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Proposition 13.34. \( \operatorname{STM}\left( {A,\text{ Ax. }{5}_{\psi }}\right) \rightarrow \left( {\forall x \in A}\right) \left\lbrack {\left\{ {y \in x \mid {\varphi }^{A}\left( y\right) }\right\} \in A}\right\rbrack \) where \( \psi \left( {a, b}\right) \) is \( \varphi \left( a\right) \land a = b \) . | Proof. If \( A \) is a transitive model of Axiom \( {5}_{\psi } \) then \( A \) is a nonempty transitive class that satisfies all of the nonlogical axioms required to prove the instance of Zermelo's Schema of Separation:\n\n\[ \left( {\forall x}\right) \left( {\exists y}\right) \left\lbrack {y = \{ z \in x \mid \varphi... | No |
Proposition 13.35. \( A \neq 0 \land \operatorname{Tr}\left( A\right) \rightarrow \operatorname{STM}\left( {A,\mathrm{{Ax}}.6}\right) \) . | Proof. Ax. \( 6 \leftrightarrow \left( {\forall x}\right) \left\lbrack {x \neq 0 \rightarrow \left( {\exists y}\right) \left\lbrack {y \in x \land y \cap x = 0}\right\rbrack }\right\rbrack \) . In particular, from Axiom 6.\n\n\[ \left( {\forall x \in A}\right) \left\lbrack {x \neq 0 \rightarrow \left( {\exists y}\right... | Yes |
Proposition 13.36. \( A \neq 0 \land \operatorname{Tr}\left( A\right) \rightarrow 0 \in A \) . | Proof. By the Axiom of Regularity\n\n\[ \nA \neq 0 \land \operatorname{Tr}\left( A\right) \rightarrow \left( {\exists x}\right) \left\lbrack {x \in A \land x \cap A = 0}\right\rbrack \n\]\n\n\[ \n\rightarrow \left( {\exists x}\right) \left\lbrack {x \in A \land x = 0}\right\rbrack \n\]\n\n\[ \n\rightarrow 0 \in A\text{... | Yes |
Proposition 13.37. \( \operatorname{STM}\left( {A,\mathrm{{Ax}}.2}\right) \land \operatorname{STM}\left( {A,\mathrm{{Ax}}.3}\right) \rightarrow \omega \subseteq A \) . | Proof (By induction). From Proposition 13.36, \( 0 \in A \) . If \( k \in A \), then since \( A \) is a model of Ax. 2, \( \{ k\} \in A \) and hence \( \{ k,\{ k\} \} \in A \) . Since \( A \) is also a model of Ax. \( 3 \cup \{ k,\{ k\} \} \in A \), that is \( k + 1 \in A \) . | Yes |
Proposition 13.38. If \( A \) is a standard transitive model of the \( {Ax} \) iom of Pairing and the Axiom of Unions then \( A \) is a standard transitive model of the Axiom of Infinity iff \( \omega \in A \) . | Proof. Since \( A \) is nonempty and transitive \( A \) is a standard transitive model of the Axiom of Infinity iff\n\n\[ \left( {\exists x \in A}\right) {\left\lbrack x = \omega \right\rbrack }^{A}. \]\n\nBut Propositions 13.37 and 13.21 establish that \( x = \omega \) is absolute w.r.t. \( A \) . Therefore \( A \) is... | Yes |
Proposition 13.39. If \( M = \{ x \mid \mathrm{{Wf}}\left( x\right) \} \) then \( M \) is a standard transitive model of \( \mathrm{{ZF}} \). | Proof. Since \( 0 \in {R}_{1}^{\mathfrak{c}}1 \) we have \( 0 \in M \), i.e., \( M \neq 0 \) . If \( x \in M \) then \( \left( {\exists \alpha }\right) \left\lbrack {x \in {R}_{1}^{\mathfrak{c}}\alpha }\right\rbrack \) . Since \( {R}_{1}^{\epsilon }\alpha \) is transitive \( x \in {R}_{1}^{\epsilon }\alpha \rightarrow ... | Yes |
