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Theorem 23.2 (Gregory). If \( \lambda \) is regular such that \( {\kappa }^{\lambda } = \kappa \) and if \( {2}^{\kappa } = {\kappa }^{ + } \) , then \( \diamond \left( {E}_{\lambda }^{{\kappa }^{ + }}\right) \) holds. | Proof. We prove the version of \( \diamondsuit \left( E\right) \) from (23.2) where \( E = {E}_{\lambda }^{{\kappa }^{ + }} \) ; by Lemma 23.1, \( \diamondsuit \left( E\right) \) follows. Let \( \left\langle {{x}_{\alpha } : \alpha < {\kappa }^{ + }}\right\rangle \) enumerate all bounded subsets of \( {\kappa }^{ + } \... | Yes |
Theorem 23.3 (Shelah). Let \( \kappa \geq {\aleph }_{3} \) be a regular uncountable cardinal, and let \( \lambda \) be a regular uncountable cardinal such that \( {\lambda }^{ + } < \kappa \) . Then there exists a sequence \( \left\langle {{c}_{\alpha } : \alpha \in {E}_{\lambda }^{\kappa }}\right\rangle \) with each \... | Proof. It suffices to find a family \( \left\{ {{c}_{\alpha } : \alpha \in {E}_{\lambda }^{\kappa }}\right\} \) such that each \( {c}_{\alpha } \) is a closed subset of \( \alpha \), and for every closed unbounded \( C \subset \kappa \), the set \( \left\{ {\alpha \in {E}_{\lambda }^{\kappa } : {c}_{\alpha }}\right. \)... | Yes |
Lemma 23.4. Let \( \kappa \) and \( \lambda \) be regular uncountable cardinals such that \( {\lambda }^{ + } < \) \( \kappa \) . For every stationary set \( E \subset {E}_{\lambda }^{\kappa } \) there exists a sequence \( \left\langle {{c}_{\alpha } : \alpha \in E}\right\rangle \) with each \( {c}_{\alpha } \subset \a... | Proof. For proof, see Gitik and Shelah [1997]. | No |
Lemma 23.6. \( {▱}_{{\omega }_{1}} \) implies that there exists a stationary set \( S \subset {E}_{{\aleph }_{0}}^{{\aleph }_{2}} \) that does not reflect. | Proof. Let \( \left\langle {{C}_{\alpha } : \alpha \in \operatorname{Lim}\left( {\omega }_{2}\right) }\right\rangle \) be a square-sequence. For each \( \alpha < {\omega }_{2} \) of cofinality \( {\omega }_{1} \), the order-type of \( {C}_{\alpha } \) is \( {\omega }_{1} \) . It follows that there exists a countable li... | No |
Lemma 23.7. Let \( \kappa \) be a regular uncountable cardinal and let \( P \) be a notion of forcing. If \( P \) is \( < \kappa \) -closed then every stationary \( S \subset \kappa \) remains stationary in \( V\left\lbrack G\right\rbrack \) . | Proof. Let \( p \Vdash \dot{C} \) is closed unbounded; we find a \( \gamma \in S \) and a \( q \leq p \) such that \( q \Vdash \gamma \in \dot{C} \) as follows: We construct an increasing continuous ordinal sequence \( \left\langle {{\gamma }_{\alpha } : \alpha < \kappa }\right\rangle \) and a decreasing sequence \( \l... | Yes |
Theorem 23.8. Let \( S \) be a stationary subset of \( {\omega }_{1} \). There is a notion of forcing \( {P}_{S} \) that adds generically a closed unbounded set \( C \subset {\omega }_{1} \) such that \( C \subset S \), and such that \( {P}_{S} \) adds no new countable sets. | Proof. \( {P}_{S} \) consists of all bounded closed sets of ordinals \( p \) such that \( p \subset S \); \( p \) is stronger than \( q \) if \( p \) is an end-extension of \( q \) (if \( q = p \cap \alpha \) for some \( \alpha \)).\n\nIf \( G \) is a generic filter, let \( C = \bigcup G \). Clearly, \( C \) is a subse... | Yes |
Lemma 23.9. \( {P}_{S} \) is \( \omega \) -distributive. | Proof. Let \( p \Vdash \dot{f} : \omega \rightarrow \) Ord; we shall find a \( q \leq p \) and some \( f \) so that \( q \Vdash \dot{f} = f. \)\n\nBy induction on \( \alpha \) we construct a chain \( \left\{ {{A}_{\alpha } : \alpha < {\omega }_{1}}\right\} \) of countable subsets of \( {P}_{S} \) . Let \( {A}_{0} = \{ ... | Yes |
Lemma 23.11. \( {Q}_{{\kappa }^{ + }} \) satisfies the \( {\kappa }^{ + } \) -chain condition. | Proof. Let \( W \) be a maximal antichain. Since \( \left| {Q}_{\alpha }\right| \leq \kappa \) for each \( \alpha < {\kappa }^{ + } \) there exists an \( \alpha < {\kappa }^{ + } \) such that for every \( q \in W \) there is some \( {q}^{\prime } \in W \cap {Q}_{\alpha } \) with \( {q}^{\prime } \leq q \upharpoonright ... | Yes |
If \( \beta < \alpha \) then \( {I}_{\beta } = {I}_{\alpha } \cap V\left\lbrack {G,\mathcal{C} \mid \beta }\right\rbrack \) . | It is clear that \( {I}_{\beta } \subset {I}_{\alpha } \) . Thus let \( \dot{X} \) be a \( P * {Q}_{\beta } \) -name for a subset of \( \kappa \), and let \( p \in P \) and \( q \in {Q}_{\alpha } \) be such that \( \left( {p, q}\right) \) forces \( \dot{X} \in {I}_{\alpha } \) and \( \dot{X} \notin {I}_{\beta } \) . By... | Yes |
Lemma 23.14. I is a normal ideal. | Proof. Let \( f \in V\left\lbrack {G,\mathcal{C}}\right\rbrack \) be a function \( f : \kappa \rightarrow \kappa \), and assume that\n\n\[\n\text{23.8)}\;\left( {p, q}\right) \Vdash \{ \alpha : f\left( \alpha \right) < \alpha \} \notin I\;\text{and}\;\left( {\forall \gamma < \kappa }\right) \{ \alpha : f\left( \alpha \... | Yes |
I is, in \( V\left\lbrack {G,\mathcal{C}}\right\rbrack \), the nonstationary ideal on \( {\aleph }_{1} = \kappa \) . | Proof. That \( \kappa = {\aleph }_{1} \) in \( V\left\lbrack {G,\mathcal{C}}\right\rbrack \) follows from the normality of \( I \) . Each \( {C}_{\alpha } \) is a closed unbounded subset of \( {\omega }_{1} \), and since \( {C}_{\alpha } \cap {A}_{\alpha } = \varnothing \) if \( {A}_{\alpha } \in I \), every set in \( ... | Yes |
Theorem 23.17 (Gitik-Shelah). For every regular cardinal \( \kappa \geq {\aleph }_{2} \), the ideal \( {I}_{\mathrm{{NS}}} \) on \( \kappa \) is not \( {\kappa }^{ + } \) -saturated. | The proof of Theorem 23.17 appears in Gitik and Shelah [1997]. Most special cases were proved earlier by Shelah, and we present this proof first, as it is somewhat easier. The complete proof will follow. | No |
Lemma 23.18. Let \( I \) be a normal \( {\kappa }^{ + } \) -saturated \( \kappa \) -complete ideal on \( \kappa \), let \( R\left( I\right) \) be the forcing with \( I \) -positive sets, let \( G \) be the \( R\left( I\right) \) -generic ultrafilter and let \( M = {\operatorname{Ult}}_{G}\left( V\right) \) . Then \( {P... | Proof. The Boolean algebra \( B = P\left( \kappa \right) /I \) is complete (see Exercise 22.9). If \( \dot{A} \) is a name for a subset \( A = {\dot{A}}^{G} \) of \( \kappa \) in \( V\left\lbrack G\right\rbrack \), let \( {S}_{\alpha } \in {I}^{ + } \) be, for each \( \alpha < \kappa \), such that \( \parallel \alpha \... | Yes |
