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Lemma 18.8. Let \( \sum \) be an E.M. set, let \( \alpha \leq \beta \), and let \( j : \alpha \rightarrow \beta \) be order-preserving. Then \( j \) can be extended to an elementary embedding of the \( \left( {\sum ,\alpha }\right) \) -model into the \( \left( {\sum ,\beta }\right) \) -model. | Proof. Extend \( j \) as in (18.9). | No |
Lemma 18.9. The following are equivalent, for an E.M. set \( \sum \) :\n\n(i) For every ordinal \( \alpha \), the \( \left( {\sum ,\alpha }\right) \) -model is well-founded.\n\n(ii) For some ordinal \( \alpha \geq {\omega }_{1} \), the \( \left( {\sum ,\alpha }\right) \) -model is well-founded.\n\n(iii) For every ordin... | Proof. (i) \( \rightarrow \) (ii) is trivial.\n\n(ii) \( \rightarrow \) (iii): If \( \left( {\mathfrak{A}, I}\right) \) is the \( \left( {\sum ,\alpha }\right) \) -model and if \( \beta \leq \alpha \), let \( J \) be the initial segment of the first \( \beta \) elements of \( I \) ; let \( \mathfrak{B} = {H}^{\mathfrak... | Yes |
Lemma 18.10. The following are equivalent, for any E.M. set \( \sum \) :\n\n(i) For all \( \alpha ,\left( {\sum ,\alpha }\right) \) is unbounded.\n\n(ii) For some \( \alpha ,\left( {\sum ,\alpha }\right) \) is unbounded.\n\n(iii) For every Skolem term \( t\left( {{v}_{1},\ldots ,{v}_{n}}\right) \) the set \( \sum \) co... | Proof. (i) \( \rightarrow \) (ii) is trivial.\n\n(ii) \( \rightarrow \) (iii): Let \( \left( {\mathfrak{A}, I}\right) \) be a \( \left( {\sum ,\alpha }\right) \) -model, where \( \alpha \) is a limit ordinal, and assume that \( I \) is unbounded in \( {\operatorname{Ord}}^{\mathfrak{A}} \) . To prove (iii), it suffices... | Yes |
Lemma 18.11. The following are equivalent for any unbounded E.M. set \( \sum \) :\n\n(i) For all \( \alpha > \omega \), the \( \left( {\sum ,\alpha }\right) \) -model is remarkable.\n\n(ii) For some \( \alpha > \omega \), the \( \left( {\sum ,\alpha }\right) \) -model is remarkable.\n\n(iii) For every Skolem term \( t\... | Proof. (i) \( \rightarrow \) (ii) is trivial.\n\n(ii) \( \rightarrow \) (iii): Let \( \alpha > \omega \) be a limit ordinal and let \( \left( {\mathfrak{A}, I}\right) \) be a remarkable \( \left( {\sum ,\alpha }\right) \) -model. To prove (iii), it suffices to show that for any \( t \) ,(18.13) is true in \( \mathfrak{... | Yes |
Lemma 18.12. If \( \left( {\mathfrak{A}, I}\right) \) is remarkable, then \( I \) is closed in \( {\operatorname{Ord}}^{\mathfrak{A}} \) . | Proof. Let \( \gamma < \alpha \) be a limit ordinal. If \( x \) is an ordinal of \( \mathfrak{A} \) less than \( {i}_{\gamma } \), then by remarkability, \( x \) is in the \( \left( {\sum ,\gamma }\right) \) -model \( \mathfrak{B} = {H}^{\mathfrak{A}}\left( \left\{ {{i}_{\xi } : \xi < \gamma }\right\} \right) \) . Howe... | Yes |
Lemma 18.13. If \( \kappa \) is an uncountable cardinal, then the universe of the \( \left( {\sum ,\kappa }\right) \) -model is \( {L}_{\kappa } \) . | Proof. The \( \left( {\sum ,\kappa }\right) \) -model is \( \left( {{L}_{\beta }, I}\right) \) for some \( \beta \) ; since \( \left| I\right| = \kappa \), we clearly have \( \beta \geq \kappa \) . To prove that \( \beta = \kappa \), assume that \( \beta > \kappa \) . Since \( I \) is unbounded in \( \beta \) and has o... | Yes |
Lemma 18.14. If \( \kappa < \lambda \) are uncountable cardinals, then \( {I}_{\lambda } \cap \kappa = {I}_{\kappa } \), and \( {H}^{{L}_{\lambda }}\left( {I}_{\kappa }\right) = {L}_{\kappa } \) | Proof. Let \( J \) be the set consisting of the first \( \kappa \) members of \( {I}_{\lambda } \) and let \( \mathfrak{A} = \) \( {H}^{{L}_{\lambda }}\left( J\right) \) . Then \( \left( {\mathfrak{A}, J}\right) \) is a \( \left( {\sum ,\kappa }\right) \) -model and the ordinals of \( \mathfrak{A} \) are an initial seg... | Yes |
Lemma 18.15. There is at most one well-founded remarkable E.M. set. | Proof. Assuming that there is one such \( \sum \), we define the class \( I \) in (18.16). Now since \( {L}_{{\aleph }_{\omega }} \) is the \( \left( {\sum ,{\aleph }_{\omega }}\right) \) -model and \( {\aleph }_{n} \in I \) for each \( n \geq 1 \), we have\n\n(18.17)\n\n\[ \varphi \left( {{v}_{1},\ldots ,{v}_{n}}\righ... | Yes |
For every regular uncountable cardinal \( \kappa \) there is at most one closed unbounded set of indiscernibles \( X \) for \( {L}_{\kappa } \) such that \( {L}_{\kappa } = {H}^{{L}_{\kappa }}\left( X\right) \) . | Let \( \sum = \sum \left( {{L}_{\kappa }, X}\right) \) . Since \( X \) is closed unbounded, it follows that \( X \cap I \) is infinite, and \( \sum \left( {{L}_{\kappa }, X}\right) = \sum \left( {{L}_{\kappa }, X \cap I}\right) = \sum \left( {{L}_{\kappa }, I \cap \kappa }\right) \) . Hence \( \sum = {0}^{\sharp } \) a... | No |
Theorem 18.20 (Kunen). The following are equivalent:\n\n(i) \( {0}^{\sharp } \) exists.\n\n(ii) There is a nontrivial elementary embedding \( j : L \rightarrow L \) . | Toward the proof of Kunen's Theorem, let us investigate elementary embeddings \( j : M \rightarrow N \) where \( M \) is a transitive model of ZFC. | No |
Lemma 18.22. If \( j : M \rightarrow N \) is an elementary embedding and \( \kappa \) is the critical point of \( j \) then \( \kappa \) is a regular uncountable cardinal in \( M \), and \( D = \) \( \left\{ {X \in {P}^{M}\left( \kappa \right) : \kappa \in j\left( X\right) }\right\} \) is a nonprincipal normal \( \kapp... | Proof. Exactly as the proof of Lemma 17.2. Note that \( \kappa \) -completeness of \( D \) implies that \( \kappa \) is regular in \( M \) . | No |
Lemma 18.23. If \( \kappa \) is a limit cardinal such that \( \operatorname{cf}\kappa > \gamma \), then \( {j}_{D}\left( \kappa \right) = \kappa \) . | Proof. Every constructible function \( f : \gamma \rightarrow \kappa \) is bounded by some \( \alpha < \) \( \kappa \) and hence \( \left\lbrack f\right\rbrack < \left\lbrack {c}_{\alpha }\right\rbrack \) (where \( {c}_{\alpha } \) is the constant function with value \( \alpha \) ). Thus \( {j}_{D}\left( \kappa \right)... | Yes |
Lemma 18.26. The set \( \\left\\{ {{\\gamma }_{\\alpha } : \\alpha < {\\omega }_{1}}\\right\\} \) is a set of indiscernibles for \( \\left( {{L}_{\\kappa }, \\in }\\right) \) . | Proof. Let \( \\varphi \) be a formula and let \( {\\alpha }_{1} < \\ldots < {\\alpha }_{n} \) and \( {\\beta }_{1} < \\ldots < {\\beta }_{n} \) be two sequences of countable ordinals. We wish to show that\n\n(18.32)\n\n\[ \n{L}_{\\kappa } \\vDash \\varphi \\left\\lbrack {{\\gamma }_{{\\alpha }_{1}},\\ldots ,{\\gamma }... | Yes |
Theorem 18.27. Let \( j : {L}_{\alpha } \rightarrow {L}_{\beta } \) be an elementary embedding and let \( \gamma \) be the critical point of \( j \) . If \( \gamma < \left| \alpha \right| \), then \( {0}^{\sharp } \) exists. | Proof. Let \( \gamma \) be the critical point of \( j \) . Since \( \gamma < \left| \alpha \right| \), every \( X \subset \gamma \) is in \( {L}_{\alpha } \) , and so \( D = \{ X \subset \gamma : \gamma \in j\left( X\right) \} \) is an \( L \) -ultrafilter.\n\nLet us consider the ultrapower \( {\operatorname{Ult}}_{D}\... | Yes |