Proposition 14.3. If \( M \) is a standard transitive model of \( \mathrm{{ZF}} \) then \( M \) is closed under the eight fundamental operations. | Proof. From Proposition 13.25\n\n\[ a \in M \land b \in M \land \operatorname{STM}\left( {M,\text{ Ax. }2}\right) \rightarrow {\mathcal{F}}_{1}\left( {a, b}\right) \in M. \]\n\nSince \( M \) is a model of Axiom \( 2, c = \langle a, b\rangle \) and \( d = \langle a, b, c\rangle \) are each absolute with respect to \( M ... | Yes |
Proposition 14.5. If \( M \) is transitive, almost universal, and closed under the eight fundamental operations and \( a, b \in M \), then\n\n(1) \( \{ a, b\} \in M \) , (3) \( a \times b \in M \) , (5) \( a \cap b \in M \) ,\n\n(2) \( \langle a, b\rangle \in M \) , (4) \( a - b \in M \) , (6) \( a \cup b \in M \) . | Proof.\n\n(1) \( \{ a, b\} = {\mathcal{F}}_{1}\left( {a, b}\right) \in M \) .\n\n(2) \( \langle a, b\rangle = \{ \{ a\} ,\{ a, b\} \} \in M \) .\n\n(3) Since \( M \) is transitive \( a \subseteq M \) and \( b \subseteq M \) . Therefore \( a \times b \subseteq M \) and \( b \times a \subseteq M \) . But \( M \) is also ... | Yes |
Proposition 14.6. If \( M \) is transitive, almost universal, and closed under the eight fundamental operations then\n\n\[ \left( {\forall {x}_{1},\ldots ,{x}_{n} \in M}\right) \left\lbrack {{x}_{1} \times {x}_{2} \times \cdots \times {x}_{n} \in M}\right\rbrack . \] | Proof. Obvious from Proposition 14.5 (3) by induction. | No |
Proposition 14.7. If \( M \) is transitive, almost universal, and closed under the eight fundamental operations and \( a \in M \), then\n\n(1) \( {a}^{n} \in M, n \geqq 1 \) , (3) \( \mathcal{D}\left( a\right) \in M \) , (5) \( {\operatorname{Cnv}}_{2}\left( a\right) \in M \) ,\n\n(2) \( {a}^{-1} \in M \) , (4) \( \mat... | Proof. (1) Obvious from Proposition 14.6.\n\n(2) Since \( M \) is transitive \( \langle b, c\rangle \in a \) and \( a \in M \) imply \( b, c \in M \) . This in turn implies that \( \langle c, b\rangle \in M \) . Therefore, \( {a}^{-1} \subseteq M \) . But since \( M \) is almost universal \( \left( {\exists x \in M}\ri... | No |
Proposition 14.8. If \( M \) is transitive, almost universal, and closed under the eight fundamental operations, if \( \left( {{i}_{1},{i}_{2},{i}_{3}}\right) \) is a permutation of 1,2,3 and if \( a \in M \) then\n\n\[ \left\{ {\left\langle {{x}_{1},{x}_{2},{x}_{3}}\right\rangle \mid \left\langle {{x}_{{i}_{1}},{x}_{{... | Proof.\n\n\[ \left\{ {\left\langle {{x}_{1},{x}_{2},{x}_{3}}\right\rangle \mid \left\langle {{x}_{1},{x}_{3},{x}_{2}}\right\rangle \in a}\right\} = {\operatorname{Cnv}}_{3}\left( a\right) \in M. \]\n\n\[ \left\{ {\left\langle {{x}_{1},{x}_{2},{x}_{3}}\right\rangle \mid \left\langle {{x}_{3},{x}_{1},{x}_{2}}\right\rangl... | Yes |
Proposition 14.9. If \( M \) is transitive, almost universal, and closed under the eight fundamental operations and \( a, b \in M \), then\n\n(1) \( \{ \langle x, y, z\rangle \mid \langle x, y\rangle \in a \land z \in b\} \in M \) ,\n\n(2) \( \{ \langle x, z, y\rangle \mid \langle x, y\rangle \in a \land z \in b\} \in ... | Proof. Obvious from Proposition 14.8 and the fact that \( a \times b \in M \) . | No |
