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Theorem 27.7 (Jensen). If \( V = L \) then for every infinite cardinal \( \kappa \) there exists a \( {\kappa }^{ + } \) -Suslin tree. | When \( \kappa \) is regular, the proof is a straightforward generalization of the construction of a Suslin tree using \( \diamond \) : instead we use \( \diamond \left( {E}_{\kappa }^{{\kappa }^{ + }}\right) \) . We construct a tree by induction on levels. At limit levels \( \alpha \) of cofinality \( < \kappa \) we e... | Yes |
Theorem 27.9 (Silver [1971c]). If there exists an inaccessible cardinal then there is a generic extension in which there are no Kurepa trees. | Proof. Let \( \lambda \) be an inaccessible cardinal. Let \( \left( {P, < }\right) \) be the Lévy collapse of \( \lambda \) to \( {\aleph }_{2} \) : forcing conditions are countable functions \( p \) on subsets of \( \lambda \times {\omega }_{1} \) such that \( p\left( {\alpha ,\xi }\right) < \alpha \) for every \( \le... | Yes |
Lemma 27.10. If \( P \) is an \( {\aleph }_{0} \) -closed notion of forcing and \( T \) is an \( {\omega }_{1} \) -tree in the ground model such that every level of \( T \) is countable, then \( T \) has no new branches in \( V\left\lbrack G\right\rbrack \) . | Proof. Assume that \( T \) has a branch \( b \in V\left\lbrack G\right\rbrack \) that is not in \( V \) ; since \( V\left\lbrack G\right\rbrack \) has no new countable sets, \( b \) has length \( {\omega }_{1} \) . There is a name \( \dot{b} \) for \( b \) and a condition \( {p}_{0} \in G \) such that \( {p}_{0} \Vdash... | Yes |
Theorem 27.11 (Hajnal). If \( V = L \) then there is no \( {\omega }_{2}{nd} \) canonical function on \( {\omega }_{1} \) . | Assume \( V = L \), and assume that there is an \( {\omega }_{2}\mathrm{{nd}} \) canonical function. This statement can be expressed in \( {L}_{{\omega }_{2}} \) :\n\n\[ \left( {\exists f : {\omega }_{1} \rightarrow {\omega }_{1}}\right) \forall \eta \left( {{f}_{\eta } < f}\right) \text{and} \]\n\n\[ \left( {\forall s... | Yes |
Theorem 27.12 (Jech-Shelah). There is a generic extension of \( L \) in which the \( {\omega }_{2}{nd} \) canonical function exists. | The model is obtained by first adding (by forcing with countable conditions) an increasing sequence \( \left\langle {{f}_{\alpha } : \alpha \leq {\omega }_{2}}\right\rangle \) of ordinal functions from \( {\omega }_{1} \) into \( {\omega }_{1} \) . Then one uses an iterated forcing, with countable supports of length \(... | Yes |
Lemma 28.2. For each \( \alpha ,{Q}_{\alpha + 1} \) is dense in \( {P}_{\alpha + 1} \), and \( {Q}_{\alpha + 1} \times {Q}_{\alpha + 1} \) is dense in \( {P}_{\alpha + 1} \times {P}_{\alpha + 1} \) . | Proof. It suffices to show that below each \( p \in {P}_{\alpha } \) there is some \( q \in {Q}_{\alpha + 1} \) . If \( p \in {P}_{\alpha } \), the \( p = {p}_{n}^{\alpha } \) for some \( n \), and \( \mathcal{F}\left( {T}_{n}\right) \subset {T}_{n}\left( \varnothing \right) = p \) . | No |
Lemma 28.3. For each \( \alpha \), if \( X \in {\mathcal{X}}_{\alpha } \) is predense in \( {P}_{\alpha } \), then \( X \) is predense in \( {P}_{\alpha + 1} \) ; if \( Y \in {\mathcal{Y}}_{\alpha } \) is predense in \( {P}_{\alpha } \times {P}_{\alpha } \), then \( Y \) is predense in \( {P}_{\alpha + 1} \times {P}_{\... | Proof. Let \( X \in {\mathcal{X}}_{\alpha } \) be predense in \( {P}_{\alpha } \) ; we have to show that for each \( p \upharpoonright n \in \) \( {Q}_{\alpha + 1} \) there is a stronger \( q \in {Q}_{\alpha + 1} \) such that \( q \leq x \) for some \( x \in X \) . Let \( p = \mathcal{F}\left( {T}_{n}\right) \) and let... | Yes |
Lemma 28.4. \( P \times P \) satisfies the countable chain condition (and hence \( P \) also satisfies the countable chain condition). | Proof. Here we use \( \diamondsuit \) . Let us assume that \( Y \subset P \times P \) is a maximal incompatible set of conditions in \( P \times P \) and that \( Y \) is uncountable. Since each \( {P}_{\alpha } \) is countable, it is easy to see that the set of all \( \alpha < {\omega }_{1} \) such that \( \tau \left( ... | Yes |
(i) If \( a \in \{ 0,1{\} }^{\omega } \), then \( a \) is \( P \) -generic over \( L \) if and only if for every \( \alpha < {\omega }_{1} \) there is some \( n \in \mathbf{N} \) such that \( a \) is a branch in \( \mathcal{F}\left( {T}_{n}^{\alpha }\right) \) . | Proof. (i) Let \( a \) be \( P \) -generic and let \( \alpha < {\omega }_{1} \) . Since \( {Q}_{\alpha + 1} \) is dense in \( {P}_{\alpha + 1} \) and because \( {Q}_{\alpha + 1} \in {\mathcal{X}}_{\alpha + 1},{Q}_{\alpha + 1} \) is predense in \( P \) . By the genericity of \( a \) , there exists a \( q \in {Q}_{\alpha... | Yes |
Corollary 28.7. If \( a \) is \( P \) -generic over \( L \), then \( L\left\lbrack a\right\rbrack \vDash a \) is the only \( P \) -generic over \( L \) . | Proof. If \( a \neq b \) and if both \( a \) and \( b \) are \( P \) -generic over \( L \), then by the Product Lemma, \( b \) is a \( P \) -generic over \( L\left\lbrack a\right\rbrack \) and hence \( b \notin L\left\lbrack a\right\rbrack \) . | Yes |
Lemma 28.8. The set \( H = \{ a : a \) is \( P \) -generic over \( L\} \) is \( {\Pi }_{1} \) over \( {HC} \) . | Proof. It follows from the construction of \( P \) that the function \( \alpha \mapsto \left\langle {T}_{n}^{\alpha }\right. \) : \( n \in \omega \rangle \) is \( {\Delta }_{1} \) over \( {L}_{{\omega }_{1}^{L}} \) . Since \( {L}_{{\omega }_{1}^{L}} \) is a \( {\sum }_{1} \) set over \( {HC} \), the function is \( {\De... | Yes |
If \( a \) is a \( P \) -generic over \( L \) and \( A = \{ n \in \mathbf{N} : a\left( n\right) = 1\} \) , then \( L\left\lbrack a\right\rbrack \vDash A \) is a \( {\Delta }_{3}^{1} \) subset of \( \mathbf{N} \) . | We have in \( L\left\lbrack a\right\rbrack \)\n\n\[ \n n \in A \leftrightarrow \left( {\exists a \in \mathcal{N}}\right) \left( {a \in H\text{ and }a\left( n\right) = 1}\right) \leftrightarrow \left( {\forall a \in \mathcal{N}}\right) \left( {a \in H \rightarrow a\left( n\right) = 1}\right) .\n\]\n\nSince \( H \) is \(... | Yes |