If there is a Jónsson cardinal, then \( {0}^{\sharp } \) exists. | Let \( \kappa \) be a Jónsson cardinal and let us consider the model \( \left( {{L}_{\kappa }, \in }\right) \) . Let \( A \) be an elementary submodel, of size \( \kappa \), such that \( A \neq {L}_{\kappa } \) . Let \( \pi \) be the transitive collapse of \( A \) ; clearly, \( \pi \left( A\right) = {L}_{\kappa } \) . ... | Yes |
Corollary 18.29. Chang’s Conjecture implies that \( {0}^{\sharp } \) exists. | Proof. Consider the model \( \left( {{L}_{{\omega }_{2}},{\omega }_{1}, \in }\right) \), and let \( \mathfrak{A} = \left( {A,{\omega }_{1} \cap A, \in }\right) \) be its elementary submodel such that \( \left| A\right| = {\aleph }_{1} \) and \( \left| {{\omega }_{1} \cap A}\right| = {\aleph }_{0} \) . Let \( \pi \) be ... | Yes |
Corollary 18.31. If \( {0}^{\sharp } \) does not exist then for every \( \lambda \geq {\aleph }_{2} \), if \( \lambda \) is a regular cardinal in \( L \) then \( \operatorname{cf}\lambda = \left| \lambda \right| \) . Consequently, every singular cardinal is a singular cardinal in \( L \) . | Proof. Let \( \lambda \) be a limit ordinal such that \( \lambda \geq {\omega }_{2} \) and that \( \lambda \) is a regular cardinal in \( L \) . Let \( X \) be an unbounded subset of \( \lambda \) such that \( \left| X\right| = \operatorname{cf}\lambda \) . By the Covering Theorem, there exists a constructible set \( Y... | Yes |
Corollary 18.32. If \( {0}^{\sharp } \) does not exist then for every singular cardinal \( \kappa \) , \( {\left( {\kappa }^{ + }\right) }^{L} = {\kappa }^{ + } \) . | Proof. Let \( \kappa \) be a singular cardinal and let \( \lambda \) be the successor cardinal of \( \kappa \) in \( L \) ; we want to show that \( \lambda = {\kappa }^{ + } \) . If not, then \( \left| \lambda \right| = \kappa \), and since \( \kappa \) is singular, we have cf \( \lambda < \kappa \) . However, this mea... | Yes |
Corollary 18.33. If \( {0}^{\sharp } \) does not exist then the Singular Cardinal Hypothesis holds. | Proof. Let \( \kappa \) be such that \( {2}^{\text{cf }\kappa } < \kappa \), and let \( A = {\left\lbrack \kappa \right\rbrack }^{\text{cf }\kappa } \) be the set of all subsets of \( \kappa \) of size cf \( \kappa \) . We shall show that \( \left| A\right| \leq {\kappa }^{ + } \) . By the Covering Theorem, for every \... | Yes |
If \( {0}^{\sharp } \) does not exist then if \( \kappa \) is a singular cardinal and if there exists a nonconstructible subset of \( \kappa \), then some \( \alpha < \kappa \) has a nonconstructible subset. | Let \( \kappa \) be a singular cardinal and assume that each \( \alpha < \kappa \) has only constructible subsets; we shall show that every subset of \( \kappa \) is constructible. It suffices to show that each subset of \( \kappa \) of size cf \( \kappa \) is constructible: If \( A \subset \kappa \) , let \( \left\{ {... | Yes |
Lemma 18.37. Let \( M \) be sufficiently closed, \( X \subset M \prec {L}_{\tau } \), such that \( \left| X\right| = \) \( \nu = \left| M\right| \), and let \( {L}_{\eta } \) be the transitive collapse of \( M \) . Then \( \eta \) is a cardinal in \( L \). | The proof is by contradiction. Assuming that \( \eta \) is not a cardinal in \( L \), we shall produce a constructible set of size \( \nu \) that covers \( X \) . It is in this proof that we need a finer analysis of constructibility. We start by refining Gödel's Condensation Lemma: | No |
Lemma 18.42. If \( \gamma \) is infinite, \( \alpha < \gamma \) and \( p \subset {L}_{\gamma + 1} \) is finite, then there exists a finite set \( q \subset {L}_{\gamma } \) such that\n\n\[{H}_{1}^{\gamma + 1}\left( {\alpha \cup q}\right) \cap {L}_{\gamma } \subset {H}^{\gamma }\left( {\alpha \cup p}\right)\] | Proof. This is quite routine when \( p = \varnothing \) . When \( p \) is nonempty, the idea is to replace members of \( p \) by the parameters used in their definitions over \( {L}_{\gamma } \) . We omit the proof. | No |
Lemma 18.43. There exists a model \( M \prec {L}_{\tau } \) such that \( X \subset M,\left| M\right| = \nu = \) \( \left| X\right| \), and if \( {j}^{-1} \) is the transitive collapse of \( M \) onto \( {L}_{\eta } \), then (18.41) for every directed system \( \left\{ {{L}_{{\eta }_{i}},{e}_{i, k} : i, k \in D}\right\}... | The construction of \( M \) proceeds in \( \nu \) steps. At each step \( \xi < \nu \) let \( \left( {\eta \left( \xi \right) ,\rho \left( \xi \right) }\right) \) be the least \( \left( {\eta ,\rho }\right) \) such that for some increasing \( {\left\{ {i}_{n}\right\} }_{n = 0}^{\infty } \subset D \) , there are ordinals... | No |
Theorem 18.30 (and its corollaries) is due to Jensen. | A proof of the theorem appeared in Devlin and Jensen [1975]. Jensen's proof makes use of his fine structure theory, see Jensen [1972]. The present proof is due to Magidor [1990]. | Yes |
Lemma 19.1. In \( L\left\lbrack U\right\rbrack ,\bar{U} \) is a \( \kappa \) -complete nonprincipal ultrafilter on \( \kappa \) . Moreover, if \( U \) is normal, then \( L\left\lbrack U\right\rbrack \vDash \bar{U} \) is normal. | Proof. A straightforward verification. For instance if \( U \) is normal and \( f \in \) \( L\left\lbrack U\right\rbrack \) is a regressive function on \( \kappa \), then for some \( \gamma < \kappa \), the set \( X = \{ \alpha \) : \( f\left( \alpha \right) = \gamma \} \) is in \( U \) ; since \( X \in L\left\lbrack U... | No |
Lemma 19.2. If \( V = L\left\lbrack A\right\rbrack \), and if \( A \subset P\left( {\omega }_{\alpha }\right) \), then \( {2}^{{\aleph }_{\alpha }} = {\aleph }_{\alpha + 1} \). | Proof. Let \( X \) be a subset of \( {\omega }_{\alpha } \). Let \( \lambda \) be a cardinal such that \( A \in {L}_{\lambda }\left\lbrack A\right\rbrack \) and \( X \in {L}_{\lambda }\left\lbrack A\right\rbrack \). Let \( M \) be an elementary submodel of \( \left( {{L}_{\lambda }\left\lbrack A\right\rbrack , \in }\ri... | Yes |
Lemma 19.4. If \( V = L\left\lbrack D\right\rbrack \) and \( D \) is a normal measure on \( \kappa \), then \( \kappa \) is the only measurable cardinal. | Proof. Let us assume that there is a measurable cardinal \( \lambda \neq \kappa \) and let us consider the elementary embedding \( {j}_{U} : V \rightarrow M \) where \( U \) is some nonprin-cipal \( \lambda \) -complete ultrafilter on \( \lambda \) . We shall prove that \( M = L\left\lbrack D\right\rbrack = V \) thus g... | Yes |
Lemma 19.5 (The Factor Lemma). Let us assume that \( {\mathrm{{Ult}}}^{\left( \alpha \right) } \) is wellfounded. Then for each \( \beta \), the iterated ultrapower \( {\operatorname{Ult}}_{{U}^{\left( \alpha \right) }}^{\left( \beta \right) }\left( {\mathrm{{Ult}}}^{\left( \alpha \right) }\right) \) taken in \( {\math... | Moreover, there is for each \( \beta \) an isomorphism \( {e}_{\beta }^{\left( \alpha \right) } \) such that if for all \( \xi \) and \( \eta ,{i}_{\xi ,\eta }^{\left( \alpha \right) } \) denotes the elementary embedding of \( {\operatorname{Ult}}_{{U}^{\left( \alpha \right) }}^{\left( \xi \right) }\left( {\mathrm{{Ult... | Yes |