Lemma 23.19. If \( \alpha < {\lambda }^{ + } \) and \( \operatorname{cf}\alpha \neq \operatorname{cf}\lambda \) then there exists no strongly almost disjoint family of subsets of \( \alpha \) . | Proof. Assume to the contrary that \( \left\{ {{X}_{\xi } : \xi < {\lambda }^{ + }}\right\} \) is a strongly almost disjoint family of subsets of \( \alpha \) . We may assume that each \( {X}_{\xi } \) has order-type cf \( \alpha \) . Let \( f \) be a function that maps \( \lambda \) onto \( \alpha \) . Since cf \( \la... | Yes |
Corollary 23.20. If \( \kappa \) is a regular cardinal and if a notion of forcing \( P \) makes \( \operatorname{cf}\kappa \neq \operatorname{cf}\left| \kappa \right| \), then \( P \) collapses \( {\kappa }^{ + } \) . | Proof. Assume that \( {\kappa }^{ + } \) is preserved; thus in \( V\left\lbrack G\right\rbrack ,{\left( {\kappa }^{ + }\right) }^{V} = {\lambda }^{ + } \) where \( \lambda = \left| \kappa \right| \) . In \( V \) there is a strongly almost disjoint family \( \left\{ {{X}_{\xi } : \xi < {\left( {\kappa }^{ + }\right) }^{... | Yes |
Corollary 23.21. If \( \kappa = {\lambda }^{ + } \), if \( \nu \neq \operatorname{cf}\lambda \) is a regular cardinal, and if \( I \) is a normal \( \kappa \) -complete \( {\kappa }^{ + } \) -saturated ideal on \( \kappa \), then \( {E}_{\nu }^{\kappa } = \{ \alpha < \kappa : \operatorname{cf}\alpha = \) \( \nu \} \in ... | Proof. Assume that \( {E}_{\nu }^{\kappa } \in {I}^{ + } \), and let \( G \) be a generic ultrafilter on \( P\left( \kappa \right) /I \) . By Lemma 23.18, all cardinals \( \leq \lambda \), as well as \( {\kappa }^{ + } \), are preserved in \( V\left\lbrack G\right\rbrack \) . If \( {E}_{\nu }^{\kappa } \in G \), then i... | Yes |
Lemma 23.22. If \( {I}_{\mathrm{{NS}}} \mid {E}_{\lambda }^{\kappa } \) is \( {\kappa }^{ + } \) -saturated then there exists a stationary set \( \widetilde{E} \subset E \) such that for every closed unbounded \( C,\widetilde{E} - \mathcal{G}\left( C\right) \) is nonstationary ( \( C \) is guessed at almost every \( \a... | Proof. If not, then for every stationary \( S \subset E \) there exists a closed unbounded set \( C \) such that \( S - \mathcal{G}\left( C\right) \) is stationary. By the \( {\kappa }^{ + } \) -saturation, there exists a collection \( \left\{ {\left( {{S}_{i},{C}_{i}}\right) : i < \kappa }\right\} \) such that \( W = ... | Yes |
Theorem 23.23 (Magidor). The following are equiconsistent:\n\n(i) the existence of a weakly compact cardinal,\n\n(ii) every stationary set \( S \subset {E}_{{\aleph }_{0}}^{{\aleph }_{2}} \) reflects at almost all \( \alpha \in {E}_{{\aleph }_{1}}^{{\aleph }_{2}} \) . | We shall prove that (ii) implies that \( {\aleph }_{2} \) is weakly compact in \( L \), and then give a brief account of the consistency proof of (ii). If every stationary set \( S \subset \) \( {E}_{{\aleph }_{0}}^{{\aleph }_{2}} \) reflects then \( {\aleph }_{2} \) is a Mahlo cardinal in \( L \) . Using Jensen’s Theo... | Yes |
Lemma 23.24. If every stationary \( S \subset {E}_{{\aleph }_{0}}^{{\aleph }_{2}} \) reflects then \( {\aleph }_{2} \) is inaccessible in \( L \) . | Proof. Let \( \kappa = {\aleph }_{2} \) . Assume that \( \kappa \) is in \( L \) the successor of some \( \lambda ,\kappa = {\left( {\lambda }^{ + }\right) }^{L} \) . In \( L \), there exists a square-sequence \( \left\langle {{C}_{\alpha } : \alpha \in \operatorname{Lim}\left( \kappa \right) }\right\rangle \), and the... | Yes |
Lemma 24.2. Assume that \( {\aleph }_{\alpha }^{{\aleph }_{1}} < {\aleph }_{{\omega }_{1}} \) for all \( \alpha < {\omega }_{1} \) . Let \( F \) be an almost disjoint family of functions\n\n\[ F \subset \mathop{\prod }\limits_{{\alpha < {\omega }_{1}}}{A}_{\alpha } \]\n\n such that \( \left| {A}_{\alpha }\right| < {\al... | Proof. We first introduce the following relation among functions \( \varphi : {\omega }_{1} \rightarrow {\omega }_{1} \)\n\n4.1) \( \varphi < \psi \) if and only if \( \left\{ {\alpha < {\omega }_{1} : \varphi \left( \alpha \right) \geq \psi \left( \alpha \right) }\right\} \) is nonstationary.\n\nSince the closed unbou... | No |
Lemma 24.3. Assume that \( {\aleph }_{\alpha }^{{\aleph }_{1}} < {\aleph }_{{\omega }_{1}} \) for all \( \alpha < {\omega }_{1} \) . Let \( \varphi : {\omega }_{1} \rightarrow {\omega }_{1} \) and let \( F \) be an almost disjoint family of functions\n\n\[ F \subset \mathop{\prod }\limits_{{\alpha < {\omega }_{1}}}{A}_... | Proof of Lemma 24.3. By induction on \( \parallel \varphi \parallel \) . If \( \parallel \varphi \parallel = 0 \), then \( \varphi \left( \alpha \right) = 0 \) on a stationary set and the statement is precisely Lemma 8.16.\n\nTo handle the case \( \parallel \varphi \parallel > 0 \), we first generalize the definition o... | Yes |
Lemma 24.5. Let \( \kappa \) be a regular uncountable cardinal. There exist ordinal functions \( {f}_{\eta },\eta < {\kappa }^{ + } \), on \( \kappa \) such that\n\n(i) \( {f}_{0}\left( \alpha \right) = 0 \) and \( {f}_{\eta + 1}\left( \alpha \right) = {f}_{\eta }\left( \alpha \right) + 1 \), for all \( \alpha < \kappa... | Proof. Let \( \left\langle {{\xi }_{\nu } : \nu < \operatorname{cf}\eta }\right\rangle \) be some sequence with limit \( \eta \) . If \( \operatorname{cf}\eta < \kappa \), let \( {f}_{\eta }\left( \alpha \right) = \sup \left\{ {{f}_{{\xi }_{\nu }}\left( \alpha \right) : \nu < \operatorname{cf}\eta }\right\} \), and if ... | Yes |
Theorem 24.8 (Shelah). Let \( \kappa \) be a strong limit cardinal of cofinality \( \omega \) . There exists an increasing sequence \( \left\langle {{\lambda }_{n} : n < \omega }\right\rangle \) of regular cardinals with limit \( \kappa \) such that the true cofinality of \( \mathop{\prod }\limits_{{n < \omega }}{\lamb... | Proof. Let \( I \) be the ideal of finite subsets of \( \omega \) . We shall find the \( {\lambda }_{n} \) ’s and a \( {\kappa }^{ + } \) -scale in \( \mathop{\prod }\limits_{n}{\lambda }_{n} \) in the partial ordering \( { < }_{I} \) .\n\nFirst we choose any increasing sequence \( {\kappa }_{n}, n < \omega \), of regu... | Yes |
There exists a function \( g : \omega \rightarrow \kappa \) that is an upper bound of \( F \) in \( { < }_{I} \), and is \( { \leq }_{I} \) -minimal among such upper bounds. | Let \( {g}_{0} = \left\langle {{\kappa }_{n} : n < \omega }\right\rangle \) ; we shall construct a maximal transfinite \( { \leq }_{I} \) - decreasing sequence \( {\left\langle {g}_{\nu }\right\rangle }_{\nu } \) of upper bounds of \( F \) . It suffices to show that the length of the sequence \( {\left\langle {g}_{\nu ... | Yes |