(1) \( \max \left( {\alpha ,\beta }\right) \leqq {J}_{0}^{\iota }\langle \alpha ,\beta \rangle \) . | Proof. (1) If \( \left( {\forall \alpha }\right) \left\lbrack {{F}^{c}\alpha \triangleq {J}_{0}^{c}\langle 0,\alpha \rangle }\right\rbrack \) then \( F \) is a strictly monotonic ordinal function and hence \( \alpha \leqq {F}^{t}\alpha \) . In particular if \( \gamma = \max \left( {\alpha ,\beta }\right) \) then\n\n\[ ... | Yes |
Corollary 15.9. If \( m < 9 \) then\n\n(1) \( {K}_{1}^{c}{J}^{c}\langle \alpha ,\beta, m\rangle = \alpha \) ,\n\n(2) \( {K}_{2}^{c}{J}^{c}\langle \alpha ,\beta, m\rangle = \beta \) ,\n\n(3) \( {K}_{3}^{c}{J}^{c}\langle \alpha ,\beta, m\rangle = m \) . | Details are left to the reader. | No |
(1) \( {K}_{1}^{c}\gamma \leqq \gamma \land {K}_{2}^{c}\gamma \leqq \gamma \) .\n\n(2) \( {K}_{3}^{2}\gamma \neq 0 \rightarrow {K}_{1}^{2}\gamma < \gamma \land {K}_{2}^{2}\gamma < \gamma \) . | Proof. Since \( \gamma = {J}^{i}\left\langle {{K}_{1}^{i}\gamma ,{K}_{2}^{i}\gamma ,{K}_{3}^{i}\gamma }\right\rangle = 9 \cdot {J}_{0}^{i}\left\langle {{K}_{1}^{i}\gamma ,{K}_{2}^{i}\gamma }\right\rangle + {K}_{3}^{i}\gamma \), it follows from properties of ordinal arithmetic (Corollary 8.5 and Proposition 8.21)\n\n\[ ... | Yes |
Proposition 15.11. \( m < 9 \land \alpha < {\aleph }_{\gamma } \land \beta < {\aleph }_{\gamma } \rightarrow {J}^{c}\langle \alpha ,\beta, m\rangle < {\aleph }_{\gamma } \) | Proof. By Proposition 15.1 we have\n\n\[ \n{J}_{0}^{c}\langle \alpha ,\beta \rangle < {\aleph }_{\gamma } \n\]\n\nIf \( {J}_{0}^{c}\langle \alpha ,\beta \rangle < {\aleph }_{0} \) then\n\n\[ \n{J}^{i}\langle \alpha ,\beta, m\rangle = 9 \cdot {J}_{0}^{i}\langle \alpha ,\beta \rangle + m < {\aleph }_{0} \leqq {\aleph }_{... | Yes |
Proposition 15.12. \( {J}^{2}\left\langle {0,{\aleph }_{\gamma },0}\right\rangle = {\aleph }_{\gamma } \) . | Proof. If it were the case that \( {K}_{1}^{i}{\aleph }_{\gamma } < {\aleph }_{\gamma } \) and \( {K}_{2}^{i}{\aleph }_{\gamma } < {\aleph }_{\gamma } \) then by Proposi-\n\nProposition 15.8 it would follow that\n\n\[ \n{\aleph }_{\gamma } = {J}^{\iota }\left\langle {{K}_{1}^{\iota }{\aleph }_{\gamma },{K}_{2}^{\iota }... | Yes |
\[ {F}^{a}\alpha = {F}^{a}\alpha \] if \( {K}_{3}^{c}\alpha = 0 \), \[ {F}^{2}\alpha = {\mathcal{F}}_{n}^{2}\left( {{F}^{2}{K}_{1}^{2}\alpha ,{F}^{2}{K}_{2}^{2}\alpha }\right) \;\text{ if }{K}_{3}^{2}\alpha = n \neq 0. \] | Proof. Since \( \mathcal{D}\left( {F \upharpoonright \alpha }\right) = \alpha \) we have \[ {F}^{c}\alpha = {G}^{c}\left( {F \upharpoonright \alpha }\right) = \mathcal{W}\left( {F \upharpoonright \alpha }\right) = {F}^{\alpha }\alpha \] if \( {K}_{3}^{c}\alpha = 0 \), \[ {F}^{\epsilon }\alpha = {G}^{\epsilon }\left( {F... | Yes |
Proposition 15.17.\n\n\[ \n{F}_{a}^{c}\alpha = \alpha ,\alpha \leqq \omega \]\n\n\[ \n= a,\alpha = \omega + 1 \]\n\n\[ \n= {F}_{a}^{a}\alpha ,\alpha > \omega + 1 \land {K}_{3}^{a}\alpha = 0 \]\n\n\[ \n= {\mathcal{F}}_{n}\left( {{F}_{a}^{\epsilon }{K}_{1}^{\epsilon }\alpha ,{F}_{a}^{\epsilon }{K}_{2}^{\epsilon }\alpha }... | The proof is left to the reader. | No |