Lemma 28.11. There exists some \( g : \omega \rightarrow \{ 0,1\} \) such that \( T\left( g\right) \) contains a perfect subtree. | Proof. Let us assume that no \( T\left( g\right) \) has a perfect subtree. Then by (28.8) there exists, for each \( g : \omega \rightarrow \{ 0,1\} \), a function \( {h}_{g} : T\left( g\right) \rightarrow {\omega }_{3} \) such that \( {h}_{g}\left( s\right) \geq {h}_{g}\left( t\right) \) whenever \( s \subset t \), and... | Yes |
Theorem 28.12 (Shelah). If \( r \) is a Cohen real over \( V \) then in \( V\left\lbrack r\right\rbrack \) there exists a Suslin tree. | Proof. We start with an alternative construction of an Aronszajn tree, a modification of the construction in Theorem 9.16. | No |
Lemma 28.13. There exists an \( {\omega }_{1} \) -sequence of functions \( \left\langle {{e}_{\alpha } : \alpha < {\omega }_{1}}\right\rangle \) such that\n\n(i) \( {e}_{\alpha } \) is a one-to-one function from \( \alpha \) into \( \omega \), for each \( \alpha < {\omega }_{1} \) ;\n\n(ii) for all \( \alpha < \beta < ... | Proof. Exercise 28.1 (or see Kunen [1980], Theorem II.5.9). | No |
Theorem 28.14 (Laver). Assuming GCH there is a generic extension \( V\left\lbrack G\right\rbrack \) in which \( {2}^{{\aleph }_{0}} = {\aleph }_{2} \) and Borel’s Conjecture holds. | Laver’s proof uses the countable support iteration (of length \( {\omega }_{2} \) ) of a forcing notion that adds a Laver real. We shall now describe this forcing. (Subsequently, Laver proved that an iteration of Mathias forcing also yields Borel's Conjecture).\n\nDefinition 28.15. A tree \( p \subset \) Seq is a Laver... | Yes |
Lemma 28.17. If \( p \Vdash \dot{X} : \omega \rightarrow V \) then there exists a \( q{ \leq }_{0}p \) and a countable \( A \) such that \( q \Vdash \dot{X} \subset A \) . | Proof. Let \( {\left\{ {u}_{n}\right\} }_{n} \) be a sequence of natural numbers such that each number appears infinitely often. We shall construct a fusion sequence \( {\left\{ {p}_{n}\right\} }_{n} \) with \( {p}_{0} = p \), and finite sets \( {A}_{n} \) so that the fusion forces \( \dot{X} \subset \mathop{\bigcup }\... | Yes |
Lemma 28.19. Let \( p \Vdash {\varphi }_{1} \vee \ldots \vee {\varphi }_{k} \) . Then there exists some \( q{ \leq }_{0}p \) such that\n\n(28.20)\n\n\[ \exists i \leq {kq} \Vdash {\varphi }_{i} \] | Proof. Assume to the contrary that the lemma fails. Let \( s \) be the stem of \( p \) ; there are only finitely many \( a \in {S}^{p}\left( s\right) \) such that some \( q{ \leq }_{0}p \upharpoonright \left( {{s}^{ \frown }a}\right) \) satisfies (28.20). By removing the part of \( p \) above these finitely many nodes ... | Yes |
Lemma 28.21. Let \( p \) be a Laver tree with stem \( s \) and let \( \dot{x} \) be a name for a real in \( \left\lbrack {0,1}\right\rbrack \) . Then there exist a condition \( q{ \leq }_{0}p \) and a real \( u \) such that for every \( \varepsilon > 0 \) ,\n\n\[ q \upharpoonright \left( {s \cap a}\right) \Vdash \left|... | Proof. Let \( {\left\{ {t}_{n}\right\} }_{n} \) be an enumeration of \( \left\{ {{s}^{ \frown }a : a \in {S}^{p}\left( s\right) }\right\} \) . For each \( n \) we find, by Lemma 28.19, a condition \( {q}_{n}{ \leq }_{0}p \upharpoonright {t}_{n} \) and an interval \( {J}_{n} = \left\lbrack {\frac{m}{n},\frac{m + 1}{n}}\... | Yes |
Lemma 28.22. Let \( p \) be a condition with stem \( s \) and let \( \left\langle {{\dot{x}}_{n} : n < \omega }\right\rangle \) be a sequence of names for reals. Then there exist a condition \( q{ \leq }_{0}p \) and a set of reals \( \left\{ {{u}_{t} : t \in q, t \supset s}\right\} \) such that for every \( \varepsilon... | Proof. Using Lemma 28.21 we get \( {p}_{1}{ \leq }_{0}p \) and \( {u}_{s} \) . For every immediate successor \( t \) of \( s \) in \( {p}_{1} \), we get \( {q}_{t}{ \leq }_{0}{p}_{1} \mid t \) and \( {u}_{t} \), and let \( {p}_{2} = \mathop{\bigcup }\limits_{t}{q}_{t} \) . By repeating this argument, we build a fusion ... | Yes |
Theorem 28.23 (Silver). If there exists no \( {\aleph }_{2} \) -Aronszajn tree then \( {\aleph }_{2} \) is a weakly compact cardinal in \( L \) . | Proof. If \( {\aleph }_{2} \) is a successor cardinal in \( L \), then there exists some \( A \subset {\omega }_{1} \) such that \( {\aleph }_{1}^{L\left\lbrack A\right\rbrack } = {\aleph }_{1} \) and \( {\aleph }_{2}^{L\left\lbrack A\right\rbrack } = {\aleph }_{2} \) . In \( L\left\lbrack A\right\rbrack ,{2}^{{\aleph ... | Yes |
Theorem 29.1 (Hindman). If \( N \) is partitioned into finitely many pieces then one of the pieces \( A \) contains an infinite set \( H \) such that \( {a}_{1} + \ldots + {a}_{n} \in A \) whenever \( {a}_{1},\ldots ,{a}_{n} \) are distinct members of \( H \) . | For the proof, we introduce the concept of an idempotent ultrafilter. If \( U \) and \( V \) are ultrafilters on \( \mathbf{N} \), let\n\n(29.1)\n\n\[ U + V = \{ X \subset \mathbf{N} : \{ m \in \mathbf{N} : X - m \in V\} \in U\} \]\n\nwhere \( X - m = \{ n : m + n \in X\} \) . See Exercises 29.1 and 29.2 for an alterna... | No |
Lemma 29.2. There exists a nonprincipal ultrafilter \( U \) on \( \mathbf{N} \) such that \( U + \) \( U = U \) . | An ultrafilter \( U \) such that \( U + U = U \) is idempotent. While Glazer’s Lemma can be proved directly, it can be deduced from a more general result on topological semigroups. Let \( \beta \mathbf{N} \) be the space of all ultrafilters on \( \mathbf{N} \), the Stone-Čech compactification on \( \mathbf{N} \) . The ... | No |
Theorem 29.3 (van der Waerden). If \( N \) is partitioned into finitely many pieces then one of the pieces contains arbitrarily long arithmetic progressions. | For the proof of Theorem 29.3, we fix an integer \( k \geq 1 \) and consider the space \( {\left( \beta \mathbf{N}\right) }^{k} \) . Let \( I \) be the set of all arithmetic progressions of length \( k \), and let \( \bar{I} \) be its closure in \( {\left( \beta \mathbf{N}\right) }^{k} \) . Arguments similar to those i... | No |