Lemma 24.10. If \( \lambda > {2}^{\left| A\right| } \) is a regular cardinal then every \( { < }_{I} \) -increasing \( \lambda \) -sequence of ordinal functions on \( A \) has an exact upper bound. | Proof. Let \( F = \left\langle {{f}_{\alpha } : \alpha < \lambda }\right\rangle \) be \( { < }_{I} \) -increasing. Let \( M \) be an elementary submodel of \( {H}_{\vartheta } \) for a sufficiently large \( \vartheta \) such that \( I \in M, F \in M,\left| M\right| = {2}^{\left| A\right| } \) and \( {M}^{\left| A\right... | Yes |
Corollary 24.11. If \( \lambda > {2}^{\left| A\right| } \) is regular, \( F = \left\langle {{f}_{\alpha } : \alpha < \lambda }\right\rangle \) is \( { < }_{I} \) -increasing and \( g \) is an upper bound of \( F \), then either \( F \) is bounded below \( g \), or \( F \) is cofinal in \( g \), or \( A = X \cup Y \) wi... | Proof. Let \( f \) be an exact upper bound of \( F \) and let \( X = \{ a \in A : f\left( a\right) < \) \( g\left( a\right) \} \) . | No |
Corollary 24.12. Let \( \lambda > {2}^{\left| A\right| } \) be a regular cardinal, let \( {\gamma }_{a}, a \in A \), be limit ordinals, and assume that \( \mathop{\prod }\limits_{{a \in A}}{\gamma }_{a} \) is \( \lambda \) -directed in \( { < }_{I} \) . Then either \( \mathop{\prod }\limits_{{a \in A}}{\gamma }_{a} \) ... | Proof. Assume that \( \mathop{\prod }\limits_{{a \in A}}{\gamma }_{a} \) is \( \lambda \) -directed but not \( {\lambda }^{ + } \) -directed, and let \( S \subset \) \( \mathop{\prod }\limits_{{a \in A}}{\gamma }_{a} \) be such that \( \left| S\right| = \lambda \) and \( S \) is not bounded. Using the \( \lambda \) -di... | Yes |
Lemma 24.14. Let \( F = \left\langle {{f}_{\alpha } : \alpha < \lambda }\right\rangle \) be \( \gamma \) -rapid, with \( \gamma > \left| A\right| \) . For each \( a \in A \), let \( {S}_{a} \subset \lambda \) be such that \( \left| {S}_{a}\right| < \gamma \) . Then there exists an \( \alpha < \lambda \) with the proper... | Proof. Assume by contradiction that for every \( \alpha < \lambda \) there exists an \( h \in \mathop{\prod }\limits_{{a \in A}}{S}_{a} \) such that \( h{ > }_{I}{f}_{\alpha } \) but \( h \) is not an upper bound of \( F \) . By induction, we construct a continuous increasing sequence \( {\alpha }_{\xi },\xi < \gamma \... | Yes |
Corollary 24.15. If \( F = \left\langle {{f}_{\alpha } : \alpha < \lambda }\right\rangle \) is \( \gamma \) -rapid, with \( \left| A\right| < \gamma < \lambda \), and if \( f \) is the least upper bound of \( F \), then cf \( f\left( a\right) \geq \gamma \) for \( I \) -almost all \( a \in A \) . | Proof. Let \( f \) be an upper bound of \( F \), and assume that \( B = \{ a \in A \) : cf \( f\left( a\right) < \gamma \} \in {I}^{ + } \) . We shall find an upper bound \( h \) of \( F \) such that \( h{ < }_{I}f \) on \( B \) .\n\nFor \( a \in B \), let \( {S}_{a} \) be a cofinal subset of \( f\left( a\right) \) of ... | Yes |
Theorem 24.16 (Shelah). Let \( \kappa \) be a regular uncountable cardinal, and let \( I = {I}_{\mathrm{{NS}}} \) be the nonstationary ideal on \( \kappa \) . Let \( \left\langle {{\eta }_{\xi } : \xi < \kappa }\right\rangle \) be a continuous increasing sequence with limit \( \eta \) . Then \( \mathop{\prod }\limits_{... | We shall prove this theorem only under the assumption \( {2}^{\kappa } < {\aleph }_{\eta } \) (we only need the weaker version for the proof of Theorem 24.33). For the general proof, see Burke and Magidor [1990].\n\nProof. Let \( \lambda = {\aleph }_{\eta + 1} \) . We wish to find a \( \lambda \) -scale. It is not diff... | Yes |
Lemma 24.19. Let \( A \) be an interval of regular cardinals such that \( \min A = \) \( {\left( {2}^{\left| A\right| }\right) }^{ + } \) . Then \( \operatorname{pcf}A \) is an interval. | Proof. Let \( D \) be an ultrafilter on \( A \) and let \( \lambda \) be a regular cardinal such that \( \min A \leq \lambda < \operatorname{cof}D \) . We shall find an ultrafilter \( E \) on \( A \) such that \( \operatorname{cof}E = \lambda \) .\n\nLet \( \left\{ {{f}_{\alpha } : \alpha < \operatorname{cof}D}\right\}... | Yes |
Corollary 24.20. If \( {\aleph }_{\omega } \) is a strong limit cardinal, then \( \operatorname{pcf}{\left\{ {\aleph }_{n}\right\} }_{n = 0}^{\infty } \) is an interval and \( \operatorname{suppcf}{\left\{ {\aleph }_{n}\right\} }_{n = 0}^{\infty } < {\aleph }_{{\aleph }_{\omega }} \) . | Proof. Apply Lemma 24.19 to the interval \( A = \left\lbrack {{\left( {2}^{{\aleph }_{0}}\right) }^{ + },{\aleph }_{\omega }}\right) \), and use \( \left| {\operatorname{pcf}A}\right| \leq {2}^{{2}^{{\aleph }_{0}}} < {\aleph }_{\omega } \) | No |
Lemma 24.21. There exists a family \( F \) of functions in \( \mathop{\prod }\limits_{{n = 0}}^{\infty }{\aleph }_{n},\left| F\right| = \lambda \) , such that for every \( g \in \mathop{\prod }\limits_{{n = 0}}^{\infty }{\aleph }_{n} \) there is some \( f \in F \) with \( g\left( n\right) \leq f\left( n\right) \) for a... | Proof. For every ultrafilter \( D \) on \( \omega \) choose a sequence \( \left\langle {{f}_{\alpha }^{D} : \alpha < \operatorname{cof}D}\right\rangle \) that is cofinal in \( \mathop{\prod }\limits_{{n = 0}}^{\infty }{\aleph }_{n}/D \), and let \( F \) be the set of all \( f = \max \left\{ {{f}_{{\alpha }_{1}}^{{D}_{1... | Yes |
Lemma 24.22. If \( a \) and \( b \) are countable subsets of \( {\aleph }_{\omega } \) and if \( {\chi }^{a} = {\chi }^{b} \), then \( {M}^{a} \cap {\aleph }_{\omega } = {M}^{b} \cap {\aleph }_{\omega } \) | Proof. By induction on \( n \) we show that \( {M}^{a} \cap {\aleph }_{n} = {M}^{b} \cap {\aleph }_{n} \), for all \( n \geq k \) . This is true for \( n = k \) ; thus assume that this is true for \( n \) and prove it for \( n + 1 \) . Both \( {M}^{a} \cap {\aleph }_{n + 1} \) and \( {M}^{b} \cap {\aleph }_{n + 1} \) c... | Yes |
Lemma 24.23. There exists a family \( {F}_{\lambda } \) of \( \lambda \) subsets of \( \lambda \), each of size \( {\aleph }_{k} \) , such that for every subset \( Z \subset \lambda \) of size \( {\aleph }_{k} \) there exists an \( X \in {F}_{\lambda } \) such that \( X \subset Z \) . | Proof. We prove (by induction on \( \alpha \) ) that for every ordinal \( \alpha \) such that \( {2}^{{\aleph }_{k}} \leq \alpha \leq \lambda \) there is a family \( {F}_{\alpha } \subset {\left\lbrack \alpha \right\rbrack }^{{\aleph }_{k}},\left| {F}_{\alpha }\right| \leq \left| \alpha \right| \) such that for every \... | Yes |
Lemma 24.24. If \( \left| {\operatorname{pcf}A}\right| < \min A \) then \( \operatorname{pcf}\left( {\operatorname{pcf}A}\right) = \operatorname{pcf}A \) . | Proof. Let \( B = \operatorname{pcf}A \) . For each \( \lambda \in B \) choose \( {D}_{\lambda } \) on \( A \) such that \( \operatorname{cof}{D}_{\lambda } = \lambda \) , and let \( \left\langle {{f}_{\alpha }^{\lambda } : \alpha < \lambda }\right\rangle \) be cofinal in \( \prod A/{D}_{\lambda } \) . Let \( \mu \in \... | Yes |