For every limit ordinal \( \lambda \), if \( {\mathrm{{Ult}}}^{\left( \lambda \right) } \) is well-founded then \( {\mathrm{{Ult}}}^{\left( \lambda \right) } \subset {\mathrm{{Ult}}}^{\left( \alpha \right) } \) for all \( \alpha < \lambda \) . | Proof. \( {\mathrm{{Ult}}}^{\left( \lambda \right) } \) is a class in \( {\mathrm{{Ult}}}^{\left( \alpha \right) } \) ; it is the iterated ultrapower \( {\mathrm{{Ult}}}_{{U}^{\left( \alpha \right) }}^{\left( \beta \right) }\left( {\mathrm{{Ult}}}^{\left( \alpha \right) }\right) \) where \( \alpha + \beta = \lambda \) ... | No |
(i) If \( \gamma < {\kappa }^{\left( \alpha \right) } \), then \( {i}_{\alpha ,\beta }\left( \gamma \right) = \gamma \) for all \( \beta \geq \alpha \) . | Proof. By the Factor Lemma, it suffices to give the proof for \( \alpha = 0 \) . (i) As we know, \( {i}_{0,1}\left( \gamma \right) = \gamma \) for all \( \gamma < \kappa \) . By induction on \( \beta \), if \( {i}_{0,\beta }\left( \gamma \right) = \gamma \) , then \( {i}_{0,\beta + 1}\left( \gamma \right) = {i}_{\beta ... | Yes |
Lemma 19.9. The sequence \( \left\langle {{\kappa }^{\left( \alpha \right) } : \alpha \in \text{Ord}}\right\rangle \) is normal; i.e., increasing and continuous. | Proof. For each \( \alpha ,{\kappa }^{\left( \alpha + 1\right) } = {i}_{\alpha ,\alpha + 1}\left( {\kappa }^{\left( \alpha \right) }\right) > {\kappa }^{\left( \alpha \right) } \) . To show that the sequence is continuous, let \( \lambda \) be a limit ordinal; we want to show that \( {\kappa }^{\left( \lambda \right) }... | Yes |
Let \( D \) be a normal measure on \( \kappa \), and let for each \( \alpha ,{\mathrm{{Ult}}}^{\left( \alpha \right) } \) be the \( \alpha \) th iterated ultrapower \( {\;\operatorname{mod}\;D},{\kappa }^{\left( \alpha \right) } = {i}_{0,\alpha }\left( \kappa \right) \), and \( {D}^{\left( \alpha \right) } = {i}_{0,\al... | Proof. Since for no \( X \) can both \( X \) and its complement contain a final segment of the sequence \( \left\langle {{\kappa }^{\left( \gamma \right) } : \gamma < \lambda }\right\rangle \), it suffices to show that if \( X \in {D}^{\left( \lambda \right) } \), then there is an \( \alpha \) such that \( {\kappa }^{\... | Yes |
If \( E \subset F \) are finite sets of ordinals, then for each \( X \subset {\kappa }^{E} \), \[ X \in {U}_{E}\;\text{ if and only if }\;{\operatorname{in}}_{E, F}\left( X\right) \in {U}_{F}. \] | Proof. By induction on \( \left( {m, n}\right) \) where \( m = \left| E\right| \) and \( n = \left| F\right| \) . Let \( E \subset F \) be finite sets of ordinals. Let \( a \) be the least element of \( F \), and let us assume that \( a \in E \) (if \( a \notin E \), then the proof is similar). Let \( {E}^{\prime } = E... | Yes |
Lemma 19.13 (The Representation Lemma). For every \( \alpha \), the model \( \left( {{\operatorname{Ult}}_{{U}_{\alpha }}\left( V\right) ,{E}_{\alpha }}\right) \) is (isomorphic to) the \( \alpha \) th iterated ultrapower \( {\operatorname{Ult}}_{U}^{\left( \alpha \right) }\left( V\right) \), and the canonical embeddin... | Proof. By induction on \( \alpha \) . The induction step from \( \alpha \) to \( \alpha + 1 \) follows closely the proof of Lemma 19.11; thus let us describe only how to assign to \( {\left\lbrack f\right\rbrack }_{{U}_{\alpha + 1}} \) the corresponding \( {\left\lbrack \widetilde{f}\right\rbrack }_{{U}^{\left( \alpha ... | Yes |
Theorem 19.14 (Kunen).\n\n(i) If \( V = L\left\lbrack D\right\rbrack \) and \( D \) is a normal measure on \( \kappa \), then \( \kappa \) is the only measurable cardinal and \( D \) is the only normal measure on \( \kappa \) . | The proof of Theorem 19.14 uses iterated ultrapowers. The following lemma uses the representation of iterated ultrapowers.\n\nLemma 19.15. Let \( U \) | No |
Lemma 19.15. Let \( U \) be a \( \kappa \) -complete nonprincipal ultrafilter on \( \kappa \) and let, for each \( \alpha ,{i}_{0,\alpha } : V \rightarrow {\mathrm{{Ult}}}^{\left( \alpha \right) } \) be the embedding of \( V \) in its \( \alpha \) th iterated ultrapower.\n\n(i) If \( \alpha \) is a cardinal and \( \alp... | Proof. It follows from the Representation Lemma that for all \( \xi ,\eta \), the ordinals below \( {i}_{0,\xi }\left( \eta \right) \) are represented by functions with finite support from \( {\kappa }^{\xi } \) into \( \eta \) and hence \( \left| {{i}_{0,\xi }\left( \eta \right) }\right| \leq \left| \xi \right| \cdot ... | Yes |
Lemma 19.16. Assume that in \( L\left\lbrack D\right\rbrack, D \) is a normal measure on \( \kappa \) . Let \( A \) be a set of ordinals of size at least \( {\kappa }^{ + } \) and let \( \theta \) be a cardinal such that \( D \in {L}_{\theta }\left\lbrack D\right\rbrack \) and \( A \subset {L}_{\theta }\left\lbrack D\r... | Proof. Let \( \pi \) be the transitive collapse of \( M \) . We have \( \pi \left( M\right) = {L}_{\alpha }\left\lbrack D\right\rbrack \) for some \( \alpha \), and since \( A \subset M \), we have necessarily \( \alpha \geq {\kappa }^{ + } \) . By Lemma 19.2, every \( X \subset \kappa \) in \( L\left\lbrack D\right\rb... | Yes |
Lemma 19.17. Let \( D \subset P\left( \kappa \right) \) be such that \( D \in L\left\lbrack D\right\rbrack \) and \[ L\left\lbrack D\right\rbrack \vDash D\text{is a normal measure on}\kappa \text{.} \] For each \( \alpha \), let \( {\operatorname{Ult}}_{D}^{\left( \alpha \right) }\left( {L\left\lbrack D\right\rbrack }\... | Proof. First, we have \( {i}_{0,\lambda }\left( \kappa \right) = \lambda \) by Lemma 19.15(i) because \( \lambda > {\kappa }^{ + } \geq \) \( {\left( {\kappa }^{ + }\right) }^{L\left\lbrack D\right\rbrack } = {\left( {2}^{\kappa }\right) }^{L\left\lbrack D\right\rbrack } \) . Let \( {D}^{\left( \lambda \right) } = {i}_... | Yes |
Lemma 19.20. Let \( U \) be a nonprincipal \( \kappa \) -complete ultrafilter on \( \kappa \). Then \( L\left\lbrack U\right\rbrack = L\left\lbrack D\right\rbrack \) where \( D \) is the normal measure on \( \kappa \) in \( L\left\lbrack D\right\rbrack \). | Proof. By the absoluteness of \( L\left\lbrack D\right\rbrack \), we have \( L\left\lbrack D\right\rbrack \subset L\left\lbrack U\right\rbrack \) because \( L\left\lbrack U\right\rbrack \) satisfies that \( \kappa \) is measurable. Thus it suffices to prove that \( U \cap L\left\lbrack D\right\rbrack \in L\left\lbrack ... | Yes |