Theorem 29.4 (Hales-Jewett). If \( W \) is partitioned into finitely many pieces then there is a variable word \( x \in V \) whose all instances lie in the same piece of the partition. | We refer the reader to Todorčević's book for a proof using topological semigroups. | No |
Lemma 29.6. If \( {f}_{0} < {f}_{1} < \ldots < {f}_{n} < \ldots < {g}_{m} < \ldots < {g}_{1} < {g}_{0} \), then there exists an \( h \) between \( {\left\{ {f}_{n}\right\} }_{n} \) and \( {\left\{ {g}_{m}\right\} }_{m} \) . | Proof. For each \( k \) there exists an \( {n}_{k} \) such that for all \( n \geq {n}_{k},{m}_{k}\left( n\right) = \) \( \max \left\{ {{f}_{0}\left( n\right) ,\ldots ,{f}_{k}\left( n\right) }\right\} \leq \min \left\{ {{g}_{0}\left( n\right) ,\ldots ,{g}_{k}\left( n\right) }\right\} = {M}_{k}\left( n\right) \) . Choose... | Yes |
Theorem 29.8 (Todorčević). If OCA holds then \( \mathfrak{b} = {\aleph }_{2} \) . | First we show that under OCA, \( \mathfrak{b} > {\omega }_{1} \) . | No |
Theorem 29.13.\n\n(i) If \( \operatorname{cof}\left( {\omega }^{{\omega }_{1}}\right) < {2}^{{\aleph }_{1}} \) then \( {2}^{{\aleph }_{0}} \geq {\aleph }_{3} \) .\n\n(ii) If \( {2}^{{\aleph }_{0}} < {2}^{{\aleph }_{1}} \) and \( {2}^{{\aleph }_{0}} < {\aleph }_{{\omega }_{1}} \) then \( \operatorname{cof}\left( {\omega... | The theorem is a consequence of this lemma:\n\nLemma 29.14. If there e | No |
Lemma 29.15. There exists a \( \sigma \) -ideal \( I \) on \( {\omega }_{1} \) such that there exist \( {\aleph }_{2}I \) - disjoint functions from \( {\omega }_{1} \) into \( \omega \) . | Proof. We find such an \( I \) on \( {\omega }_{1} \times {\omega }_{1} \) : Let \( I = {I}_{0} \times {I}_{0} \) where \( {I}_{0} \) is the \( \sigma \) - ideal of all countable subsets of \( {\omega }_{1} \) (each \( X \in I \) is included in the union of \( \omega \) vertical lines and the set under the graph of a f... | No |
Lemma 29.16. If there exist \( {2}^{{\aleph }_{1}} \) almost disjoint functions from \( {\omega }_{1} \) into \( {\omega }_{2} \) then there exists a \( \sigma \) -ideal \( J \) on \( {\omega }_{1} \) such that there are \( {2}^{{\aleph }_{1}}J \) -disjoint functions from \( {\omega }_{1} \) into \( \omega \) . | Proof. We find such a \( J \) on \( {\omega }_{1} \times {\omega }_{1} \) : Let \( J = {I}_{0} \times I \) where \( {I}_{0} \) is the ideal of countable sets and \( I \) is the ideal given by Lemma 29.15. Let \( \left\{ {{g}_{\alpha } : \alpha < {2}^{{\aleph }_{1}}}\right\} \) be a family of almost disjoint functions f... | Yes |
Theorem 30.1 (Maharam). Every infinite homogeneous measure algebra is the unique measure algebra of its weight. | If \( A \) and \( B \) are infinite homogeneous measure algebras of the same weight and if \( \mu \) and \( \nu \) are strictly positive probabilistic measures on \( A \) and \( B \), then there exists an isomorphism \( f \) between \( A \) and \( B \) such that \( \nu \left( {f\left( a\right) }\right) = \mu \left( a\r... | Yes |
Lemma 30.3. If \( \nu \) is a signed measure on \( B \) that satisfies c.c.c. then there exists an \( a \in B \) such that \( \nu \left( x\right) \geq 0 \) for all \( x \leq a \) and \( \nu \left( x\right) \leq 0 \) for all \( x \leq - a \) . | Proof. First we claim that when \( \nu \left( a\right) > 0 \) then there exists some \( b \leq a \) such that\n\n(30.1)\n\n\[ \nu \left( b\right) > 0\text{, and}\nu \left( x\right) \geq 0\text{for all}x \leq b\text{.} \]\n\nIf (30.1) fails then for every \( b \leq a, b \neq 0 \), there exists an \( x \leq a, x \neq 0 \... | Yes |
Lemma 30.4. Let \( \mu \) and \( \nu \) be measures on \( B \) and let \( a \in B \) be such that \( \nu \left( a\right) > 0 \) . Then there exist \( {ab} \leq a, b \neq 0 \), and a number \( \varepsilon > 0 \) such that \( \nu \left( x\right) \geq \varepsilon \cdot \mu \left( x\right) \) for all \( x \leq b \) . | Proof. Let \( \varepsilon > 0 \) be such that \( \nu \left( a\right) > \varepsilon \cdot \mu \left( a\right) \) and consider the signed measure \( \nu - {\varepsilon \mu } \) on \( B \mid a \) . By Lemma 30.3 there exists a \( b \leq a \) such that \( \left( {\nu - {\varepsilon \mu }}\right) \left( x\right) \geq 0 \) f... | Yes |
Lemma 30.5 (Fremlin [1989]). Let \( A \) be a measure algebra and let \( \mu \) be a strictly positive measure on \( A \). Let \( B \) be a complete subalgebra of \( A \) and let \( \nu \) be a measure on \( B \) such that \( \nu \left( b\right) \leq \mu \left( b\right) \) for all \( b \in B \). Assume that \( A \uphar... | Proof. For each \( a \in A \), let \( {\nu }_{a} \) denote the measure on \( B \) defined by (30.3): \( {\nu }_{a}\left( b\right) = \mu \left( {a \cdot b}\right) \). We first prove the following consequence of (30.2): For every \( a \in {A}^{ + } \) and every \( \varepsilon > 0 \) there exists a \( c \in {\left( A \mid... | Yes |
Lemma 30.6. Let \( {A}_{1} \) and \( {A}_{2} \) be homogeneous measure algebras, both of the same weight \( \kappa \), and let \( {\mu }_{1} \) and \( {\mu }_{2} \) be probabilistic measures on \( {A}_{1} \) and \( {A}_{2} \). Let \( {B}_{1} \) and \( {B}_{2} \) be complete subalgebras of \( {A}_{1} \) and \( {A}_{2} \... | Proof. First we note that since every \( {A}_{1} \upharpoonright a \) has weight \( \kappa \), the subalgebra \( {B}_{1} \) satisfies (30.2); similarly for \( {A}_{2} \) and \( {B}_{2} \). Let \( {a}_{1} \in {A}_{1} \) ; if we let \( \nu \left( {f\left( b\right) }\right) = \) \( {\mu }_{1}\left( {{a}_{1} \cdot b}\right... | Yes |