Theorem 24.25 (Shelah). If \( A \) is a set of regular cardinals such that \( {2}^{\left| A\right| } < \) \( \min A \), then there exist sets \( {B}_{\lambda } \subset A,\lambda \in \operatorname{pcf}A \), such that for every \( \lambda \in \operatorname{pcf}A \)\n\n(a) \( \lambda = \max \operatorname{pcf}{B}_{\lambda ... | (To see that \( {J}_{\lambda } \) is an ideal, we observe that if \( X \in {J}_{\lambda } \) then \( X \subset {B}_{{\nu }_{1}} \cup \) \( \ldots \cup {B}_{{\nu }_{k}} \), hence \( \operatorname{pcf}X \subset \operatorname{pcf}{B}_{{\nu }_{1}} \cup \ldots \cup \operatorname{pcf}{B}_{{\nu }_{k}} \) and so by (a), \( \la... | No |
Corollary 24.26. If \( {2}^{\left| A\right| } < \min A \) then \( \left| {\operatorname{pcf}A}\right| \leq {2}^{\left| A\right| } \) . | Proof. The number of generators is at most \( {2}^{\left| A\right| } \) . | No |
Corollary 24.27. If \( {\aleph }_{\omega } \) is strong limit then \( {2}^{{\aleph }_{\omega }} < {\aleph }_{{\left( {2}^{{\aleph }_{0}}\right) }^{ + }} \) . | Proof. Corollary 24.26, Corollary 24.20 and Theorem 24.18. | No |
Corollary 24.28. If \( {2}^{\left| A\right| } < \min A \) then \( \operatorname{pcf}A \) has a greatest element. | Proof. Assume that \( \operatorname{pcf}A \) does not have a greatest element. Then the set \( \left\{ {A - {B}_{\lambda } : \lambda \in \operatorname{pcf}A}\right\} \) has the finite intersection property, and so extends to an ultrafilter \( D \) . By (b), \( {B}_{\operatorname{cof}D} \in D \), a contradiction. | No |
Let \( {B}_{\lambda },\lambda \in \operatorname{pcf}A \), be generators of \( \operatorname{pcf}A \) . For every \( X \subset \) A there exists a finite set \( \left\{ {{\nu }_{1},\ldots ,{\nu }_{k}}\right\} \subset \operatorname{pcf}X \) such that \( X \subset {B}_{{\nu }_{1}} \cup \ldots \cup {B}_{{\nu }_{k}} \) . | Proof. Assume the contrary. Then \( \left\{ {X - {B}_{\nu } : \nu \in \operatorname{pcf}X}\right\} \) has the finite intersection property and there exists an ultrafilter \( D \) on \( X \) such that \( {B}_{\nu } \notin \) \( D \) for all \( \nu \in \operatorname{pcf}X \) . If \( \lambda = \operatorname{cof}D \) then ... | Yes |
Let \( \kappa \) be a regular uncountable cardinal, and let \( {\aleph }_{\eta } \) be a singular cardinal of cofinality \( \kappa \) such that \( {2}^{\kappa } < {\aleph }_{\eta } \) . Then there is a closed unbounded set \( C \subset \eta \) such that \( \max \left( {\operatorname{pcf}\left\{ {{\aleph }_{\alpha + 1} ... | Proof. Let \( {C}_{0} \) be any closed unbounded subset of \( \eta \) of order-type \( \kappa \) such that \( {2}^{\kappa } < {\aleph }_{{\alpha }_{0}} \) where \( {\alpha }_{0} = \min {C}_{0} \) . Let \( {A}_{0} = \left\{ {{\aleph }_{\alpha + 1} : \alpha \in {C}_{0}}\right\} \), let \( \lambda = {\aleph }_{\eta + 1} \... | Yes |
Lemma 24.32 (Localization). Let \( A \) be a set of regular cardinals such that \( {2}^{\left| \operatorname{pcf}A\right| } < \min A \), let \( X \subset \operatorname{pcf}A \) and let \( \lambda \in \operatorname{pcf}X \) . There exists a set \( W \subset X \) such that \( \left| W\right| \leq \left| A\right| \) and s... | Proof. First, since \( {2}^{\left| X\right| } < \min X \), there exist generators for \( \operatorname{pcf}X \), and in particular there exists a set \( Y \subset X \) with \( \max \left( {\operatorname{pcf}Y}\right) = \lambda \) . Let \( \bar{A} = \operatorname{pcf}A \) . By (24.7)(vii) we have \( \operatorname{pcf}\b... | Yes |
Theorem 24.33 (Shelah). If \( {\aleph }_{\omega } \) is a strong limit cardinal then \( {2}^{{\aleph }_{\omega }} < {\aleph }_{{\omega }_{4}} \) . | Proof. Let us assume that \( {\aleph }_{\omega } \) is strong limit. We already know, by Corollary 24.27, that \( {2}^{{\aleph }_{\omega }} = \max \operatorname{pcf}{\left\{ {\aleph }_{n}\right\} }_{n = 0}^{\infty } < {\aleph }_{{\aleph }_{\omega }} \) . We shall prove that\n\n\[ \max \operatorname{pcf}{\left\{ {\aleph... | Yes |
Lemma 24.34. There exists an ordinal function on \( P\left( \vartheta \right) \) with the following properties:\n\n(24.18)\n\n(i) If \( X \subset Y \) then \( F\left( X\right) \leq F\left( Y\right) \) .\n\n(ii) For every limit ordinal \( \eta < \vartheta \) of uncountable cofinality there is a closed unbounded set \( C... | Proof. Let \( X \subset \vartheta \) and consider the set \( A = \left\{ {{\aleph }_{\xi + 1} : \xi \in X}\right\} \) . As \( {2}^{\left| A\right| } = {\aleph }_{k} \) for some finite \( k,\max \left( {\operatorname{pcf}A}\right) \) exists and is equal to some \( {\aleph }_{\gamma + 1} \) . We define \( F\left( X\right... | Yes |
Lemma 25.2. Let \( n \geq 1 \) . (i) If \( A, B \) are \( {\sum }_{n}^{1}\left( a\right) \) relations, then so are \( \exists {xA}, A \land B, A \vee B,\exists {mA} \) , \( \forall {mA} \) . | Proof. We prove the lemma for \( n = 1 \) ; the general case follows by induction. Moreover, clauses (ii)-(v) follow from (i). First, let \( A \in {\sum }_{1}^{1}\left( a\right) \) and let us show that \( \exists {xA} \) is \( {\sum }_{1}^{1}\left( a\right) \) . We have \[ \left( {x, y}\right) \in A \leftrightarrow \ex... | Yes |
Theorem 25.3 (Normal Form for \( {\Pi }_{1}^{1} \) ). A set \( A \subset \mathcal{N} \) is \( {\Pi }_{1}^{1} \) if and only if there exists a recursive mapping \( x \mapsto T\left( x\right) \) such that each \( T\left( x\right) \) is a sequential tree, and | \[ x \in A\text{if and only if}T\left( x\right) \text{is well-founded.} \] | Yes |
Theorem 25.4 (Mostowski’s Absoluteness). If \( P \) is a \( {\mathbf{\sum }}_{1}^{1} \) property then \( P \) is absolute for every transitive model that is adequate for \( P \) . | Proof. \ | No |
Lemma 25.6. If \( S \) and \( T \) are well-founded trees and \( \parallel S\parallel \leq \parallel T\parallel \) then there exists an order-preserving map \( f : S \rightarrow T \) . | Proof. By induction on \( \parallel T\parallel \) . For each \( \langle a\rangle \in S,\parallel S/\langle a\rangle \parallel < \parallel S\parallel \leq \parallel T\parallel \) and there exists a \( {t}_{a} \neq \varnothing \) such that \( \parallel S/\langle a\rangle \parallel \leq \begin{Vmatrix}{T/\left\langle {t}_... | Yes |