(1) \( x \in L \leftrightarrow x = {F}^{2}{\mathrm{{Od}}}^{2}x \) . | Proof. Definition 15.18. | No |
(1) \( \left( {\forall \alpha }\right) \left\lbrack {{F}^{2}\alpha \subseteq {F}^{\alpha }\alpha }\right\rbrack \) . | Proof. (1) (By transfinite induction). If \( \beta = {K}_{1}^{i}\alpha ,\gamma = {K}_{2}^{i}\alpha \) and \( n = {K}_{3}^{i}\alpha \) then\n\n\[ \alpha = {J}^{\iota }\langle \beta ,\gamma, n\rangle \]\n\nIf \( n = 0 \) then \( {F}^{c}\alpha = {F}^{\alpha }\alpha \) and hence \( {F}^{c}\alpha \subseteq {F}^{\alpha }\alp... | Yes |
(1) \( \operatorname{Tr}\left( {{F}^{\alpha }\alpha }\right) \) . | If \( x \in {F}^{\alpha }\alpha \), then \( \left( {\exists \beta < \alpha }\right) \left\lbrack {x = {F}^{\iota }\beta }\right\rbrack \) . But from Proposition 15.20 and the fact that \( \beta < \alpha \) we have \[ x = {F}^{c}\beta \subseteq {F}^{\alpha }\beta \subseteq {F}^{\alpha }\alpha . \] | No |
(1) \( \operatorname{Tr}\left( L\right) \) . | If \( x \in L \), then \( \left( {\exists \alpha }\right) \left\lbrack {x = {F}^{\prime }\alpha }\right\rbrack \) . Therefore\n\n\[ x = {F}^{a}\alpha \in {F}^{a}\left( {\alpha + 1}\right) . \]\n\nSince \( {F}^{\alpha }\left( {\alpha + 1}\right) \) is transitive\n\n\[ x \subseteq {F}^{\alpha }\left( {\alpha + 1}\right) ... | No |
(1) \( x \in L \land y \in L \land x \in y \rightarrow {\mathrm{{Od}}}^{4}x < {\mathrm{{Od}}}^{4}y \) . | (1) \( x \in y \land y \in L \rightarrow x \in {F}^{c}{\operatorname{Od}}^{c}y \)\n\n\[ \rightarrow x \in {F}^{a}{\operatorname{Od}}^{c}y \]\n\n\[ \rightarrow \left( {\exists \beta < {\operatorname{Od}}^{2}y}\right) \left\lbrack {x = {F}^{2}\beta }\right\rbrack \]\n\n\[ \rightarrow {\mathrm{{Od}}}^{2}x < {\mathrm{{Od}}... | No |
\[ \left( {\forall x \in {L}_{a}}\right) \left( {\exists \alpha > \omega }\right) \left\lbrack {x = {F}_{a}^{ * }\alpha }\right\rbrack . \] | The proof is left to the reader. | No |
(1) \( \left( {\forall x, y \in L}\right) \left\lbrack {{\mathcal{F}}_{n}\left( {x, y}\right) \in L}\right\rbrack, n = 1,\ldots ,8 \) . | Proof. (1) If \( \alpha = {\operatorname{Od}}^{\iota }x \) and \( \beta = {\operatorname{Od}}^{\iota }y \) then \( x = {F}^{\iota }\alpha \) and \( y = {F}^{\iota }\beta \) . Let \( \gamma = \)\n\n\( {J}^{\iota }\langle \alpha ,\beta, n\rangle \) . Then\n\n\[{\mathcal{F}}_{n}\left( {x, y}\right) = {\mathcal{F}}_{n}\lef... | Yes |
(1) \( b \subseteq L \rightarrow \left( {\exists x \in L}\right) \left\lbrack {b \subseteq x}\right\rbrack \) . | Proof. (1) Since Od is a function from \( V \) into On, Od“ \( b \) is a set of ordinals. Therefore\n\n\[ \left( {\exists \alpha }\right) \left\lbrack {{\mathrm{{Od}}}^{\alpha }b \leqq \alpha }\right\rbrack \text{.}\]\n\nLet\n\n\[ \beta = {J}^{\iota }\langle 0,\alpha ,0\rangle \]\n\nThen \( {K}_{3}^{c}\beta = 0 \) and ... | Yes |