Let \( B \) be an infinite Boolean algebra of uniform density. \( B \) is a Cohen algebra if and only if the set \( \left\{ {A \in {\left\lbrack B\right\rbrack }^{\omega } : A{ \leq }_{\text{reg }}B}\right\} \) contains a closed unbounded set \( C \) with the property\n\n(30.8)\n\n\[ \text{if}{A}_{1},{A}_{2} \in C\text... | First we prove the forward direction of the theorem: If \( B \) is a dense subalgebra of \( {\mathbf{C}}_{\kappa } \), then \( B \) has the property stated in Theorem 30.10. (In particular, \( {\mathbf{C}}_{\kappa } \) itself has the property.) Let \( B \) be a dense subalgebra of \( {\mathbf{C}}_{\kappa } \) . For eve... | No |
Lemma 30.11. The set \( C \) is closed unbounded in \( {\left\lbrack B\right\rbrack }^{\omega } \), satisfies (30.8), and every \( A \in C \) is a regular subalgebra of \( B \) . | Proof. Let \( A \in C \) and let \( S \) be a countable subset of \( \kappa \) such that (30.9) holds. Since \( B \cap {\mathbf{C}}_{S} \) is dense in \( {\mathbf{C}}_{S} \) and \( {\mathbf{C}}_{S}{ \leq }_{\text{reg }}{\mathbf{C}}_{\kappa } \), we have \( B \cap {\mathbf{C}}_{S}{ \leq }_{\text{reg }}{\mathbf{C}}_{\kap... | Yes |
Lemma 30.12. Let \( D \) be a complete Boolean algebra of uniform density and let \( A \) be a complete subalgebra of \( D \) of smaller density. For every \( v \in D \) there exists some \( u \in D \) independent over \( A \) such that \( v \in A\left( u\right) \) . | Proof. Exercises 30.11 and 30.12. | No |
Theorem 30.14. If \( B \) is a semi-Cohen algebra and if \( A \) is a regular subalgebra of \( B \) of uniform density then \( A \) is semi-Cohen. | Proof. \( {\left\lbrack B\right\rbrack }^{\omega } \) has a closed unbounded subset of regular subalgebras of \( B \) . Since \( A{ \leq }_{\text{reg }}B \), there exists for every \( b \in {B}^{ + } \) some \( a \in {A}^{ + } \) such that there is no \( x \in {A}^{ + } \) with \( x \leq a - b \) . Let \( F : {B}^{ + }... | Yes |
Lemma 30.17. A Boolean algebra B of uniform density is semi-Cohen if and only if \( B \) is Cohen in \( {V}^{P} \), where \( P \) is the collapse of \( \left| B\right| \) onto \( {\aleph }_{1} \) with countable conditions. | Proof. As \( \left| B\right| = {\aleph }_{1} \) in \( {V}^{P} \), it suffices to show that \( B \) is semi-Cohen if and only if it is semi-Cohen in \( {V}^{P} \) . As \( P \) does not add new countable sets, \( {\left\lbrack B\right\rbrack }^{\omega } \) remains the same in \( {V}^{P} \) . By property (iii) of Lemma 30... | Yes |
Corollary 30.18. B is semi-Cohen if and only if its completion is semi-Cohen. | Proof. Let \( B \) be semi-Cohen and let \( A = \bar{B} \) . Let \( P \) be the \( \omega \) -closed collapse of \( \left| A\right| \) to \( {\aleph }_{1} \) . In \( {V}^{P}, A \) has a dense subalgebra \( B \) that is a Cohen algebra, hence \( A \) itself is Cohen. Therefore \( A \) is semi-Cohen.\n\nThe converse foll... | Yes |
Theorem 30.22. An atomless complete Boolean algebra is simple if and only if it is rigid and minimal. | An example of a rigid and minimal algebra is \( {B}_{P} \) where \( P \) is Jensen’s forcing from Theorem 28.1 that produces a minimal \( {\Delta }_{3}^{1} \) real. \( {B}_{P} \) is minimal because the generic real has minimal degree of constructibility, and rigid because it is definable. It follows that if \( V = L \)... | No |
Lemma 30.23. Player I has a winning strategy in \( \mathcal{G} \) if and only if \( B \) is not \( \omega \) -distributive. | Proof. Let \( \sigma \) be a winning strategy for I. Let \( {a}_{0} = \sigma \left( {\langle \rangle }\right) \) ; we shall find partitions \( {W}_{n} \) of \( {a}_{0} \) without a common refinement. Let \( {W}_{0} = \left\{ {a}_{0}\right\} \) . Having constructed \( {W}_{0},\ldots ,{W}_{n} \), consider all finite sequ... | Yes |
Corollary 30.26. \( B \) is strategically \( \omega \) -closed if and only if \( B \) is a regular subalgebra of some algebra that has an \( \omega \) -closed dense subset. | Proof. Sufficiency follows from Exercise 30.18. Thus assume that \( B \) is strategically \( \omega \) -closed and let \( \sigma \) be a winning strategy for II. Let \( P \) be the collapse with countable conditions of \( \left| B\right| \) to \( {\aleph }_{1} \) . In \( {V}^{P},\sigma \) is still a winning strategy, a... | No |
If \( P \) satisfies the countable chain condition then for every uncountable \( \lambda \), every closed unbounded set \( C \subset {\left\lbrack \lambda \right\rbrack }^{\omega } \) in \( V\left\lbrack G\right\rbrack \) has a subset \( D \in V \) that is closed unbounded in \( V \) . Hence every stationary set \( S \... | Proof. Let \( p \Vdash \dot{C} \) is closed unbounded; let \( \dot{F} \) be a name for a function from \( {\lambda }^{ < \omega } \) into \( \lambda \) such that \( p \Vdash {C}_{\dot{F}} \subset \dot{C} \) (where \( {C}_{\dot{F}} \) is the set of all closure points of \( \dot{F} \) -see Theorem 8.28). Let \( f : {\lam... | Yes |
Lemma 31.3. If \( P \) is \( \omega \) -closed then every stationary set \( S \subset {\left\lbrack \lambda \right\rbrack }^{\omega } \) remains stationary in \( V\left\lbrack G\right\rbrack \) . | Proof. Let \( p \Vdash \dot{F} : {\lambda }^{ < \omega } \rightarrow \lambda \) . We shall find a condition \( q \leq p \) and some \( x \in S \) such that \( q \Vdash \dot{F}\left( {x}^{ < \omega }\right) \subset x \) .\n\nConsider the model \( \left( {{H}_{\kappa }, \in ,\left( {P, < }\right), p,\dot{F}, \Vdash }\rig... | Yes |
Lemma 31.4. If \( P \) is proper then every countable set of ordinals in \( V\left\lbrack G\right\rbrack \) is included in a set in \( V \) that is countable in \( V \) . | Proof. Let \( X \) be a countable set of ordinals in \( V\left\lbrack G\right\rbrack \) and let \( \lambda \) be uncountable in \( V \) such that \( X \subset \lambda \) . The set \( {\left( {\left\lbrack \lambda \right\rbrack }^{\omega }\right) }^{V} \) remains stationary in \( V\left\lbrack G\right\rbrack \) and ther... | Yes |
Theorem 31.9. A forcing notion \( P \) is proper if and only if for every \( p \in P \) , II has a winning strategy for the proper game. | Proof. Exercise 31.3. | No |