Lemma 25.9. The sets WF and WO are \( {\Pi }_{1}^{1} \) . | Proof. We prove in some detail that WF is \( {\Pi }_{1}^{1}.{E}_{x} \) is well-founded if and only if there is no \( z : \mathbf{N} \rightarrow \mathbf{N} \) such that \( z\left( {k + 1}\right) {E}_{x}z\left( k\right) \) for all \( k \) . Thus\n\n\[ x \in \mathrm{{WF}} \leftrightarrow \forall z\exists k\neg z\left( {k ... | Yes |
For each \( \alpha < {\omega }_{1} \), the sets \[ {\mathrm{{WF}}}_{\alpha } = \{ x \in \mathrm{{WF}} : \parallel x\parallel \leq \alpha \} ,\;{\mathrm{{WO}}}_{\alpha } = \{ x \in \mathrm{{WO}} : \parallel x\parallel \leq \alpha \} \] are Borel sets. | Note that the set \( \left\{ {\left( {x, n}\right) : n \in \operatorname{field}\left( {E}_{x}\right) }\right\} \) is arithmetical (and hence Borel). Let us prove the lemma first for \( {\mathrm{{WO}}}_{\alpha } \). For each \( \alpha < {\omega }_{1} \), let \[ {B}_{\alpha } = \left\{ {\left( {x, n}\right) : {E}_{x}}\ri... | Yes |
Corollary 25.11. The sets \( \{ x \in \mathrm{{WF}} : \parallel x\parallel = \alpha \} \) and \( \{ x \in \mathrm{{WF}} : \parallel x\parallel < \alpha \} \) are Borel (similarly for WO). | Proof. \( \{ x \in \mathrm{{WF}} : \parallel x\parallel < \alpha \} = \mathop{\bigcup }\limits_{{\beta < \alpha }}{\mathrm{{WF}}}_{\beta } \) . | Yes |
If \( C \) is a \( {\mathbf{\Pi }}_{1}^{1} \) set, then there exists a continuous function \( f : \mathcal{N} \rightarrow \mathcal{N} \) such that \( C = {f}_{-1}\left( \mathrm{{WF}}\right) \), and there exists a continuous function \( g : \mathcal{N} \rightarrow \mathcal{N} \) such that \( C = {g}_{-1}\left( \mathrm{{... | We shall give the proof for WF; the proof for WO is similar. Let \( T \subset {\operatorname{Seq}}_{2} \) be such that\n\n\[ x \in C \leftrightarrow T\left( x\right) \text{is well-founded.} \]\n\nLet \( \left\{ {{t}_{0},{t}_{1},\ldots ,{t}_{n},\ldots }\right\} \) be an enumeration of the set Seq. For each \( x \in \mat... | Yes |
Corollary 25.13. WF is not \( {\mathbf{\sum }}_{1}^{1} \) ; WO is not \( {\mathbf{\sum }}_{1}^{1} \) . | Proof. Otherwise every \( {\mathbf{\Pi }}_{1}^{1} \) set would be the inverse image by a continuous function of an analytic set and hence analytic; however, there are \( {\mathbf{\Pi }}_{1}^{1} \) sets that are not analytic. | Yes |
Corollary 25.14 (Boundedness Lemma). If \( B \subset \mathrm{{WO}} \) is \( {\mathbf{\sum }}_{1}^{1} \), then there is an \( \alpha < {\omega }_{1} \) such that \( \parallel x\parallel < \alpha \) for all \( x \in B \) . | Proof. Otherwise we would have\n\n\[ \mathrm{{WO}} = \{ x \in \mathcal{N} : \exists z\left( {z \in B \land \parallel x\parallel \leq \parallel z\parallel }\right) \} .\n\]\n\nHence \( \parallel x\parallel \leq \parallel z\parallel \) for \( x, z \in \mathcal{N} \) means: Either \( z \notin \) WO or \( \parallel x\paral... | No |
Every \( {\mathbf{\Pi }}_{1}^{1} \) set is the union of \( {\aleph }_{1} \) Borel sets. | Proof. If \( C \) is \( {\mathbf{\Pi }}_{1}^{1} \), then \( C = {f}_{-1}\left( \mathrm{{WF}}\right) \) for some continuous \( f \) . But \( \mathrm{{WF}} = \) \( \mathop{\bigcup }\limits_{{\alpha < {\omega }_{1}}}{\mathrm{{WF}}}_{\alpha } \), and hence\n\n\[ C = \mathop{\bigcup }\limits_{{\alpha < {\omega }_{1}}}{f}_{-... | Yes |
Every \( {\sum }_{1}^{1} \) set is the union of \( {\aleph }_{1} \) Borel sets. | Let \( A \) be a \( {\mathbf{\sum }}_{1}^{1} \) set. Let \( T \subset {\operatorname{Seq}}_{2} \) be a tree such that \( A = p\left\lbrack T\right\rbrack \) . We prove by induction on \( \alpha \) that for each \( t \in {Seq} \) and every \( \alpha < {\omega }_{1} \), the set\n\n(25.16)\n\n\[ \{ x \in \mathcal{N} : \pa... | Yes |
Theorem 25.19 (Sierpiński). Every \( {\mathbf{\sum }}_{2}^{1} \) set is the union of \( {\aleph }_{1} \) Borel sets. | Proof. Let \( A \) be a \( {\mathbf{\sum }}_{2}^{1} \) set. By Theorem 25.18 there is a tree \( T \) on \( \omega \times {\omega }_{1} \) such that \( A = p\left\lbrack T\right\rbrack \) . For each \( \gamma < {\omega }_{1} \) let \( {T}^{\gamma } = \{ \left( {s, h}\right) \in T : h \in \operatorname{Seq}\left( \gamma ... | Yes |
Theorem 25.20 (Shoenfield’s Absoluteness Theorem). Every \( {\sum }_{2}^{1}\left( a\right) \) relation and every \( {\Pi }_{2}^{1}\left( a\right) \) relation is absolute for all inner models \( M \) of \( \mathrm{{ZF}} + \) DC such that \( a \in M \) . In particular, \( {\mathbf{\sum }}_{2}^{1} \) and \( {\mathbf{\Pi }... | Proof. Let \( a \in \mathcal{N} \) and let \( A \) be a \( {\sum }_{2}^{1}\left( a\right) \) subset of \( \mathcal{N} \) ; let \( A = \{ x : A\left( x\right) \} \) where \( A\left( x\right) \) is a \( {\sum }_{2}^{1}\left( a\right) \) property. Let \( M \) be an inner model of \( \mathrm{{ZF}} + \mathrm{{DC}} \) such t... | Yes |
Lemma 25.22. Let \( S \) be a set of countable ordinals such that the set \( A = \) \( \{ x \in \mathrm{{WO}} : \parallel x\parallel \in S\} \) is \( {\sum }_{2}^{1} \). Then \( S \) is constructible. (And more generally, if A is \( {\sum }_{2}^{1}\left( a\right) \), then \( S \in L\left\lbrack a\right\rbrack \) .) | Proof. Let \( A\left( x\right) \) be the \( {\sum }_{2}^{1} \) property such that \( A = \{ x : A\left( x\right) \} \). For each countable ordinal \( \alpha \), let \( {P}_{\alpha } \) be the notion of forcing that collapses \( \alpha \) ; i.e., the elements of \( {P}_{\alpha } \) are finite sequences of ordinals less ... | Yes |
Lemma 25.25. A set \( A \subset \mathcal{N} \) is \( {\sum }_{2}^{1} \) if and only if it is \( {\sum }_{1} \) over \( \left( {{HC}, \in }\right) \) . | Proof. If \( A \) is \( {\sum }_{1} \) over \( {HC} \), there exists a \( {\sum }_{0} \) formula \( \varphi \) such that\n\n\[ x \in A \leftrightarrow {HC} \vDash \exists {u\varphi }\left( {u, x}\right) \leftrightarrow \left( {\exists u \in {HC}}\right) {HC} \vDash \varphi \left\lbrack {u, x}\right\rbrack .\n\]\n\nSinc... | Yes |
Theorem 25.26 (Gödel). The set of all constructible reals is a \( {\sum }_{2}^{1} \) set. The ordering \( { < }_{L} \) is a \( {\sum }_{2}^{1} \) relation. | We proved in Chapter 13 that \ | No |
Lemma 25.27. The following relation \( R \) on \( \mathcal{N} \) is \( {\sum }_{2}^{1} \) :\n\n\[ \n\left( {z, x}\right) \in R \leftrightarrow \left\{ {{z}_{n} : n \in \mathbf{N}}\right\} = \left\{ {y : y{ < }_{L}x}\right\} .\n\] | Proof. Since the relation \( \left\{ {{z}_{n} : n \in \mathbf{N}}\right\} \subset \left\{ {y : y{ < }_{L}x}\right\} \) is clearly \( {\sum }_{2}^{1} \), it suffices to show that\n\n(25.26)\n\n\[ \n\forall y{ < }_{L}x\exists n\left( {y = {z}_{n}}\right)\n\]\n\nis \( {\sum }_{2}^{1} \) . There is a sentence \( \Theta \) ... | Yes |
