\right) \) satisfies the following \( {\Pi }_{1}^{1} \) sentence:\n\n\( \neg \exists F\left( {F\text{is a function from some}\lambda < \kappa \text{cofinally into}\kappa }\right) \)\n\nand \... | Yes |
Corollary 17.20. If \( \kappa \) is weakly compact and if \( S \subset \kappa \) is stationary, then there is a regular uncountable \( \lambda < \kappa \) such that \( S \cap \lambda \) is stationary in \( \lambda \) . | Proof. \ | No |
Lemma 17.21. If \( \kappa \) is weakly compact and if \( A \subset \kappa \) is such that \( A \cap \alpha \in L \) for every \( \alpha < \kappa \), then \( A \) is constructible. | Proof. Let \( A \subset \kappa \) be such that \( A \cap \alpha \in L \) for all \( \alpha < \kappa \) . By Lemma 17.17 there is a transitive model \( \left( {M, \in ,{A}^{\prime }}\right) \succ \left( {{V}_{\kappa }, \in, A}\right) \) such that \( \kappa \in M \) . Consider the sentence \( \forall \alpha \exists x \) ... | No |
Theorem 17.22. If \( \kappa \) is weakly compact then \( \kappa \) is weakly compact in \( L \) . | Proof. In \( L \), let \( T = \left( {\kappa ,{ < }_{T}}\right) \) be a tree of height \( \kappa \) such that each level of \( T \) has size less than \( \kappa \) . If \( \kappa \) is weakly compact then \( T \) has a branch \( B \) (in the universe), and by Lemma 17.21, \( B \in L \) . Hence \( \kappa \) has the tree... | Yes |
Lemma 17.24. Let \( \kappa \) be an infinite cardinal and assume that \[ \kappa \rightarrow {\left( \alpha \right) }_{{2}^{\lambda }}^{ < \omega } \] where \( \alpha \) is a limit ordinal and \( \lambda \) is an infinite cardinal. Let \( \mathcal{L} \) be a language of size \( \leq \lambda \) and let \( \mathfrak{A} \)... | Proof. Let \( \Phi \) be the set of all formulas of the language \( \mathcal{L} \) . We consider the function \( F : {\left\lbrack \kappa \right\rbrack }^{ < \omega } \rightarrow P\left( \Phi \right) \) defined as follows: If \( x \in {\left\lbrack \kappa \right\rbrack }^{n} \) and \( x = \left\{ {{\alpha }_{1},\ldots ... | Yes |
Lemma 17.25. If \( \kappa \rightarrow {\left( \kappa \right) }^{ < \omega } \) and if \( \lambda < \kappa \) is a cardinal, then \( \kappa \rightarrow {\left( \kappa \right) }_{\lambda }^{ < \omega } \) . | Proof. Let \( F : {\left\lbrack \kappa \right\rbrack }^{ < \omega } \rightarrow \lambda \) be a partition into \( \lambda < \kappa \) pieces. We consider the following partition \( G \) of \( {\left\lbrack \kappa \right\rbrack }^{ < \omega } \) into two pieces: If \( {\alpha }_{1} < \ldots < {\alpha }_{k} < {\alpha }_{... | Yes |
Lemma 17.29. If \( \kappa \rightarrow {\left( \alpha \right) }^{ < \omega } \), then \( \kappa \rightarrow {\left( \alpha \right) }_{{2}^{{\aleph }_{0}}}^{ < \omega } \) . | Proof. Let \( f \) be a partition, \( f : {\left\lbrack \kappa \right\rbrack }^{ < \omega } \rightarrow \{ 0,1{\} }^{\omega } \) . For each \( n < \omega \), let \( {f}_{n} = \) \( f \upharpoonright {\left\lbrack \kappa \right\rbrack }^{n} \), and for each \( \kappa < \omega \), let \( {f}_{n, k} : {\left\lbrack \kappa... | Yes |
Every Erdős cardinal \( {\eta }_{\alpha } \) is inaccessible, and if \( \alpha < \beta \) then \( {\eta }_{\alpha } < {\eta }_{\beta } \) | Proof. First we claim that \( {\eta }_{\alpha } \) is a strong limit cardinal. If \( \kappa < {\eta }_{\alpha } \) then because \( {2}^{\kappa } \nrightarrow {\left( \alpha \right) }_{\kappa }^{2} \) (by Lemma 9.3) and \( {\eta }_{\alpha } \rightarrow {\left( \alpha \right) }_{\kappa }^{2} \), we have \( {2}^{\kappa } ... | Yes |