Lemma 31.11. If \( P \) satisfies Axiom A then \( P \) is proper. | Proof. Let \( P \) satisfy Axiom A and let \( p \in P \) . The following is a winning strategy for II in the game from Exercise 31.2: When I plays \( {\dot{\alpha }}_{n} \), let II find a condition \( {p}_{n}{ \leq }_{n - 1}{p}_{n - 1} \) (with \( {p}_{0} \leq p \) ) and a countable set \( {B}_{n} \) such that \( {p}_{... | No |
The notions of forcing that add a Sacks real, a Mathias real or a Laver real satisfy Axiom A. | For Sacks reals, see (15.26). For Laver forcing, see (28.17); Mathias forcing is similar. | No |
Theorem 31.15 (Shelah). If \( {P}_{\alpha } \) is a countable support iteration of \( \left\{ {\dot{Q}}_{\beta }\right. \) : \( \beta < \alpha \} \) such that every \( {\dot{Q}}_{\beta } \) is a proper forcing notion in \( {V}^{{P}_{\alpha } \mid \beta } \), then \( {P}_{\alpha } \) is proper. | Toward the proof of Theorem 31.15 we first observe that the properness condition in Theorem 31.7 can be somewhat simplified: | No |
Lemma 31.18. Let \( P \) be proper, let \( \dot{Q} \in {V}^{P} \) be such that \( { \Vdash }_{P}\dot{Q} \) is proper and let \( R = P * \dot{Q} \) . Let \( M \prec {H}_{\lambda } \) be countable, with \( R \in M \) . For every \( \left( {M, P}\right) \) -generic \( {q}_{0} \in P \) and every \( \dot{p} \in {V}^{P} \) s... | Proof. To find the name \( {\dot{q}}_{1} \), let \( G \) be a generic filter on \( P \) containing \( {q}_{0} \) . Let \( p = {\dot{p}}^{G} \) and \( q = {\dot{Q}}^{G} \) ; then \( p \in M \cap R \) and \( p = \left( {{p}_{0},{\dot{p}}_{1}}\right) \) with \( {p}_{0} \in G \) . Since \( {\dot{p}}_{1} \in M \), we have \... | No |
Theorem 31.21. If there exists a supercompact cardinal then there is a generic model that satisfies PFA. | Proof. The proof follows loosely the proof of the consistency of MA. Let \( \kappa \) be a supercompact cardinal. The model is obtained by countable support iteration of length \( \kappa \) . Each notion of forcing used in the iteration is proper and has size \( < \kappa \), thus both \( {\aleph }_{1} \) and all cardin... | Yes |
Lemma 31.22. In \( V\left\lbrack G\right\rbrack \), if \( P \) is proper and \( \mathcal{D} = \left\{ {{D}_{\alpha } : \alpha < \gamma }\right\} \), with \( \gamma < \kappa \) , is a family of dense subsets of \( P \), then there exists a \( \mathcal{D} \) -generic filter on \( P \) . | Proof of Lemma 31.22. Let \( \dot{P} \) and \( \dot{\mathcal{D}} \) be \( {P}_{\kappa } \) -names for \( P \) and \( \mathcal{D} \) . Let \( \lambda > {2}^{{2}^{\left| P\right| }} \) be sufficiently large; we may also assume that \( P \subset \lambda \) . Since \( f \) is a Laver function, there exists an elementary em... | Yes |
Theorem 31.23 (Todorčević). PFA implies \( {2}^{{\aleph }_{0}} = {\aleph }_{2} \) . | As the first step we show that the Open Coloring Axiom (29.6) is a consequence of PFA. If \( {\left\lbrack X\right\rbrack }^{2} = {K}_{0} \cup {K}_{1} \) with \( {K}_{0} \) open, let us call \( Z \subset X \) 0-homogeneous if \( {\left\lbrack Z\right\rbrack }^{2} \subset {K}_{0} \) and 1-homogeneous if \( {\left\lbrack... | No |
Lemma 31.24 (Todorčević). Assume \( {2}^{{\aleph }_{0}} = {\aleph }_{1} \) . Let \( X \subset \mathbf{R} \) and \( {\left\lbrack X\right\rbrack }^{2} = \) \( {K}_{0} \cup {K}_{1} \) with \( {K}_{0} \) open, and assume that \( X \) is not the union of countably many closed 1-homogeneous sets. Then there exists an uncoun... | Proof. See Theorem 4.4 of Todorčević [1989]. (To apply the theorem, let \( F\left( x\right) \) be the closure of \( \left\{ {y \in X : x < y}\right. \) and \( \left. {\{ x, y\} \in {K}_{1}}\right) \) . | No |
Theorem 31.25. PFA implies OCA. | Proof. Let \( X \subset \mathbf{R} \) and let \( {\left\lbrack X\right\rbrack }^{2} = {K}_{0} \cup {K}_{1} \) with \( {K}_{0} \) open, and assume that \( X \) is not the union of countably many closed 1-homogeneous sets. We shall use PFA to find an uncountable 0-homogeneous set.\n\nLet \( P \) be the forcing (15.2) tha... | No |
Lemma 31.26. There exists a partition \( F : {\left\lbrack \mathfrak{b}\right\rbrack }^{2} \rightarrow \omega \) such that in \( {V}^{P} \), for every \( \dot{C} \) as above and every \( r \in {\omega }^{\omega },{\dot{Q}}_{r}\left( \dot{C}\right) \) satisfies the countable chain condition. | Proof. See Bekkali [1991], page 49. The partition \( F \) is obtained by using oscillating real numbers, cf. Chapter 1 of Todorčević [1989]. | No |
Lemma 31.27. PFA implies \( \mathfrak{b} = {2}^{{\aleph }_{0}} \) . | Proof. Let \( F : {\left\lbrack \mathfrak{b}\right\rbrack }^{2} \rightarrow \omega \) be as in Lemma 31.26. Let \( P \) be the \( \omega \) -closed forcing that adds a subset of \( {\omega }_{1} \), and let \( \dot{C} \in {V}^{P} \) be a closed unbounded subset of \( \mathfrak{b} \) , of order-type \( {\omega }_{1} \).... | Yes |
Theorem 31.28 (Todorčević). PFA implies that \( {▱}_{\kappa } \) fails for every uncountable cardinal \( \kappa \) . | Proof. Let \( \kappa \) be an uncountable cardinal, assume that \( {▱}_{\kappa } \) holds, and let \( \left\langle {{C}_{\alpha } : \alpha \in \operatorname{Lim}\left( {\kappa }^{ + }\right) }\right\rangle \) be a square-sequence (cf. (23.4)).\n\nLet \( T \) be the tree whose nodes are limit ordinals below \( {\kappa }... | Yes |
Lemma 31.29. \( P \) is proper. | Proof. We omit the proof, as it is similar to the proof of properness in Exercise 31.5 (and using the fact that \( \left( {T, \prec }\right) \) has no \( {\kappa }^{ + } \) -branch). | No |
Theorem 31.31. If PFA holds then any two normal Aronszajn trees are club-isomorphic. | Proof. Let \( {T}_{1} \) and \( {T}_{2} \) be two normal Aronszajn trees. Consider the forcing with finite conditions \( \left( {E, f}\right) \) such that\n\n(31.7)\n\n(i) \( E \) is a finite subset of \( {\omega }_{1} \) ,\n\n(ii) \( \operatorname{dom}\left( f\right) \) is a subtree of \( {T}_{1} \mid E \) in which ev... | No |