If \( V = L \) then there exists an uncountable \( {\sum }_{2}^{1} \) set without a perfect subset. | Proof. Let\n\n\[ x \in A \leftrightarrow x \in \mathrm{{WO}} \land \forall y{ < }_{L}x\left( {\neg \parallel y\parallel = \parallel x\parallel }\right) .\n\]\n\nThe set \( A \) is uncountable: \( A \) is a subset of WO and for every \( \alpha < {\omega }_{1} \) there is exactly one \( x \) in \( A \) such that \( \para... | Yes |
Lemma 25.30. If \( {0}^{\sharp } \) exists then \( {0}^{\sharp } \) is a \( {\Delta }_{3}^{1} \) real, and the singleton \( \left\{ {0}^{\sharp }\right\} \) is \( a{\Pi }_{2}^{1} \) set. | Proof. We identify \( {0}^{\sharp } \) with the set of Gödel numbers of the sentences in \( {0}^{\sharp } \) . We claim that the property \( \sum = {0}^{\sharp } \) is \( {\Pi }_{1} \) over \( \left( {{HC}, \in }\right) \), and therefore \( {\Pi }_{2}^{1} \) . We use the description (18.24) of \( {0}^{\sharp } \) and n... | Yes |
Theorem 25.32. For every \( {\Pi }_{1}^{1} \) set \( A \) there exists a norm \( \varphi \) on \( A \) with the property that there exist a \( {\Pi }_{1}^{1} \) relation \( P\left( {x, y}\right) \) and a \( {\sum }_{1}^{1} \) relation \( Q\left( {x, y}\right) \) such that for every \( y \in A \) and all \( x \) ,\n\n\[... | Proof. Let \( A \) be a \( {\Pi }_{1}^{1} \) and let \( T \) be a recursive tree on \( \omega \times \omega \) such that\n\n\[ A\left( x\right) \leftrightarrow T\left( x\right) \text{is well-founded.} \]\n\nFor each \( x \in A \) let \( \varphi \left( x\right) = \parallel T\left( x\right) \parallel \) be the height of ... | Yes |
Theorem 25.36 (Kondô). Every \( {\Pi }_{1}^{1} \) relation \( A \subset \mathcal{N} \times \mathcal{N} \) is uniformized by a \( {\Pi }_{1}^{1} \) function. | Proof. We give a proof of the following statement that easily generalizes to a proof of Kondô’s Theorem: If \( A \) is a nonempty \( {\Pi }_{1}^{1} \) subset of \( \mathcal{N} \) then there exists an \( a \in A \) such that \( \{ a\} \) is \( {\Pi }_{1}^{1} \n\nThus let \( A \) be a nonempty \( {\Pi }_{1}^{1} \) subset... | Yes |
If \( V = L \) then there exists an uncountable \( {\Pi }_{1}^{1} \) set without a perfect subset. | Proof. Let \( A \) be a \( {\sum }_{2}^{1} \) set without a perfect subset (by 25.29). Now \( A \) is the projection of some \( {\Pi }_{1}^{1} \) set \( B \subset {\mathcal{N}}^{2} \) . By the Uniformization Theorem, \( B \) has a \( {\Pi }_{1}^{1} \) subset \( f \) that is a function and has the same projection \( A \... | Yes |
Theorem 25.38. The following are equivalent:\n\n(i) For every \( a \subset \omega ,{\aleph }_{1}^{L\left\lbrack a\right\rbrack } \) is countable.\n\n(ii) Every uncountable \( {\mathbf{\Pi }}_{1}^{1} \) set contains a perfect subset.\n\n(iii) Every uncountable \( {\mathbf{\sum }}_{2}^{1} \) set contains a perfect subset... | Proof. Obviously, (iii) implies (ii). In order to show that (i) implies (iii), let us assume (i) and let \( A \) be an uncountable \( {\mathbf{\sum }}_{2}^{1} \) set. Let \( a \in \mathcal{N} \) be such that \( A \in {\sum }_{2}^{1}\left( a\right) \) . Since \( {\aleph }_{1}^{L\left\lbrack a\right\rbrack } \) is counta... | Yes |
Theorem 25.39 (Mansfield). If \( < \) is a \( {\sum }_{2}^{1} \) well-ordering of \( \mathcal{N} \) then every real is constructible. More generally, if \( < \) is \( {\sum }_{2}^{1}\left( a\right) \) then \( \mathcal{N} \subset L\left\lbrack a\right\rbrack \) . | Proof. Let \( < \) be a \( {\sum }_{2}^{1} \) well-ordering of \( \mathcal{N} \) and let us assume that there is a nonconstructible real. Let \( {T}_{0} = \operatorname{Seq}\left( {\{ 0,1\} }\right) \), and let \( \mathbf{C} = \left\lbrack {T}_{0}\right\rbrack = \{ 0,1{\} }^{\omega } \) be the Cantor space. Let us cons... | No |
Lemma 25.40. If \( T \subset {T}_{0} \) is a constructible perfect tree and if \( f : T \rightarrow {T}_{0} \) is a constructible function such that \( {f}^{ * } \) is one-to-one, then there exist a constructible perfect tree \( U \subset T \) and a constructible \( g : U \rightarrow {T}_{0} \) such that \( {g}^{ * } \... | Proof of Lemma 25.40. Let \( T \subset {T}_{0} \) be a constructible tree and let \( f : T \rightarrow {T}_{0} \) be constructible, such that \( {f}^{ * } \) is one-to-one.\n\nSince \( T \) is perfect, there exists a constructible function \( h : T \rightarrow {T}_{0} \) such that \( {h}^{ * } : \left\lbrack T\right\rb... | No |
Lemma 25.44. The set BC of all Borel codes is \( {\Pi }_{1}^{1} \) . | Proof. Let us consider the following relation \( E \) on \( \mathcal{N} \) :\n\n(25.45) \( {xEy} \) if and only if either \( y\left( 0\right) = 0 \) and \( x = u\left( y\right) \) ,\n\nor \( y\left( 0\right) = 1 \) and \( x = {v}_{i}\left( y\right) \) for some \( i \in \omega \) .\n\nThe relation \( E \) is arithmetica... | Yes |
Lemma 25.46. The following properties (of codes) are absolute for all transitive models \( M \) of \( \mathrm{{ZF}} + \mathrm{{DC}} \) :\n\n\[ \n{A}_{e} = {A}_{c} \cup {A}_{d},\;{A}_{e} = {A}_{c} \cap {A}_{d},\n\]\n\n\[ \n{A}_{e} = \mathbf{R} - {A}_{c},\;{A}_{e} = {A}_{c}\bigtriangleup {A}_{d},\;{A}_{e} = \mathop{\bigc... | Proof. If \( {c}_{0},{c}_{1},\ldots ,{c}_{n},\ldots \) is a sequence of Borel codes in \( M \), let \( c \in \mathcal{N} \) be such that \( c\left( 0\right) = 1 \) and that \( {v}_{i}\left( c\right) = {c}_{i} \) for all \( i \in \omega \) . Clearly, \( c \) is a Borel code, \( c \in M \), and \( c \) codes (both in the... | Yes |
(i) If \( G \) is an \( M \) -generic ultrafilter on \( {\mathcal{B}}_{m} \), then there is a unique real number \( {x}_{G} \) such that for all \( B \in \mathcal{B} \), \[ {x}_{G} \in {B}^{ * } \leftrightarrow {\left\lbrack B\right\rbrack }_{m} \in G. \] | The same proof works for both (i) and (ii); let \( \left\lbrack B\right\rbrack \) denote \( {\left\lbrack B\right\rbrack }_{m} \) in case (i) and \( {\left\lbrack B\right\rbrack }_{c} \) in case (ii). First we claim that there is at most one real number \( x \) that satisfies \[ x \in {B}^{ * } \leftrightarrow \left\lb... | Yes |
Lemma 26.5. Let \( S \) be a Solovay set of reals over \( M \) . There exist Borel sets \( A \) and \( B \) such that \[ S \cap R\left( M\right) = A \cap R\left( M\right) \;\text{ and }\;S \cap C\left( M\right) = B \cap C\left( M\right) . \] | Proof. Let us prove the lemma for random reals. Let us consider the forcing language in \( M \) associated with \( {\mathcal{B}}_{m} \) . Let \( \dot{G} \) be the canonical name for a generic ultrafilter on \( {\mathcal{B}}_{m} \), and let \( \dot{a} \) be the canonical name for a random real; i.e., let \( \dot{a} \) b... | Yes |