Lemma 19.21. Assume \( V = L\left\lbrack D\right\rbrack \) . If \( U \) is a nonprincipal \( \kappa \) -ultrafilter on \( \kappa \) , then there exists some \( \delta < {i}_{0,\omega }\left( \kappa \right) \) such that \[ U = \left\{ {X \subset \kappa : \delta \in {i}_{0,\omega }\left( X\right) }\right\} \] | Proof. Let \( j = {j}_{U} \) be the canonical embedding of \( V = L\left\lbrack D\right\rbrack \) in \( {\mathrm{{Ult}}}_{U} \) . We have \( j\left( \kappa \right) = {i}_{0,\alpha }\left( \kappa \right) \) for some \( \alpha \) . We shall show that \( \alpha \) is a finite number; then the lemma follows by (19.15). Fir... | Yes |
Corollary 19.22. If \( V = L\left\lbrack D\right\rbrack \), there are exactly \( {\kappa }^{ + } \) nonprincipal \( \kappa \) -complete ultrafilters on \( \kappa \) . | Proof. If \( \kappa \) is measurable, then it is easy to obtain \( {2}^{\kappa } \) nonprincipal \( \kappa \) -complete ultrafilters on \( \kappa \) (because there are \( {2}^{\kappa } \) subsets of \( \kappa \) of size \( \kappa \) such that \( \left| {X \cap Y}\right| < \) \( \kappa \) for any two of them). By Lemma ... | Yes |
Lemma 19.24. If there is a \( \kappa \) -complete nonprincipal ultrafilter \( U \) on \( \kappa \) such that \( {j}_{U}\left( \kappa \right) \geq {i}_{0,\omega }\left( \kappa \right) \), then (19.16) holds. | Proof. Let us work in the model \( M = {\operatorname{Ult}}_{U}\left( V\right) \) . The cardinal \( j\left( \kappa \right) \) is measurable while \( {i}_{0,\omega }\left( \kappa \right) \) has cofinality \( \omega \), and so \( {i}_{0,\omega }\left( \kappa \right) < j\left( \kappa \right) \) . Let \( F \) be the collec... | Yes |
Lemma 19.27. If \( {\operatorname{Ult}}_{U}\left( M\right) \) is well-founded, and \( N \) is the transitive collapse of the ultrapower, then\n\n(i) \( {P}^{M}\left( \kappa \right) = {P}^{N}\left( \kappa \right) \) .\n\n(ii) \( {j}^{u}U \) is a normal iterable \( N \) -ultrafilter on \( j\left( \kappa \right) \) . | Proof. (i) It is a routine verification by induction that \( j\left( \alpha \right) = \alpha \) for all \( \alpha < \kappa \) . For every \( X \in {P}^{M}\left( \kappa \right) \), we have \( X = j\left( X\right) \cap \kappa \), and therefore \( X \in {P}^{N}\left( \kappa \right) \) , verifying \( {P}^{M}\left( \kappa \... | Yes |
Lemma 19.32. The Mitchell order is well-founded. | Proof. Toward a contradiction, let \( \kappa \) be the least measurable cardinal on which the Mitchell order is not well-founded, and let \( {U}_{0} > {U}_{1} > \ldots > {U}_{n} > \) ... be a descending sequence of normal measures on \( \kappa \) . Let \( M = {\operatorname{Ult}}_{{U}_{0}}\left( V\right) \) and let \( ... | Yes |
Lemma 19.34. Let \( o \) be the function \( \langle o\left( \alpha \right) : \alpha < \kappa \rangle \) . If \( U \) is a normal measure on \( \kappa \) then \( o\left( U\right) = {\left\lbrack o\right\rbrack }_{U} \) . | Proof. Clearly, \( {\left\lbrack o\right\rbrack }_{U} = {o}^{M}\left( \kappa \right) \) where \( M = {\operatorname{Ult}}_{U}\left( V\right) \) . The set \( \left\{ {{U}^{\prime } : {U}^{\prime } < U}\right\} \) is the set of all normal measures in \( M \), and since \( < \) is absolute for \( M \) (see Lemma 19.32), t... | Yes |
Theorem 19.37 (Mitchell). Let \( A \subset \kappa \), and let \( \mathcal{U} = {\left\langle {U}_{\alpha },{U}^{0},{U}^{1}\right\rangle }_{\alpha \in A} \) and \( \mathcal{W} = {\left\langle {W}_{\alpha },{W}^{0},{W}^{1}\right\rangle }_{\alpha \in A} \) be such that for each \( \alpha \in A,{U}_{\alpha } \) and \( {W}_... | Proof. We use Lemma 19.35. Let \( \mathcal{D} \) be the following set of measures: The \( {U}_{\alpha } \) ’s, the \( {W}_{\alpha } \) ’s, \( {U}^{1},{U}^{0},{W}^{1},{W}^{0} \), and all the normal measures on \( \kappa \) in \( {j}_{{U}^{1}}\left( {\mathcal{U} \cup \mathcal{W}}\right) \) and \( {j}_{{W}^{1}}\left( {\ma... | Yes |
Theorem 19.3 (the proof of the GCH in \( L\left\lbrack D\right\rbrack \) ) | is due to Silver [1971d]. | No |
Lemma 20.2. The following are equivalent, for any regular cardinal \( \kappa \) :\n\n(i) For any set \( S \), every \( \kappa \) -complete filter on \( S \) can be extended to a \( \kappa \) - complete ultrafilter on \( S \) .\n\n(ii) For any \( A \) such that \( \left| A\right| \geq \kappa \), there exists a fine meas... | Proof. (i) \( \rightarrow \) (ii) is clear.\n\n(ii) \( \rightarrow \) (iii): Let \( \sum \) be a set of sentences of \( {\mathcal{L}}_{\kappa ,\omega } \) and assume that every \( S \subset \sum \) of size less than \( \kappa \) has a model, say \( {\mathfrak{A}}_{S} \) . Let \( U \) be a fine measure on \( {P}_{\kappa... | Yes |
Theorem 20.4 (Kunen). If there exists a strongly compact cardinal then there exists an inner model with two measurable cardinals. | Kunen proved a stronger version (and the proof can be so modified): For every ordinal \( \vartheta \) there exists an inner model with \( \vartheta \) measurable cardinals. This was improved by Mitchell who showed that the existence of a strongly compact cardinal leads to an inner model that has a measurable cardinal \... | Yes |
Lemma 20.5. Let \( \kappa \) be an inaccessible cardinal. There exists a family \( \mathcal{G} \) of functions \( g : \kappa \rightarrow \kappa \) such that \( \left| \mathcal{G}\right| = {2}^{\kappa } \), and whenever \( \mathcal{H} \subset \mathcal{G} \) is a subfamily of size \( < \kappa \) and \( \left\{ {{\beta }_... | Proof. Let \( \mathcal{A} \) be a family of almost disjoint subsets of \( \kappa \) (i.e., \( \left| A\right| = \kappa \) for each \( A \in \mathcal{A} \) and \( \left| {A \cap B}\right| < \kappa \) for any distinct \( A, B \in \mathcal{A} \) ), such that \( \left| \mathcal{A}\right| = {2}^{\kappa } \) . For each \( A ... | Yes |
Lemma 20.6. Let \( \kappa \) be a strongly compact cardinal. For every \( \delta < {\left( {2}^{\kappa }\right) }^{ + } \) there exists a \( \kappa \) -complete ultrafilter \( U \) on \( \kappa \) such that \( {j}_{U}\left( \kappa \right) > \delta \) . | Proof. Let \( \delta < {\left( {2}^{\kappa }\right) }^{ + } \) . Let \( \mathcal{G} \) be a family of functions \( g : \kappa \rightarrow \kappa \) of size \( \left| \delta \right| \) with the property stated in Lemma 20.5; let us enumerate \( \mathcal{G} = \left\{ {{g}_{\alpha } : \alpha \leq \delta }\right\} \) . For... | Yes |
Theorem 20.8 (Solovay). If \( \kappa \) is a strongly compact cardinal, then the Singular Cardinal Hypothesis holds above \( \kappa \) . That is, if \( \lambda > \kappa \) is a singular cardinal, then \( {2}^{\operatorname{cf}\lambda } < \lambda \) implies \( {\lambda }^{\operatorname{cf}\lambda } = {\lambda }^{ + } \)... | We shall prove the theorem in a sequence of lemmas. An ultrafilter on \( \lambda \) is uniform if every set in the ultrafilter has size \( \lambda \) . | No |
Lemma 20.9. If \( \kappa \) is a strongly compact cardinal and \( \lambda > \kappa \) is a regular cardinal, then there exists a \( \kappa \) -complete uniform ultrafilter \( D \) on \( \lambda \) with the property that almost all \( \left( {\;\operatorname{mod}\;D}\right) \) ordinals \( \alpha < \lambda \) have cofina... | Proof. Let \( U \) be a fine measure on \( {P}_{\kappa }\left( \lambda \right) \) . Since \( U \) is fine, every \( \alpha < \lambda \) belongs to almost all \( \left( {\;\operatorname{mod}\;U}\right) x \in {P}_{\kappa }\left( \lambda \right) \) . Let us consider the ultrapower \( {\operatorname{Ult}}_{U}\left( V\right... | Yes |