Theorem 17.33. If \( {\eta }_{\omega } \) exists then there exists a weakly compact cardinal below \( {\eta }_{\omega } \) . | Proof. Let \( {h}_{\varphi },\varphi \in \) Form, be Skolem functions for the language \( \{ \in \} \) of set theory, and let us consider the model \( \mathfrak{A} = {\left( {V}_{{\eta }_{\omega }}, \in ,{h}_{\varphi }^{\mathfrak{A}}\right) }_{\varphi \in \text{ Form }} \) where for each \( \varphi ,{h}_{\varphi }^{\ma... | Yes |
Theorem 17.34. If \( \kappa \rightarrow {\left( \omega \right) }^{ < \omega } \) then \( L \vDash \kappa \rightarrow {\left( \omega \right) }^{ < \omega } \) . | Proof. Let \( f \) be a constructible partition \( f : {\left\lbrack \kappa \right\rbrack }^{ < \omega } \rightarrow \{ 0,1\} \) . We claim that if there is an infinite homogeneous set for \( f \), then there is one in \( L \) . Let \( T \) be the set of all finite increasing sequences \( t = \left\langle {{\alpha }_{0... | Yes |
Corollary 18.2. Every constructible set definable in \( L \) is countable. | Proof. If \( x \in L \) is definable in \( L \) by a formula \( \varphi \), then the same formula defines \( x \) in \( {L}_{{\aleph }_{1}} \) and hence \( x \in {L}_{{\aleph }_{1}} \) . | No |
Corollary 18.3. Every uncountable cardinal is inaccessible in \( L \) . | Proof. Since \( L \vDash {\aleph }_{1} \) is regular, we have\n\n\[ L \vDash {\aleph }_{\alpha }\text{is regular} \]\n\nfor every \( \alpha \geq 1 \) . Similarly, \( L \vDash {\aleph }_{\omega } \) is a limit cardinal, and hence\n\n\[ L \vDash {\aleph }_{\alpha }\text{is a limit cardinal} \]\n\nfor every \( \alpha \geq... | Yes |
Corollary 18.4. Every uncountable cardinal is a Mahlo cardinal in L. | Proof. By Corollary 18.3, every Silver indiscernible is an inaccessible cardinal in \( L \) . Since \( I \cap {\omega }_{1} \) is closed unbounded in \( {\omega }_{1},{\aleph }_{1} \) is a Mahlo cardinal in \( L \) . | No |
Corollary 18.5. For every \( \alpha \geq \omega ,\left| {{V}_{\alpha } \cap L}\right| \leq \left| \alpha \right| \) . In particular, the set of all constructible reals is countable. | Proof. The set \( {V}_{\alpha } \cap L \) is definable in \( L \) from \( \alpha \) . Thus \( {V}_{\alpha } \cap L \) is also definable from \( \alpha \) in \( {L}_{\kappa } \) where \( \kappa \) is the least cardinal \( > \alpha \) . Hence \( {V}_{\alpha } \cap L \subset {L}_{\beta } \) for some \( \beta \) such that ... | Yes |
If \( \alpha \) and \( \beta \) are limit ordinals and if \( j : {L}_{\alpha } \rightarrow {L}_{\beta } \) is an elementary embedding of \( \left( {{L}_{\alpha }, \in }\right) \) in \( \left( {{L}_{\beta }, \in }\right) \), then for each formula \( \varphi \) and all \( {x}_{1},\ldots ,{x}_{n} \in {L}_{\alpha } \) | \[ {h}_{\varphi }^{{L}_{\beta }}\left( {j\left( {x}_{1}\right) ,\ldots, j\left( {x}_{n}\right) }\right) = j\left( {{h}_{\varphi }^{{L}_{\alpha }}\left( {{x}_{1},\ldots ,{x}_{n}}\right) }\right) . \] Hence \( j \) remains elementary with respect to the augmented language \( {\mathcal{L}}^{ * } = \) \( \{ \in \} \cup \le... | Yes |
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