Lemma 26.7. Let \( \left( {Q, < }\right) \) be a notion of forcing such that \( \left| Q\right| = \lambda > {\aleph }_{0} \) and such that \( Q \) collapses \( \lambda \) onto \( {\aleph }_{0} \), i.e., \[ \parallel \check{\lambda }\text{is countable}{\parallel }_{B\left( Q\right) } = 1\text{.} \] Then \( B\left( Q\rig... | Proof. Without loss of generality we may assume that \( \left( {Q, < }\right) \) is a separative partial ordering. We shall find a dense subset of \( Q \) isomorphic to \( {P}_{\lambda } \). Let \( B = B\left( Q\right) \), and let \( \dot{G} \) be the canonical name for the generic filter on \( Q \). Let \( \dot{f} \in... | Yes |
Corollary 26.8 (Kripke). If \( B \) is a complete Boolean algebra and \( \left| B\right| \leq \lambda \) then \( B \) embeds as a complete subalgebra of \( \operatorname{Col}\left( {{\aleph }_{0},\lambda }\right) \) . | Proof. Let \( B \) be a complete Boolean algebra, \( \left| B\right| \leq \lambda \) . The notion of forcing \( Q = {B}^{ + } \times {P}_{\lambda } \) has cardinality \( \lambda \) and collapses \( \lambda \) . By Lemma 26.7, \( B\left( Q\right) = \) \( \operatorname{Col}\left( {{\aleph }_{0},\lambda }\right) \) . In o... | Yes |
Lemma 26.9. Let \( B \) be a complete Boolean algebra, \( \left| B\right| = \lambda \) . Let \( C \) be a complete subalgebra of \( B \) such that \( \left| C\right| < \lambda \), and let \( {h}_{0} \) be an embedding of \( C \) in \( \operatorname{Col}\left( {{\aleph }_{0},\lambda }\right) \) . Then there exists an em... | Proof. Let \( D \) be the image of \( C \) under the embedding \( {h}_{0} \) . Let Col be an abbreviation for \( \operatorname{Col}\left( {{\aleph }_{0},\lambda }\right) \) ; let \( {\operatorname{Col}}^{C} \) and \( {\operatorname{Col}}^{D} \) denote, respectively, the \( \left( {{\aleph }_{0},\lambda }\right) \) - co... | Yes |
Let \( G \) be a generic filter on \( {P}_{\lambda } \) and let \( X \) be a set of ordinals in \( V\left\lbrack G\right\rbrack \) . Then either \( V\left\lbrack X\right\rbrack = V\left\lbrack G\right\rbrack \) or there exists a \( V\left\lbrack X\right\rbrack \) -generic filter \( H \) on \( {P}_{\lambda } \) such tha... | Proof. If \( \lambda \) is uncountable in \( V\left\lbrack X\right\rbrack \), then \( V\left\lbrack G\right\rbrack \) is a generic extension of \( V\left\lbrack X\right\rbrack \) by \( {\operatorname{Col}}^{V\left\lbrack X\right\rbrack }\left( {{\aleph }_{0},\kappa }\right) \), where \( \kappa = {\left| \lambda \right|... | Yes |
Corollary 26.11 (The Factor Lemma). Let \( G \) be a generic filter on the Lévy collapse \( P \), and let \( X \) be a countable set of ordinals in \( V\left\lbrack G\right\rbrack \) . Then there exists a \( V\left\lbrack X\right\rbrack \) -generic filter \( H \) on \( P \) such that \( V\left\lbrack X\right\rbrack \le... | Proof. For each \( \nu < \lambda \) we have a decomposition of \( P \) into \( {P}_{\nu } \times {P}^{\nu } \) where \( {P}_{\nu } = \{ p \in P : \operatorname{dom}p \subset \nu \times \omega \} \) and \( {P}^{\nu } = \{ p \in P : \operatorname{dom}p \subset \left( {\lambda - \nu }\right) \times \omega \} \) . Note tha... | Yes |
Theorem 26.12 (The Homogeneity of the Lévy Collapse). Let \( B = \) \( \operatorname{Col}\left( {{\aleph }_{0}, < \lambda }\right) \) . If \( A \) and \( {A}^{\prime } \) are isomorphic complete subalgebras of \( B \) such that \( \left| A\right| = \left| {A}^{\prime }\right| < \left| B\right| \) and if \( {\pi }_{0} \... | Proof. First we construct increasing sequences of complete subalgebras \( {A}_{0} \subset \) \( {A}_{1} \subset \ldots \subset {A}_{n} \subset \ldots \), and \( {A}_{0}^{\prime } \subset {A}_{1}^{\prime } \subset \ldots \subset {A}_{n}^{\prime } \subset \ldots \), as follows: We let \( {A}_{0} = A \) and \( {A}_{0}^{\p... | Yes |
Corollary 26.18. In \( M\left\lbrack G\right\rbrack \) every set of reals definable from a sequence of ordinals (and in particular, every projective set of reals) is Lebesgue measurable and has the property of Baire. | Proof. This follows from Lemmas 26.5 26.16, and 26.17. | No |
Corollary 26.21. If \( {\omega }_{1}^{L\left\lbrack a\right\rbrack } < {\omega }_{1} \), then every \( {\sum }_{2}^{1}\left( a\right) \) set of reals is Lebesgue measurable and has the Baire property. | Proof. Under the assumption, each \( L\left\lbrack a\right\rbrack \) has only countably many reals and hence only countably many Borel codes, and it follows that almost all reals are random over \( L\left\lbrack a\right\rbrack \) . Similarly for Cohen reals. | No |
Lemma 26.28. There is an \( X \) that accepts or rejects each of its finite subset. | Proof. Let \( {X}_{0} \) be such that \( {X}_{0} \) either accepts or rejects \( \varnothing \) (if no \( X \) accepts \( \varnothing \) then \( {X}_{0} = \omega \) rejects \( \varnothing \) ). Let \( {a}_{0} \) be the least element of \( {X}_{0} \) . Let \( {X}_{1} \subset X \) be such that \( {X}_{1} \) either accept... | Yes |
Lemma 26.29. There is a \( Y \) that either accepts \( \varnothing \) or rejects each of its finite subsets. | Proof. Let \( X \) be as in Lemma 26.28, and assume that it rejects \( \varnothing \) . We construct \( Y = \left\{ {{a}_{0},{a}_{1},\ldots }\right\} \subset X \) as follows: Assume we have constructed \( {a}_{0},\ldots ,{a}_{n - 1} \) such that \( X \) rejects each subset of \( \left\{ {{a}_{0},\ldots ,{a}_{n - 1}}\ri... | Yes |
Lemma 26.30. Every open set is Ramsey. | Proof. Let \( S \) be open, and let \( X \) be as in Lemma 26.29. If \( X \) accepts \( \varnothing \) then \( {\left\lbrack X\right\rbrack }^{\omega } = {\left\lbrack \varnothing, X\right\rbrack }^{\omega } \subset S \) . If \( X \) rejects each of its finite subsets, we claim that \( {\left\lbrack X\right\rbrack }^{\... | Yes |
Lemma 26.31. Every open set is completely Ramsey. | Proof. Let \( S \) be open and let \( \left( {s, A}\right) \) be arbitrary. Let \( f : \omega \rightarrow A \) be a one-toone increasing enumeration of \( A \), and for each \( X \in {\left\lbrack \omega \right\rbrack }^{\omega } \), let \( {f}^{ * }\left( X\right) = s \cup {f}^{\alpha }X \) . The function \( {f}^{ * }... | Yes |
Lemma 26.32. Every nowhere dense set is Ramsey null. | Proof. Let \( S \) be nowhere dense; we may also assume that \( S \) is closed. Let \( \left( {s, A}\right) \) be arbitrary. By Lemma 26.31 there is an \( H \subset A \) such that either \( {\left\lbrack s, H\right\rbrack }^{\omega } \subset S \) or \( {\left\lbrack s, H\right\rbrack }^{\omega } \cap S = \varnothing \)... | Yes |