Lemma 20.10. If \( \kappa \) is strongly compact and \( \lambda > \kappa \) is a regular cardinal, then there exist a \( \kappa \) -complete nonprincipal ultrafilter \( D \) on \( \lambda \) and a collection \( \left\{ {{M}_{\alpha } : \alpha < \lambda }\right\} \) such that\n\n(20.19)\n\n(i) \( \left| {M}_{\alpha }\ri... | Proof. Let \( D \) be the ultrafilter on \( \lambda \) constructed in Lemma 20.9. It follows from the construction of \( D \) that \( {\left\lbrack d\right\rbrack }_{D} = \mathop{\lim }\limits_{{\gamma \rightarrow \lambda }}{j}_{D}\left( \gamma \right) \) . For almost all \( \alpha \) \( \left( {\;\operatorname{mod}\;D... | Yes |
If \( \kappa \) is strongly compact and \( \lambda > \kappa \) is a regular cardinal, then there exists a collection \( \left\{ {{M}_{\alpha } : \alpha < \lambda }\right\} \subset {P}_{\kappa }\left( \lambda \right) \) such that \( {P}_{\kappa }\left( \lambda \right) = \mathop{\bigcup }\limits_{{\alpha < \lambda }}P\le... | Let \( \left\{ {{M}_{\alpha } : \alpha < \lambda }\right\} \) be as in Lemma 20.10. If \( x \) is a subset of \( \lambda \) of size less than \( \kappa \), then by (20.19)(ii) and by \( \kappa \) -completeness of \( D, x \subset {M}_{\alpha } \) for almost all \( \alpha \) . Hence \( x \in P\left( {M}_{\alpha }\right) ... | No |
Lemma 20.13. If \( U \) is a normal measure on \( {P}_{\kappa }\left( \lambda \right) \), then \( \left\lbrack d\right\rbrack = \{ j\left( \gamma \right) : \gamma < \lambda \} = {j}^{u}\lambda \), and hence for every \( X \subset {P}_{\kappa }\left( \lambda \right) \), \( X \in U\;\text{ if and only if }\;{j}^{\alpha }... | Proof. On the one hand, if \( \gamma < \lambda \), then \( \gamma \in x \) for almost all \( x \) and hence \( j\left( \gamma \right) \in \left\lbrack d\right\rbrack \) . On the other hand, if \( \left\lbrack f\right\rbrack \in \left\lbrack d\right\rbrack \), then \( f\left( x\right) \in x \) for almost all \( x \) and... | Yes |
Lemma 20.14. Let \( \lambda \geq \kappa \) . A normal measure on \( {P}_{\kappa }\left( \lambda \right) \) exists if and only if there exists an elementary embedding \( j : V \rightarrow M \) such that\n\n\( \left( {20.24}\right) \)\n\n(i) \( j\left( \gamma \right) = \gamma \) for all \( \gamma < \kappa \) ;\n\n(ii) \(... | Proof. (a) Let \( U \) be a normal measure on \( {P}_{\kappa }\left( \lambda \right) \) . We let \( M = {\operatorname{Ult}}_{U}\left( V\right) \) and let \( j \) be the canonical elementary embedding \( j : V \rightarrow \) Ult. We have already proved (i) and (ii). To prove (iii), it suffices to show that whenever \( ... | Yes |
Lemma 20.15. If \( \kappa \) is \( \lambda \) -supercompact and \( {2}^{\alpha } = {\alpha }^{ + } \) for every \( \alpha < \kappa \), then \( {2}^{\alpha } = {\alpha }^{ + } \) for every \( \alpha \leq \lambda \) . | Proof. Let \( j : V \rightarrow M \) witness that \( \kappa \) is \( \lambda \) -supercompact. If \( \alpha \leq \lambda \), then because \( \lambda < j\left( \kappa \right) \) and by elementarity, \( {\left( {2}^{\alpha }\right) }^{M} = {\left( {\alpha }^{ + }\right) }^{M} \) . Now \( {M}^{\lambda } \subset M \) impli... | Yes |
Lemma 20.16. If \( \kappa \) is supercompact, then there exists a normal measure \( D \) on \( \kappa \) such that almost every \( \alpha < \kappa \left( {\;\operatorname{mod}\;D}\right) \) is measurable. In particular, \( \kappa \) is the \( \kappa \) th measurable cardinal. | Proof. Let \( \lambda = {2}^{\kappa } \) and let \( j : V \rightarrow M \) witness the \( \lambda \) -supercompactness of \( \kappa \) . Let \( D \) be defined by \( D = \{ X : \kappa \in j\left( X\right) \} \), and let \( {j}_{D} : V \rightarrow {\operatorname{Ult}}_{D} \) be the corresponding elementary embedding. Le... | Yes |
Lemma 20.17. Let \( \kappa \) be a measurable cardinal such that there are \( \kappa \) strongly compact cardinals below \( \kappa \) . Then \( \kappa \) is strongly compact. | Proof. Let \( F \) be a nonprincipal \( \kappa \) -complete ultrafilter on \( \kappa \) such that \( C \in F \) where \( C = \{ \alpha < \kappa : \alpha \) is strongly compact \( \} \) . Let \( A \) be such that \( \left| A\right| \geq \kappa \) ; we shall show that there is a fine measure on \( {P}_{\kappa }\left( A\r... | Yes |
If there exists a measurable cardinal that is a limit of strongly compact cardinals, then the least such cardinal is strongly compact but not supercompact. | Let \( \kappa \) be the least measurable limit of compact cardinals. By Lemma 20.17, \( \kappa \) is strongly compact. Let us assume that \( \kappa \) is supercompact. Let \( \lambda = {2}^{\kappa } \) and let \( j : V \rightarrow M \) be an elementary embedding such that \( \kappa \) is the least ordinal moved, and th... | Yes |
(i) If \( \lambda \geq \kappa \) and if \( \kappa \) is \( \mu \) -supercompact, where \( \mu = {2}^{{\lambda }^{ < \kappa }} \), then for every \( \mathcal{X} \subset P\left( {{P}_{\kappa }\left( \lambda \right) }\right) \) there exists a normal measure on \( {P}_{\kappa }\left( \lambda \right) \) such that \( \mathca... | Proof. (i) Assume on the contrary that there exists some \( \mathcal{X} \subset P\left( {{P}_{\kappa }\left( \lambda \right) }\right) \) such that \( \varphi \left( {\mathcal{X},\kappa ,\lambda }\right) \) where \( \varphi \) is the statement\n\n(20.28)\n\n\[ \mathcal{X} \notin {\operatorname{Ult}}_{U}\text{for every n... | Yes |
Theorem 20.21 (Laver). Let \( \kappa \) be a supercompact cardinal. There exists a function \( f : \kappa \rightarrow {V}_{\kappa } \) such that for every set \( x \) and every \( \lambda \geq \kappa \) such that \( \lambda \geq \) \( \left| {\mathrm{{TC}}\left( x\right) }\right| \) there exists a normal measure \( U \... | Proof. Assume that the theorem is false. For each \( f : \kappa \rightarrow {V}_{\kappa } \), let \( {\lambda }_{f} \) be the least cardinal \( {\lambda }_{f} \geq \kappa \) for which there exists an \( x \) with \( \left| {\operatorname{TC}\left( x\right) }\right| \leq {\lambda }_{x} \) such that \( {j}_{U}\left( f\ri... | Yes |
Lemma 20.23. Let \( \lambda \geq \kappa \) be a regular cardinal and let \( \kappa \) be \( \lambda \) -supercompact. Let \( \alpha < \kappa \) . If \( \alpha \) is \( \gamma \) -supercompact for all \( \gamma < \kappa \), then \( \alpha \) is \( \lambda \) -supercompact. | Proof. Let \( U \) be a normal measure on \( {P}_{\kappa }\left( \lambda \right) \), and let us consider \( {j}_{U} : V \rightarrow \) \( {\operatorname{Ult}}_{U} \) . Since \( j\left( \alpha \right) = \alpha \), we have Ult \( \vDash \left( {\alpha \text{is}\gamma \text{-supercompact for all}\gamma < j\left( \kappa \r... | Yes |