Theorem 31.32. If PFA holds then there are no \( {\aleph }_{2} \) -Aronszajn trees. | Proof. Assume that \( T \) is an \( {\aleph }_{2} \) -Aronszajn tree. Let \( P \) be the forcing that adds a subset of \( {\omega }_{1} \) with countable conditions. Since \( {2}^{{\aleph }_{0}} = {\aleph }_{2}, P \) collapses \( {\omega }_{2} \) and so there is in \( {V}^{P} \) a closed unbounded subset \( \dot{C} \) ... | Yes |
Lemma 32.2. Let \( E \) be a meager equivalence relation on \( \mathcal{N} \). Then there exist a perfect set of inequivalent reals. | Proof. Let \( {\left\{ {D}_{n}\right\} }_{n} \) be dense open sets in \( \mathcal{N} \times \mathcal{N} \) such that \( {\mathcal{N}}^{2} - E \supset \mathop{\bigcap }\limits_{{n = 0}}^{\infty }{D}_{n} \). We construct a binary tree of finite sequences \( \left\{ {{u}_{s} : s \in \operatorname{Seq}\left( {\{ 0,1\} }\ri... | Yes |
Lemma 32.4. The \( {\sum }_{1}^{1} \) -topology satisfies the Baire Category Theorem. | Proof. Exercise 32.3. | No |
Lemma 32.5. If \( X \) is comeager in the \( {\sum }_{1}^{1} \) -topology then for every nonempty \( {\sum }_{1}^{1} \) subset \( A \) of \( \mathcal{N} \times \mathcal{N}, A \cap \left( {X \times X}\right) \neq \varnothing \) . | Proof. The lemma states that \( X \times X \) is dense in the \( {\sum }_{1}^{1} \) -topology on \( \mathcal{N} \times \mathcal{N} \) (which is larger than the product of the \( {\sum }_{1}^{1} \) -topology). If \( D \) is a dense open set in the \( {\sum }_{1}^{1} \) -topology then \( D \times \mathcal{N} \) is dense ... | Yes |
Lemma 32.6. \( H \) is a \( {\sum }_{1}^{1} \) set. | Proof. First note that if an equivalence class \( A \) of \( E \) contains a nonempty \( {\sum }_{1}^{1} \) set \( U \) then \( A \) is \( {\Pi }_{1}^{1} \) :\n\n\[ x \in A \leftrightarrow \forall y\left( {y \in U \rightarrow {xEy}}\right) . \]\n\nThen by the separation principle there exists a \( {\Delta }_{1}^{1} \) ... | No |
Lemma 32.7. For every \( a \in \mathcal{N},{E}_{a} \cap H \) is meager in the \( {\sum }_{1}^{1} \) -topology, where \( {E}_{a} = \{ b : \left( {a, b}\right) \in E\} . | Proof. If \( H = \varnothing \) then there is nothing to prove; thus assume \( H \neq \varnothing \) . The set \( {E}_{a} \) is \( {\Pi }_{1}^{1} \) and therefore has the Baire property in the \( {\sum }_{1}^{1} \) -topology. If \( {E}_{a} \cap H \) is not meager then there exists a nonempty \( {\sum }_{1}^{1} \) set \... | Yes |
Lemma 32.8. \( E \cap \left( {H \times H}\right) \) is meager (in the product of the \( {\sum }_{1}^{1} \) -topology). | Proof. By Lemma 32.7 and Lemma 11.16. | No |
Theorem 32.9. If \( E \) is a \( {\mathbf{\sum }}_{1}^{1} \) equivalence relation on \( \mathcal{N} \) then either \( E \) has at most \( {\aleph }_{1} \) equivalence classes or there exists a perfect set of mutually inequivalent reals. | Proof. Let \( E \) be a \( {\mathbf{\sum }}_{1}^{1} \) equivalence relation. There exists a tree \( T \) on \( \left( {\omega \times \omega }\right) \times \omega \) such that for all \( a, b \in \mathcal{N} \)\n\n(32.3)\n\n\[ \n{aEb} \leftrightarrow T\left( {a, b}\right) \text{is ill-founded.} \n\]\n\nWe define, for e... | No |
Lemma 32.10. There is a closed unbounded set \( C \subset {\omega }_{1} \) such that for each \( \alpha \in C,{E}^{\alpha } \) is an equivalence relation. | Proof. If \( T\left( {x, y}\right) \) is well-founded then so is \( T\left( {y, x}\right) \) (by the symmetry of \( E \) ) and so for every \( \alpha < {\omega }_{1} \) the set \( \{ T\left( {y, x}\right) : \parallel T\left( {x, y}\right) \parallel < \alpha \} \) is a set of well-founded trees. The set is \( {\mathbf{\... | Yes |
Theorem 32.11. If there exists a nonconstructible real then the set \( \mathbf{R} \cap L \) does not have a perfect subset. | Proof. As a first step we show that \( \mathbf{R} \cap L \) does not have a superperfect subset. A tree \( T \subset \) Seq is superperfect if for every \( t \in T \) there exists an \( s \supset t \) in \( T \) such that \( {s}^{ \frown }k \in T \) for infinitely many \( k \in \omega \) . (We call \( s \) an \( \omega... | No |
Theorem 32.14 (Martin and Solovay [1969], Mansfield [1971]). If there exists a measurable cardinal then for every \( {\mathbf{\sum }}_{3}^{1} \) set \( A \) there exists a tree \( T \) on \( \omega \times \lambda \) (for some \( \lambda \) ) such that \( A = p\left\lbrack T\right\rbrack \) . | Proof. Let \( \kappa \) be a measurable cardinal and let \( U \) be a normal measure on \( \kappa \) . For each \( n \), let \( {U}_{n} \) be the ultrafilter \( \left\{ {X \subset {\kappa }^{n} : X \supset {\left\lbrack Z\right\rbrack }^{n}\text{for some}Z \in U}\right\} \), and let \( {j}_{n} = {i}_{n, n + 1} \) be th... | No |
Theorem 32.16 (Magidor [1980]). Let us assume that there exists a measurable cardinal, and that \( {\omega }_{1} \) carries a precipitous ideal. Then every \( {\mathbf{\sum }}_{3}^{1} \) set is Lebesgue measurable, has the Baire property, and is either countable or contains a perfect subset. | Proof. Let \( A \) be a \( {\mathbf{\sum }}_{3}^{1} \) set and let \( A = p\left\lbrack T\right\rbrack \) where \( T \) is the tree defined in the proof of Theorem 32.14. We shall prove that under the given assumptions,\n\n(32.12)\n\n\( \mathbf{R} \cap L\left\lbrack T\right\rbrack \) is countable.\n\nThen the statement... | Yes |
Lemma 32.17. \( i\left( U\right) = \bar{U} \cap M, i\left( {U}_{n}\right) = \overline{{U}_{n}} \cap M \) . | Proof. It suffices to show that \( i\left( U\right) \subset \bar{U} \cap M \) ; if \( X \in i\left( U\right) \) we want a \( W \in U \) such that \( X \supset W.X \) is represented by \( \left\langle {{X}_{\xi } : \xi < {\omega }_{1}}\right\rangle \), so let \( Y = \mathop{\bigcap }\limits_{{\xi < {\omega }_{1}}}{X}_{\... | No |