Lemma 26.33. If \( S = \mathop{\bigcup }\limits_{{n = 0}}^{\infty }{S}_{n} \) and each \( {S}_{n} \) is Ramsey null then \( S \) is Ramsey null. | Proof. Let \( \left( {s, A}\right) \) be arbitrary. We construct an infinite \( H = \left\{ {{a}_{0},{a}_{1},{a}_{2},\ldots }\right\} \subset \) \( A \) as follows: Let \( {X}_{0} \subset A \) be such that \( {\left\lbrack s,{X}_{0}\right\rbrack }^{\omega } \cap {S}_{0} = \varnothing \), and let \( {a}_{0} \) be the le... | Yes |
Lemma 26.34. Let \( \sigma \) be a sentence of the forcing language and let \( \left( {s, A}\right) \) be a condition. Then there exists an infinite set \( B \subset A \) such that \( \left( {s, B}\right) \) decides \( \sigma \) . | Proof. Let \( {Q}^{ + } = \{ p : p \Vdash \sigma \} ,{Q}^{ - } = \{ p : p \Vdash \neg \sigma \} ,{S}^{ + } = \bigcup \left\{ {{\left\lbrack t, X\right\rbrack }^{\omega } : \left( {t, X}\right) \in }\right. \) \( \left. {Q}^{ + }\right\} \) and \( {S}^{ - } = \bigcup \left\{ {{\left\lbrack t, X\right\rbrack }^{\omega } ... | Yes |
Theorem 26.35 (Mathias). Let \( M \) be a transitive model of ZFC. An infinite set \( x \subset \omega \) is a Mathias real over \( M \) if and only if for every maximal almost disjoint family \( \mathcal{A} \in M \) of subsets of \( \omega \), there exists an \( X \in \mathcal{A} \) such that \( x - X \) is finite. | Proof. The condition is necessary: If \( \mathcal{A} \) is a maximal almost disjoint family then \( D = \{ \left( {s, A \smallsetminus s}\right) : s \in {\left\lbrack \omega \right\rbrack }^{ < \omega }, A \in \mathcal{A}\} \) is a predense set of forcing conditions, and it follows that if \( x \) is a Mathias real the... | No |
Lemma 26.36. For every infinite set \( A \subset \omega \) and for every finite \( s \subset \omega \) there exists an infinite set \( X \subset A \smallsetminus s \) such that \( X \) captures \( \left( {s, D}\right) \) . | Proof. We construct a sequence \( {Y}_{0} \supset {Y}_{1} \supset \ldots \supset {Y}_{n} \supset \ldots \) of infinite sets and a sequence \( {m}_{0} < {m}_{1} < \ldots < {m}_{n} < \ldots \) such that \( {m}_{n} = \min {Y}_{n} \), as follows: Let \( {Y}_{0} = A \smallsetminus s \) . Given \( {Y}_{n} \), we can find \( ... | Yes |
For every infinite \( A \subset \omega \) there exists an \( X \subset A \) such that for every \( s, X \smallsetminus s \) captures \( \left( {s, D}\right) \) . | By Lemma 26.36 there exist sets \( {X}_{s} \subset A \) such that for each \( s \) , \( {X}_{s} \) captures \( \left( {s, D}\right) \) . We construct \( {X}_{0} \supset {X}_{1} \supset \ldots \supset {X}_{n} \supset \ldots \) and \( {m}_{0} < {m}_{1} < \ldots < {m}_{n} < \ldots \) such that \( {m}_{n} = \min {X}_{n} \)... | Yes |
If \( x \) is a Mathias real over \( M \) and \( y \subset x \) is infinite, then \( y \) is Mathias over \( M \) . | If \( y \) is an infinite subset of \( x \) then by Corollary 26.38, \( y \) is a Mathias real over \( M\left\lbrack u\right\rbrack \) . Since \( y \subset A \) and \( \left( {\varnothing, A}\right) \Vdash \varphi \left( \dot{x}\right) \), we have \( M\left\lbrack u\right\rbrack \left\lbrack y\right\rbrack \vDash \varp... | No |
Theorem 26.46 (Bartoszyński, Raisonnier-Stern). (i) \( \operatorname{add}\left( \mathrm{{LM}}\right) \leq \operatorname{add}\left( \mathrm{{BP}}\right) \) . (ii) \( \operatorname{cof}\left( \mathrm{{BP}}\right) \leq \operatorname{cof}\left( \mathrm{{LM}}\right) \) . | It is no accident that each result is accompanied by a dual version: There is a general theory that explains this duality. (For details, see Bartoszyński's Handbook article.) For instance, consider Theorem 26.46. Both (i) and (ii) can be proved from this general result (see Exercise 26.11): | No |
Theorem 26.49. add(LM) is the least cardinality of a family \( F \subset {\omega }^{\omega } \) such that\n\n(26.32)\n\n\[ \forall \varphi \in S\exists f \in F{\exists }^{\infty }{nf}\left( n\right) \notin \varphi \left( n\right) ,\]\n\nwhere \( S \) is the set of all functions \( \varphi : \omega \rightarrow {\left\lb... | See Exercise 26.13 for the proof of add(LM) \( \leq \operatorname{cov}\left( \mathrm{{BP}}\right) \) . | No |
Lemma 26.50. If there exists a nonconstructible real, then:\n\n(i) \( \mathbf{R} \cap L \) is either null or not Lebesgue measurable.\n\n(ii) \( \mathbf{R} \cap L \) is either meager or does not have the Baire property. | Proof. (i) Let \( S \) be the set of all constructible reals in the unit interval \( \left\lbrack {0,1}\right\rbrack \) . Let \( a \) be a nonconstructible real. For each \( n > 0 \), let \( {S}_{n} = \{ x + \left( {a/n}\right) : x \in S\} \) . The sets \( {S}_{n} \) are pairwise disjoint, \( \mu \left( {S}_{n}\right) ... | Yes |
Example 26.52 (A model where \( {2}^{{\aleph }_{0}} > {\aleph }_{1} \) and the set of all constructible reals does not have the property of Baire). Let \( \lambda \) be a regular uncountable ordinal and let \( P \) be the notion of forcing that adjoins \( \lambda \) Cohen reals: A condition is a finite \( 0 - 1 \) func... | Let us consider the generic extension of the constructible universe by \( P \) . In \( L\left\lbrack G\right\rbrack ,{2}^{{\aleph }_{0}} = \lambda \) . We shall show that in \( L\left\lbrack G\right\rbrack \) the set of all constructible reals does not have the Baire property.\n\nIn view of Lemma 26.50, it suffices to ... | Yes |
Lemma 26.53. If \( L\left\lbrack G\right\rbrack \vDash \mathbf{R} \cap L \) is meager, then there exists a countable \( S \subset \lambda \) (in \( L \) ) such that \( L\left\lbrack {G}_{S}\right\rbrack \vDash \mathbf{R} \cap L \) is meager. | Proof. Let \( {I}_{k}, k \in \mathbf{N} \), be an enumeration of all open intervals with rational endpoints. If \( L\left\lbrack G\right\rbrack \vDash \mathbf{R} \cap L \) is meager, then there exists a sequence \( \left\langle {U}_{n}\right. \) : \( n \in \mathbf{N}\rangle \in L\left\lbrack G\right\rbrack \) such that... | Yes |
Lemma 26.54. If \( P \) is a countable notion of forcing in \( L \) and if \( G \) is an \( L \)-generic filter on \( P \), then \( L\left\lbrack G\right\rbrack \vDash \mathbf{R} \cap L \) is not meager. | Proof. It suffices to show that if \( \left\langle {{U}_{n} : n \in \mathbf{N}}\right\rangle \) is (in \( L\left\lbrack G\right\rbrack \) ) a sequence of dense open sets of reals, then there is a constructible real \( a \) such that \( a \in \mathop{\bigcap }\limits_{{n = 0}}^{\infty }{U}_{n} \) . Let \( {\dot{U}}_{n} ... | Yes |
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