Lemma 20.25. If Vopěnka’s Principle holds, then there exists an extendible cardinal. | Proof. Let \( A \) be the class of all limit ordinals \( \alpha \) such that cf \( \alpha = \omega \) and that for every \( \kappa < \alpha \), if \( {V}_{\alpha } \vDash \) ( \( \kappa \) is extendible), then \( \kappa \) is extendible; and for \( \kappa < \gamma < \alpha \) , if there is an elementary embedding \( j ... | Yes |
Lemma 20.27. If \( \kappa \) is a huge cardinal, then Vopěnka’s Principle is consistent: \( \left( {{V}_{\kappa }, \in }\right) \) is a model of VP. | Proof. We shall show that if \( C \) is a set of models and \( \operatorname{rank}\left( C\right) = \kappa \), then there exist two members \( \mathfrak{A},\mathfrak{B} \in C \) and an elementary embedding \( h : \mathfrak{A} \rightarrow \mathfrak{B} \) .\n\nLet \( j : V \rightarrow M \) be such that \( \kappa \) is th... | Yes |
(i) \( k\left( \alpha \right) = \alpha \) for all \( \alpha < \lambda \) . | For each \( a \in {\left\lbrack \lambda \right\rbrack }^{ < \omega } \), let \( {j}_{a,\infty } : {\mathrm{{Ult}}}_{{E}_{a}} \rightarrow {\mathrm{{Ult}}}_{E} \) be the direct limit embedding such that \( {j}_{a,\infty } \circ {j}_{a} = {j}_{E} \) ; then \( k \circ {j}_{a,\infty } = {k}_{a} \) . If \( x \in {\operatorna... | Yes |
Theorem 21.1 (Lévy-Solovay). Let \( \kappa \) be a measurable cardinal in the ground model. Let \( \left( {P, < }\right) \) be a notion of forcing such that \( \left| P\right| < \kappa \) . Then \( \kappa \) is measurable in the generic extension. | Proof. We give a proof using elementary embeddings since similar arguments will be used in subsequent constructions; for a direct proof, see Exercise 21.1. Let \( B = B\left( P\right) \) ; since \( \left| B\right| < \kappa \), we may assume \( B \in {V}_{\kappa } \) . We can also assume that \( P \) is a dense subset o... | No |
Theorem 21.2. Let \( \kappa \) be an infinite cardinal, and let \( \left( {P, < }\right) \) be a notion of forcing such that \( \left| P\right| < \kappa \) . Let \( G \) be a \( V \) -generic filter on \( P \) . Then \( \kappa \) is inaccessible (Mahlo, weakly compact, Ramsey, measurable, strongly compact, supercompact... | Proof. If \( \kappa \) is inaccessible in \( V \), then firstly \( \kappa \) is regular in \( V\left\lbrack G\right\rbrack \) because all cardinals and cofinalities above \( \left| P\right| \) are preserved. Secondly, if \( \alpha < \kappa \), then \( {\left( {2}^{\alpha }\right) }^{V\left\lbrack G\right\rbrack } \leq ... | Yes |
Theorem 21.3 (Kunen-Paris). Assume GCH and let \( \kappa \) be a measurable cardinal. Let \( D \) be a normal measure on \( \kappa \) and let \( A \) be a set of regular cardinals below \( \kappa \) such that \( A \notin D \) ; let \( F \) be a function on \( A \) such that \( F\left( \alpha \right) < \kappa \) for all... | Proof. Let \( j : V \rightarrow M \) be the elementary embedding given by the ultra-power \( {\mathrm{{Ult}}}_{D} \) . As we assume that \( A \notin D \), we have \( \kappa \notin j\left( A\right) \).\n\nLet \( \left( {P, < }\right) \) be the Easton product of \( {P}_{\alpha },\alpha \in A \), where each \( {P}_{\alpha... | Yes |
Theorem 21.4 (Silver). If there exists a supercompact cardinal \( \kappa \), then there is a generic extension in which \( \kappa \) is a measurable cardinal and \( {2}^{\kappa } > {\kappa }^{ + } \) . | Silver’s construction uses iterated forcing. As \( {2}^{\kappa } > {\kappa }^{ + } \) for a measurable cardinal implies that \( {2}^{\alpha } > {\alpha }^{ + } \) for many \( \alpha \) below \( \kappa \), the iteration adjoins not only subsets of \( \kappa \), but, iteratively, subsets of regular cardinals below \( \ka... | No |
(i) If \( P \) is \( \lambda \) -directed closed, and if \( { \Vdash }_{P}\dot{Q} \) is \( \lambda \) -directed closed, then \( P * \dot{Q} \) is \( \lambda \) -directed closed. | Proof. (i) Let \( D = \left\{ {\left( {{p}_{\alpha },{\dot{q}}_{\alpha }}\right) : \alpha < \lambda }\right\} \) be a directed subset of \( P * \dot{Q} \) . Clearly, \( {D}_{1} = \left\{ {{p}_{\alpha } : \alpha < \lambda }\right\} \) is a directed subset of \( P \) and hence there is \( p \in P \) stronger than all \( ... | Yes |
Lemma 21.8 (The Factor Lemma). Let \( {P}_{\alpha + \beta } \) be a forcing iteration of \( \left\langle {{\dot{Q}}_{\xi } : \xi < \alpha + \beta }\right\rangle \), where each \( {P}_{\xi },\xi \leq \alpha + \beta \) is either a direct limit or inverse limit. In \( {V}^{{P}_{\alpha }} \), let \( {\dot{P}}_{\beta }^{\le... | Proof. By induction on \( \beta \) . Let \( \beta \) be an ordinal number; we shall construct an isomorphism \( \pi \) between \( {P}_{\alpha } * {\dot{P}}_{\beta }^{\left( \alpha \right) } \) and \( {P}_{\alpha + \beta } \) .\n\nIf \( \beta = 0 \), then \( {P}_{\alpha } * {\dot{P}}_{\beta }^{\left( \alpha \right) } = ... | Yes |
Lemma 21.9. \( {\left( M\left\lbrack G\right\rbrack \right) }^{\lambda } \cap V\left\lbrack G\right\rbrack \subset M\left\lbrack G\right\rbrack \) . | Proof. It suffices to show that if \( f \in V\left\lbrack G\right\rbrack \) is a function from \( \lambda \) into ordinals, then \( f \in M\left\lbrack G\right\rbrack \) . Let \( \dot{f} \) be a name for \( f \) and let \( {p}_{0} \in G \) be a condition that\n\nforces that \( \dot{f} \) is a function from \( \lambda \... | Yes |
Theorem 21.10 (Prikry). Let \( \kappa \) be a measurable cardinal. There is a generic extension in which \( \operatorname{cf}\kappa = \omega \) and no cardinals are collapsed. Moreover, every bounded subset of \( \kappa \) in \( V\left\lbrack G\right\rbrack \) is in the ground model. | Proof. Let \( \kappa \) be a measurable cardinal and let \( D \) be a normal measure on \( \kappa \) . Let \( \left( {P, < }\right) \) be the following notion of forcing. A forcing condition is a pair \( p = \left( {s, A}\right) \) where \( s \in {\left\lbrack \kappa \right\rbrack }^{ < \omega } \), i.e., \( s \) is a ... | Yes |
Lemma 21.11. Let \( \sigma \) be a sentence of the forcing language. There exists a set \( A \in D \) such that the condition \( \left( {\varnothing, A}\right) \) decides \( \sigma \), i.e., either \( \left( {\varnothing, A}\right) \Vdash \sigma \) , or \( \left( {\varnothing, A}\right) \Vdash \neg \sigma \) . | Proof. Let \( {S}^{ + } be the set of all \( s \in {\left\lbrack \kappa \right\rbrack }^{ < \omega } \) such that \( \left( {s, X}\right) \Vdash \sigma \) for some \( X \) and let \( {S}^{ - } = \{ s : \exists X\left( {s, X}\right) \Vdash \neg \sigma \} \) . Let \( T = {\left\lbrack \kappa \right\rbrack }^{ < \omega } ... | Yes |