Lemma 32.18. Let \( h \in V\left\lbrack G\right\rbrack \) be a function \( h : \kappa \rightarrow V \) . Then there exists a function \( H \in V \) such that \( h\left( \alpha \right) = H\left( \alpha \right) \) a.e. \( {\;\operatorname{mod}\;U} \) . Similarly for \( h : {\kappa }^{n} \rightarrow \) \( V\left( {\text{a... | Proof. For each \( \alpha < \kappa \) there is a maximal antichain \( {W}_{\alpha } \) in \( P \) and a set \( \left\{ {{x}_{p}^{\alpha } : p \in {W}_{\alpha }}\right\} \) such that \( p \Vdash \dot{h}\left( \alpha \right) = {x}_{p}^{\alpha } \) . Let \( W \) be such that \( {W}_{\alpha } = W \) for \( U \) -almost all... | Yes |
Lemma 32.19. Let \( f \in V \) be a function \( f : \kappa \rightarrow \) Ord. Then there exists a function \( g \in M \) such that \( f\left( \alpha \right) = g\left( \alpha \right) \) a.e. \( {\;\operatorname{mod}\;U} \) . Similarly for \( f : {\kappa }^{n} \rightarrow \) Ord. | Proof. Every ordinal \( \beta \) is represented in \( M \) by some \( {h}_{\beta } : {\omega }_{1} \rightarrow \) Ord, \( {h}_{\beta } \in V \) . For each \( \alpha < \kappa \), pick (in \( V\left\lbrack G\right\rbrack \) ) some \( {h}_{f\left( \alpha \right) } : {\omega }_{1} \rightarrow \) Ord that represents \( f\le... | Yes |
Lemma 33.1. Assuming the Axiom of Choice, there exists \( A \subset {\omega }^{\omega } \) such that the game \( {G}_{A} \) is not determined. | Proof. Let \( \left\{ {{\sigma }_{\alpha } : \alpha < {2}^{{\aleph }_{0}}}\right\} \) and \( \left\{ {{\tau }_{\alpha } : \alpha < {2}^{{\aleph }_{0}}}\right\} \) enumerate all strategies for I and all strategies for II. We construct sets \( X = \left\{ {{x}_{\alpha } : \alpha < {2}^{{\aleph }_{0}}}\right\} \) and \( Y... | Yes |
Lemma 33.2. The Axiom of Determinacy implies that every countable family of nonempty sets of real numbers has a choice function. | Proof. We prove that if \( \mathcal{X} = \left\{ {{X}_{n} : n \in \omega }\right\} \) is a family of nonempty subsets of \( \mathcal{N} \), then there exists \( f \) on \( \mathcal{X} \) such that \( f\left( {X}_{n}\right) \in {X}_{n} \) for all \( n \) . Let us consider the following game: If I plays \( \left\langle {... | Yes |
Theorem 33.3. Assume the Axiom of Determinacy. Then:\n\n(i) Every set of reals is Lebesgue measurable. | Proof. (i) It suffices to prove the following lemma:\n\nLemma 33.4. Assuming AD | No |
Let \( X \subset \mathcal{N} \). Player I has a winning strategy in the Banach-Mazur game if and only if for some \( s \in {Seq}, O\left( s\right) - X \) is meager. | Proof. Note that I has a winning strategy if and only if there exists \( s \in {Seq} \) (the first move of I) such that player II has a winning strategy in the following game: I plays \( {t}_{0} \supset s \), II plays \( {s}_{0} \supset {t}_{0} \), I plays \( {t}_{1} \supset {s}_{0} \), etc.; and I wins if \( {t}_{0} \... | Yes |
Lemma 33.10. Let \( X \subset C \) . If II has a winning strategy in the perfect set game, then \( X \) is countable. | Proof. Let \( \tau \) be a winning strategy for II. A correct position is a finite sequence \( \left\langle {{s}_{0},{n}_{0},\ldots ,{s}_{k},{n}_{k}}\right\rangle \) such that \( {n}_{0} = \tau \left( \left\langle {s}_{0}\right\rangle \right) ,{n}_{1} = \tau \left( \left\langle {{s}_{0},{n}_{0},{s}_{1}}\right\rangle \r... | Yes |
Theorem 33.12 (Solovay). The Axiom of Determinacy implies that:\n\n(i) \( {\aleph }_{1} \) is a measurable cardinal, and moreover, the closed unbounded filter on \( {\aleph }_{1} \) is an ultrafilter.\n\n(ii) \( {\aleph }_{2} \) is a measurable cardinal. | Proof. (i) We first show that AD implies that \( {\omega }_{1} \) is measurable. We already know that \( {\omega }_{1} \) is inaccessible in every \( L\left\lbrack a\right\rbrack, a \subset \omega \) .\n\nLet us consider the following partial ordering of the Baire space:\n\n(33.7)\n\n\[ x \preccurlyeq y\;\text{ if and ... | Yes |
AD implies that for every \( \equiv \) -closed \( A \subset \mathcal{N} \), either \( A \) or its complement contains a cone. Hence \( \mathcal{F} \) is a \( \sigma \) -complete ultrafilter on \( \mathcal{B} \) . | We show that if I has a winning strategy in the game \( {G}_{A} \), then \( A \) contains a cone (and similarly, if II has a winning strategy, then \( \mathcal{N} - A \in \mathcal{F} \) ). Let \( \sigma \) be a winning strategy for I. It suffices to show that \( A \) contains the cone \( \{ x \in \mathcal{N} : \sigma \... | Yes |
Lemma 33.14. Assume AD. Then for every \( S \subset {\omega }_{1} \), the set \( \{ x \in \mathrm{{WO}} \) : \( \parallel x\parallel \in S\} \) is \( {\mathbf{\Pi }}_{1}^{1} \). Consequently, there is some \( a \subset \omega \) such that \( S \in L\left\lbrack a\right\rbrack \) . | Proof. If \( x \in \mathcal{N} \), then for each \( n \in \mathbf{N} \) we let \( {x}_{n} \in \mathcal{N} \) be such that \( {x}_{n}\left( m\right) = \) \( x\left( {\langle n, m\rangle }\right) \) for all \( m \in \mathbf{N} \) . We consider the following game:\n\n33.15. The Solovay Game. Let \( S \subset {\omega }_{1}... | Yes |
Lemma 33.17. If \( A \subset \mathcal{N} \) is an open set, then \( {G}_{A} \) is determined. | Proof. Player I plays \( \left\langle {{a}_{0},{a}_{1},\ldots }\right\rangle \), player II plays \( \left\langle {{b}_{0},{b}_{1},\ldots }\right\rangle \), and I wins if \( \left\langle {{a}_{0},{b}_{0},{a}_{1},{b}_{1},\ldots }\right\rangle \in A \) . Let us assume that player I does not have a winning strategy, and le... | Yes |
Theorem 33.18 (Martin [1975]). All Borel games are determined. | We shall not give a proof. It can be found either in Martin's paper [1975], or in the survey article [1980] by Kechris and Martin; furthermore, Martin gives a simplification of his proof in [1985]. | No |
Corollary 33.24. Assume PD. The classes \( {\Pi }_{{2n} + 1}^{1}\left( a\right) \) and \( {\sum }_{{2n} + 2}^{1}\left( a\right) \) have the prewellordering property and the uniformization property and satisfy the reduction principle; the classes \( {\sum }_{{2n} + 1}^{1}\left( a\right) \) and \( {\Pi }_{{2n} + 2}^{1}\l... | The scale property generalizes the prewellordering property, and implies uniformization (using the proof of Kondô’s Theorem 25.26; cf. Exercise 33.4). The prewellordering property implies the reduction principle (as in Exercise 25.7 ; see Exercise 33.5), which in turn implies the separation principle for the dual class... | No |