Lemma 21.12. Let \( \sigma \) be a sentence of the forcing language and let \( \left( {{s}_{0},{A}_{0}}\right) \) be a condition. Then there exists a set \( A \subset {A}_{0} \) in \( D \) such that the condition \( \left( {{s}_{0}, A}\right) \) decides \( \sigma \) . | Proof. A slight modification of the preceding proof; we may assume that \( \min \left( {A}_{0}\right) > \max \left( {s}_{0}\right) \) . Let \( {S}^{ + } \) be the set of all \( s \in {\left\lbrack {A}_{0}\right\rbrack }^{ < \omega } \) such that \( \left( {{s}_{0} \cup s, X}\right) \Vdash \sigma \) for some \( X \subse... | Yes |
Corollary 21.13. It is consistent (relative to the existence of a supercompact cardinal) that there is a strong limit singular cardinal \( \kappa \) such that \( {2}^{\kappa } > {\kappa }^{ + } \) . | Proof. Let \( \kappa \) be a supercompact cardinal. First we construct a generic extension in which \( \kappa \) is measurable and \( {2}^{\kappa } > {\kappa }^{ + } \) . Then we extend the model further to make \( \kappa \) a singular cardinal. The new model still satisfies \( {2}^{\kappa } > {\kappa }^{ + } \) , and ... | Yes |
Theorem 21.14 (Mathias). Let \( M \) be a transitive model of ZFC, let \( U \) be, in \( M \), a normal measure on \( \kappa \), and let \( P \) be the Prikry forcing defined from \( U \). For every set \( S \subset \kappa \) of order-type \( \omega, S \) is \( P \)-generic over \( M \) if and only if for every \( X \i... | Proof. In the easy direction, let \( G \) be a generic filter on \( P \) and let \( S = \) \( \bigcup \{ s : \left( {s, A}\right) \in G\} \). For every \( X \in U, S - X \) is finite because for every condition \( \left( {s, A}\right) \), the stronger condition \( \left( {s, A \cap X}\right) \) forces that every \( \al... | Yes |
Lemma 21.17. Let \( \kappa \) be measurable in \( M \), and let \( N \) be a symmetric extension of \( M \) (via \( B, G,\mathcal{F},\mathcal{G} \) ). If every symmetric subset of \( B \) has size \( < \kappa \) , then \( \kappa \) is measurable in \( N \) . | Proof. Let \( U \) be, in \( M \), a \( \kappa \) -complete nonprincipal ultrafilter on \( \kappa \) . We show that \( U \) generates a \( \kappa \) -complete nonprincipal ultrafilter on \( \kappa \) in \( N \) . It suffices to show that if \( \gamma < \kappa \) and \( \left\{ {{X}_{\alpha } : \alpha < \gamma }\right\}... | Yes |
Lemma 22.4. Let \( I \) be a normal \( \sigma \) -saturated \( \kappa \) -complete ideal on \( \kappa \) . If \( S \) is a set of positive measure and \( f : S \rightarrow \kappa \) is regressive on \( S \), then \( f \) is bounded almost everywhere on \( S \) ; that is, there exists \( \gamma < \kappa \) such that \( ... | Proof. For every \( X \subset S \) of positive measure there exists \( Y \subset X \) of positive measure such that \( f \) is constant on \( Y \) . Thus let \( W \) be a maximal disjoint family of sets \( X \subset S \) of positive measure such that \( f \) is constant on \( X \) . Let \( T = \bigcup \{ X : X \in W\} ... | Yes |
If \( \kappa \) is real-valued measurable (or if \( \kappa \) carries a \( \sigma \) -saturated \( \kappa \) -complete ideal), then \( \kappa \) is a weakly Mahlo cardinal. | Let \( I \) be a normal \( \sigma \) -saturated \( \kappa \) -complete ideal on \( \kappa \) . Since \( I \) is normal, every closed unbounded set has \( I \) -measure one (see Lemma 8.11). Because \( \kappa \) is weakly inaccessible, it suffices to show that the set of all regular cardinals \( \alpha < \kappa \) has m... | Yes |
Lemma 22.6. Let \( I \) be a normal \( \sigma \) -saturated \( \kappa \) -complete ideal on \( \kappa \), and let \( \lambda \) be an infinite cardinal less than \( \kappa \) . Let \( \mathfrak{A} = \left( {A,\ldots }\right) \) be a model of a language \( \mathcal{L} \) such that \( \left| \mathcal{L}\right| \leq \lamb... | The proof of Lemma 22.6 uses Skolem functions and arguments similar to those in Theorem 17.27 and Lemma 17.36. The key ingredient is the following lemma: Lemma 22.7. Let \( I \) | No |
Lemma 22.8. Let \( \nu < \kappa \) be a regular uncountable cardinal, and let \( I \) be a normal \( \nu \) -saturated \( \kappa \) -complete ideal on \( \kappa \) ; let \( F \) be the dual filter. Then in \( L\left\lbrack F\right\rbrack, F \cap L\left\lbrack F\right\rbrack \) is a normal measure on \( \kappa \) (and \... | Proof. It is easy to verify that \( L\left\lbrack F\right\rbrack = L\left\lbrack I\right\rbrack \), and that in \( L\left\lbrack F\right\rbrack, I \cap L\left\lbrack I\right\rbrack \) is a normal \( \nu \) -saturated \( \kappa \) -complete ideal on \( \kappa \) . Thus we may assume that \( V = \) \( L\left\lbrack F\rig... | Yes |
Example 22.10 (A model in which \( {2}^{{\aleph }_{0}} \) carries a \( \sigma \) -saturated ideal). Let \( \kappa \) be a measurable cardinal, and let \( \lambda \geq \kappa \) be a cardinal such that \( {\lambda }^{{\aleph }_{0}} = \lambda \). We shall construct a generic extension that satisfies \( {2}^{{\aleph }_{0}... | Let \( P \) be the notion of forcing that adjoins \( \lambda \) Cohen reals; i.e., a condition is a finite \( 0 - 1 \) function \( p \) with \( \operatorname{dom}\left( p\right) \subset \lambda \). If \( G \) is a generic filter on \( P \), then \( V\left\lbrack G\right\rbrack \vDash {2}^{{\aleph }_{0}} = \lambda \), a... | Yes |
Lemma 22.11. Let \( \kappa \) be a measurable cardinal and let \( I \) be a nonprincipal \( \kappa \) -complete prime ideal on \( \kappa \) . Let \( P \) be a notion of forcing that satisfies the countable chain condition. Then in \( V\left\lbrack G\right\rbrack \), the ideal \( J \) generated by \( I \) is a \( \sigma... | Proof. Let \( J \) be the ideal in \( V\left\lbrack G\right\rbrack \) defined as follows:\n\n\[ X \in J\text{if and only if}X \subset Y\text{for some}Y \in I\text{.}\]\n\nFirst we show that \( J \) is \( \kappa \) -complete. Let \( \mathcal{X} = \left\{ {{X}_{\xi } : \xi < \gamma }\right\} \) be a family of fewer than ... | Yes |
Lemma 22.12. Let \( \kappa \) carry a \( \sigma \) -saturated \( \kappa \) -complete ideal, and let \( \lambda < \kappa \) be a regular uncountable cardinal. If \( \mathcal{F} \) is a family of almost disjoint functions \( f : \lambda \rightarrow \kappa \), then \( \left| \mathcal{F}\right| \leq \kappa \). | Proof. If \( \left| \mathcal{F}\right| > \kappa \), then because every \( f : \lambda \rightarrow \kappa \) is bounded by some \( \beta < \kappa \) , there exist some \( \mathcal{G} \subset \mathcal{F} \) and some \( \beta < \kappa \) such that \( \left| \mathcal{G}\right| = \kappa \) and every \( f \in \mathcal{G} \) ... | Yes |
Lemma 22.20. Let \( \kappa \) be a regular uncountable cardinal. The ideal \( I = \) \( \{ X \subset \kappa : \left| X\right| < \kappa \} \) is not precipitous. | Proof. Let \( I = \{ X \subset \kappa : \left| X\right| < \kappa \} \) . A set \( X \subset \kappa \) has positive measure just in case \( \left| X\right| = \kappa \) . For each such \( X \), let \( {f}_{X} \) be the unique order-preserving function from \( X \) onto \( \kappa \) .\n\nFor each set \( X \) of positive m... | Yes |