Lemma 33.30. If \( A \subset \mathcal{N} \) is \( {\mathbf{\Pi }}_{1}^{1} \) and \( \kappa \) is a measurable cardinal then \( A \) is \( \kappa \) -homogeneously Suslin. | Proof. Exercise 33.10. | No |
Theorem 33.34 (Martin and Steel [1988]). If \( A \subset \mathcal{N} \) is \( {\delta }^{ + } \) -weakly homogeneously Suslin, where \( \delta \) is a Woodin cardinal, then \( \mathcal{N} - A \) is homogeneously Suslin. | As for Theorem 33.34, assume that \( A = p\left| T\right| \) where \( T \) is weakly homogeneous. Then one constructs a tree \( \widetilde{T} \) such that \( \mathcal{N} - A = p\left\lbrack \widetilde{T}\right\rbrack \) in a manner similar to the tree representation for \( {\mathbf{\Pi }}_{2}^{1} \) sets in Theorem 32.... | Yes |
Theorem 33.3(i) | Theorem 33.3(i) is due to Mycielski and Świerczkowski [1964]; the present proof (and the covering game) is due to Harrington. | No |
Lemma 34.2. The following are equivalent:\n\n(i) For every \( A \subset {V}_{\delta } \) there exists a \( \kappa < \delta \) that is \( \lambda \) -strong for \( A \) for all \( \lambda < \delta \).\n\n(ii) For every \( A \subset {V}_{\delta } \) the set of all \( \kappa < \delta \) that are \( \lambda \) -strong for ... | Proof. It suffices to show that (i) implies (iv) and that (iii) implies (ii).\n\nAssume that (i) holds, and let \( f : \delta \rightarrow \delta \) . By (i) there exists a \( \kappa < \delta \) that is \( \lambda \) -strong for \( A \) for all \( \lambda < \delta \) . Let \( \lambda < \delta \) be sufficiently large, a... | Yes |
Lemma 34.12. Each of the following two properties is equivalent to semi-properness of \( A \) :\n\n(i) There is a closed unbounded set of countable \( M \prec {V}_{\kappa + 1} \) such that some countable \( N \prec {V}_{\kappa + 1} \) satisfies (34.11).\n\n(ii) For all sufficiently large \( \lambda \), for every counta... | Proof. For the nontrivial implication (i) \( \Rightarrow \) (ii) see Exercise 34.10. | No |
Lemma 34.13. Let \( \kappa < \delta \) be such that \( A \cap {Q}_{ < \kappa } \) is dense in \( {Q}_{ < \kappa } \) and that \( \kappa \) is \( \lambda \) -strong for \( A \) for all \( \lambda < \delta \) . Then \( A \cap {Q}_{ < \kappa } \) is semiproper in \( {Q}_{ < \kappa } \) . | Proof. Toward a contradiction, assume that the set\n\n\[ S = \left\{ {M \prec {V}_{\kappa + 1} : }\right. \text{there is no}\left. {N \prec {V}_{\kappa + 1}\text{such that (34.11) holds}}\right\} \]\n\nis stationary. Let \( \lambda > \kappa + 1\left( {\lambda < \delta }\right) \) be such that \( \left( {{V}_{\lambda },... | Yes |
Theorem 34.14 (Woodin, [1988]). Let \( \delta \) be a Woodin cardinal and let \( {Q}_{ < \delta } \) be the stationary tower forcing. Let \( G \) be a generic filter on \( {Q}_{ < \delta } \), and let \( j : V \rightarrow {\mathrm{{Ult}}}_{G} \) be the canonical elementary embedding into the generic ultrapower. Then\n\... | Proof. (i) If \( A \) is a dense set and \( N \) is a countable model, we say that \( N \) captures \( A \) if (34.11)(ii) holds. First we claim that if \( A \subset {Q}_{ < \kappa } \) is semiproper then for every condition \( p \in {Q}_{ < \kappa } \) there is a stronger condition \( q \) such that every \( N \in q \... | Yes |
Corollary 34.15. If \( \delta \) is a Woodin cardinal and a limit of Woodin cardinals, if \( P \) is a forcing notion such that \( \left| P\right| < \delta \), and if \( G \) is a generic filter on \( P \) , then the model \( L{\left( \mathbf{R}\right) }^{V\left\lbrack G\right\rbrack } \) is elementarily equivalent to ... | Proof. As \( \delta \) remains a Woodin cardinal in \( V\left\lbrack G\right\rbrack \), we can find a \( V \) -generic filter \( H \) on \( \operatorname{Col}\left( {\omega , < \delta }\right) \) such that \( V\left\lbrack G\right\rbrack \subset V\left\lbrack H\right\rbrack \) and \( V\left\lbrack H\right\rbrack \) is ... | Yes |
Lemma 35.7. A mouse exists if and only if \( {0}^{\sharp } \) exists. | Proof. If a mouse exists at \( \kappa \), then the iterates \( {\kappa }^{\left( \alpha \right) } \) are indiscernibles for \( L \) .\n\nConversely, let \( {0}^{\sharp } \) exist and let \( {i}_{\alpha } \) be the Silver indiscernibles. For each \( \alpha \) , let \( {j}_{\alpha } : L \rightarrow L \) be the unique ele... | Yes |
Lemma 35.9. \( < \) is a well-ordering of mice, and if \( M \leq {M}^{\prime } \) then \( M \in L\left\lbrack {M}^{\prime }\right\rbrack \) . | Proof. If \( \beta < {\beta }^{\prime } \) then \( {J}_{\beta }^{{\mathcal{C}}_{\lambda }} \in {J}_{{\beta }^{\prime }}^{{\mathcal{C}}_{\lambda }} \), and \( M \in L\left\lbrack {J}_{\beta }^{{\mathcal{C}}_{\lambda }}\right\rbrack \) . | No |
Lemma 35.10. If mice exist then \( K = \bigcup \{ M : M \) is a mouse \( \} \) . | Proofs in the core model theory such as the proof of Lemma 35.10 involve iterations of mice. One of the difficulties is that since mice do not satisfy \( {\mathrm{{ZF}}}^{ - } \) , the resulting embeddings are not fully elementary. It is easy to verify that \( {i}_{0,1} : M \rightarrow {\operatorname{Ult}}_{U}M \) is \... | No |
Corollary 35.18 (Mitchell). Assume that \( \kappa \) is a measurable cardinal and \( {2}^{\kappa } > {\kappa }^{ + } \) . Then there is an inner model with a measurable \( \lambda \) of order \( {\lambda }^{+ + } \) . | Proof. If there is no such model then (iii) and (iv) hold. Let \( D \) be a normal measure on \( \kappa \) and \( {j}_{D} : V \rightarrow M = {\operatorname{Ult}}_{D}\left( V\right) \) ; let \( j = {j}_{D} \upharpoonright {K}^{m} : {K}^{m} \rightarrow N \) . By (iv), \( j \) is an iterated ultrapower, \( j = {i}_{0,\va... | Yes |
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