I is precipitous if and only if Empty has no winning strategy in the game \( {\mathcal{G}}_{I} \) . | If \( I \) is not precipitous then there is a set \( S \) of positive measure and a sequence of functionals on \( S \) such that \( {F}_{0} > {F}_{1} > \ldots > {F}_{n} > \ldots \) Empty chooses \( {S}_{0} = S \) for his first move. When Nonempty plays \( {S}_{{2n} - 1} \), Empty finds some \( f \in {F}_{n} \) such tha... | Yes |
Lemma 22.22. Let \( \kappa \) be a regular uncountable cardinal. Every \( {\kappa }^{ + } \) -saturated \( \kappa \) -complete ideal on \( \kappa \) is precipitous. | Proof. Let \( I \) be a \( {\kappa }^{ + } \) -saturated \( \kappa \) -complete ideal on \( \kappa \) . Let \( S \) be a set of positive measure and let \( {W}_{0} \geq {W}_{1} \geq \ldots \) be \( I \) -partitions of \( S \) . We shall find \( {X}_{0} \supset {X}_{1} \supset \ldots \) in \( {W}_{0},{W}_{1},\ldots \) s... | Yes |
Lemma 22.25. Let \( \kappa \) be a regular uncountable cardinal. Let \( V\left\lbrack G\right\rbrack \) be a generic extension of \( V \) by a \( \kappa \) -c.c. notion of forcing. Then every closed unbounded \( C \subset \kappa \) in \( V\left\lbrack G\right\rbrack \) has a closed unbounded subset \( D \in V \) . Cons... | Proof. Let \( \dot{C} \) be a name such that every condition forces that \( \dot{C} \) is a closed unbounded subset of \( \kappa \) . Let \( D = \{ \alpha : \parallel \alpha \in \dot{C}\parallel = 1\} \) . Clearly, \( D \) is a subset of \( C \) and is closed; we have to prove only that \( D \) is unbounded.\n\nLet \( ... | Yes |
Theorem 22.26 (Solovay). Let \( \kappa \) be a regular uncountable cardinal and assume that \( \kappa \) carries a \( \kappa \) -saturated ideal.\n\n(i) \( \kappa \) is weakly Mahlo;\n\n(ii) \( \{ \alpha < \kappa : \alpha \) is weakly Mahlo \( \} \) is stationary;\n\n(iii) if \( X \subset \kappa \) has measure one in a... | Proof. If there exists a \( \kappa \) -saturated ideal on \( \kappa \), then \( \kappa \) is weakly inaccessible by Lemma 22.24, and there exists a normal \( \kappa \) -saturated ideal on \( \kappa \) (by Lemma 22.23). Let \( I \) be a normal \( \kappa \) -saturated ideal on \( \kappa \) . We first prove:\n\nLemma 22.2... | No |
Lemma 22.27. If \( S \subset \kappa \) is stationary, then for I-almost all \( \alpha < \kappa, S \cap \alpha \) is stationary in \( \alpha \) . | Proof. If not, then there is a set \( X \) of positive measure such that \( S \cap \alpha \) is not stationary in \( \alpha \) (or cf \( \alpha = \omega \) ) for all \( \alpha \in X \) . Let \( G \) be a generic ultrafilter on \( \kappa \) (corresponding to \( I \) ) such that \( X \in G \) . Let \( N = {\operatorname{... | Yes |
Lemma 22.28. Let \( I \) be a \( {\kappa }^{ + } \) -saturated \( \kappa \) -complete ideal on \( \kappa \). (i) There exists a least unbounded function, i.e., a function \( f : \kappa \rightarrow \kappa \) such that for any \( \gamma < \kappa \) there is no \( S \) of positive measure such that \( f\left( \alpha \righ... | Proof. By Lemma 22.22 \( I \) is precipitous. Since \( I \) is \( {\kappa }^{ + } \) -saturated, the Boolean-valued names for functions on \( \kappa \) in the ground model can be represented not by functionals but by ordinary functions: Let \( F \) be a functional (on \( \kappa \) ). Let \( W = \{ \operatorname{dom}\le... | Yes |
Lemma 22.29. Let \( \kappa \) be a regular uncountable cardinal and let \( I \) be a \( {\kappa }^{ + } \) - saturated ideal on \( \kappa \) . If \( {2}^{\lambda } = {\lambda }^{ + } \) for all \( \lambda < \kappa \), then \( {2}^{\kappa } = {\kappa }^{ + } \) . | Proof. Let \( M \) be the ground model. Let \( P \) be the notion of forcing corresponding to \( I \), let \( G \) be generic on \( P \), and let \( N = {\operatorname{Ult}}_{G}\left( M\right) \) . Since \( I \) is \( {\kappa }^{ + } \) -saturated, \( N \) is well-founded and hence we identify it with a transitive mode... | Yes |
Lemma 22.30. Let \( I \) be an \( {\aleph }_{2} \) -saturated ideal on \( {\omega }_{1} \) . Then\n\n(i) If \( {2}^{{\aleph }_{0}} = {\aleph }_{1} \), then \( {2}^{{\aleph }_{1}} = {\aleph }_{2} \) . | Proof. (i) and (ii) are as in Corollary 22.17.\n\nLet \( G \) be a generic ultrafilter on \( {\omega }_{1} \), let \( N = {\operatorname{Ult}}_{G}\left( M\right) \) and let \( j : M \rightarrow N \) . \( N \) is a transitive model, \( N \subset M\left\lbrack G\right\rbrack \) .\n\nLet us denote \( \kappa = {\omega }_{1... | Yes |
Lemma 22.31. Let \( I \) be a \( {\kappa }^{ + } \) -saturated \( \kappa \) -complete ideal on \( \kappa \) . Let \( G \) be a corresponding generic ultrafilter and let \( N = {\operatorname{Ult}}_{G}\left( M\right) \) be the generic ultra-power. Then every \( s : \kappa \rightarrow M \) in \( M\left\lbrack G\right\rbr... | Proof. Let \( \dot{s} \) be a name for \( s \) ; for each \( \alpha < \kappa \), let \( {\dot{s}}_{\alpha } \) be a name such that \( \begin{Vmatrix}{\dot{s}\left( \alpha \right) = {\dot{s}}_{\alpha }}\end{Vmatrix} = 1 \) . Each \( {\dot{s}}_{\alpha } \) is represented by a function \( {f}_{\alpha } \in M \) on \( \kap... | Yes |
Lemma 22.32. Let \( B \) be a complete Boolean algebra, let \( G \) be a \( V \) -generic ultrafilter on \( B \) and let \( \kappa \) be an uncountable regular cardinal. Let \( \lambda \leq {\kappa }^{ + } \) be regular and assume that \( \operatorname{sat}\left( B\right) \leq \lambda \) and \( \operatorname{sat}\left(... | Proof. Let \( J \in V\left\lbrack G\right\rbrack \) be the ideal generated by \( I \) . Since sat \( B \leq \kappa, J \) is \( \kappa \) - complete. Let \( \dot{J} \in {V}^{B} \) be the canonical name for \( J \), and let \( \dot{C} \in {V}^{B} \) be the Boolean algebra \( \dot{C} = P\left( \check{\kappa }\right) /\dot... | Yes |
(i) If \( \kappa \) is a regular uncountable cardinal that carries a precipitous ideal, then \( \kappa \) is measurable in an inner model of \( \mathrm{{ZFC}} \). | The proof of (i) uses the technique of iterated ultrapowers (compare with (20.5)-(20.8)).\n\nLet \( \kappa \) be a regular uncountable cardinal, and let \( I \) be a precipitous ideal on \( \kappa \) . Let \( C \) be the class of all strong limit cardinals \( \nu > {2}^{\kappa } \) such that cf \( \nu \geq \operatornam... | No |
Lemma 23.1. The following principle is equivalent to \( \langle \rangle \) : There exists a sequence \( \left\langle {{S}_{\alpha } : \alpha < {\omega }_{1}}\right\rangle \) of countable sets such that for each \( X \subset {\omega }_{1} \) the set \( \left\{ {\alpha < {\omega }_{1} : X \cap \alpha \in {S}_{\alpha }}\r... | Proof. Let \( \left\langle {{S}_{\alpha } : \alpha < {\omega }_{1}}\right\rangle \) be a sequence as in the lemma; we shall produce a diamond sequence. First, let \( f \) be a one-to-one mapping of \( {\omega }_{1} \) onto \( {\omega }_{1} \times \omega \) such that \( {f}^{\alpha }\alpha = \alpha \times \omega \) for ... | Yes |
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