Theorem 15.27. (1) \( L \) is a standard transitive model of \( \mathrm{{ZF}} \) and \( \mathrm{{On}} \subseteq L \) . | Proof. Propositions 15.22, 15.25, 15.26, and Theorem 14.11. | No |
Lemma 2. \( f{\operatorname{Isom}}_{S, E}\left( {{\beta }^{2} \times 9,\alpha }\right) \) Abs \( M \) . | Proof. Proposition 13.30. | No |
(1) \( \beta = {J}^{\iota }\langle \gamma ,\delta, m\rangle \) Abs \( M \) . | (1) From properties of order isomorphisms (Proposition 7.53)\n\n\[ \left( {\exists !f}\right) \left( {\exists !\alpha }\right) \left\lbrack {f{\operatorname{Isom}}_{S, E}\left( {\mu \times \mu \times 9,\alpha }\right) }\right\rbrack \text{.} \]\n\nTherefore, from the definition of \( J \)\n\n\[ \beta = {J}^{\epsilon }\... | Yes |
(1) \( {K}_{1}^{2}\alpha = \beta \) Abs \( M \) .\n\n(2) \( {K}_{2}^{\prime }\alpha = \beta \) Abs \( M \) .\n\n(3) \( {K}_{3}^{c}\alpha = \beta \) Abs \( M \) . | \[ {K}_{1}^{\prime }\alpha = \beta \leftrightarrow \left( {\exists m}\right) \left( {\exists \gamma }\right) \left\lbrack {m < 9 \land {J}^{\prime }\langle \beta ,\gamma, m\rangle = \alpha }\right\rbrack . \]\n\n\[ {K}_{2}^{c}\alpha = \beta \leftrightarrow \left( {\exists m}\right) \left( {\exists \gamma }\right) \left... | Yes |
Lemma 5. \( b = {\mathcal{F}}_{n}\left( {c, d}\right) \) Abs \( M,\;n = 1,\ldots ,8 \) . | \[ b = {\mathcal{F}}_{1}\left( {c, d}\right) \leftrightarrow b = \{ c, d\} . \] \[ b = {\mathcal{F}}_{2}\left( {c, d}\right) \leftrightarrow b = c \cap E \] \[ \leftrightarrow \left( {\forall x}\right) \left\lbrack {x \in b \leftrightarrow x \in c \land \left( {\exists y}\right) \left( {\exists z}\right) \left\lbrack {... | Yes |
Lemma 6. \( b = {G}^{c}\left( {f \upharpoonright \beta }\right) \) Abs \( M \) . | \[b = {G}^{c}\left( {f \upharpoonright \beta }\right) \leftrightarrow \left\lbrack {{K}_{3}^{c}\beta = 0 \land b = {f}^{a}\beta }\right\rbrack \vee \]\n\n\[\left\lbrack {{K}_{3}^{\prime }\beta = 1 \land b = {\mathcal{F}}_{1}\left( {{f}^{\prime }{K}_{1}^{\prime }\beta ,{f}^{\prime }{K}_{2}^{\prime }\beta }\right) }\righ... | Yes |
(1) \( b = {F}^{2}\alpha \) Abs \( M \) . | Proof. (1) From the definition of \( F \) and Corollary 7.42\n\n\[ \left( {\exists !f}\right) \left\lbrack {f{\mathcal{F}}_{n}\left( {\alpha + 1}\right) \land \left( {\forall \beta \leqq \alpha }\right) \left\lbrack {{f}^{c}\beta = {G}^{c}\left( {f \upharpoonright \beta }\right) }\right\rbrack }\right\rbrack .\n\]\n\nT... | Yes |
(1) On \( \cong M \rightarrow L \cong M \) . | Proof. (1) Since \( \left( {\forall \alpha }\right) \left( {\exists x}\right) \left\lbrack {x = {F}^{i}\alpha }\right\rbrack \), it follows that\n\n\[ \left( {\forall \alpha \in M}\right) \left( {\exists x \in M}\right) {\left\lbrack x = {F}^{c}\alpha \right\rbrack }^{M}. \]\n\nBut since \( \mathrm{{On}} \subseteq M \)... | No |
(1) \( L \) is a model of \( V = L \) . | Proof. (1) \( V = L \leftrightarrow \left( {\forall x}\right) \left( {\exists \alpha }\right) \left\lbrack {x = {F}^{c}\alpha }\right\rbrack \). From the definition of \( L \)\n\n\[ \left( {\forall x \in L}\right) \left( {\exists \alpha }\right) \left\lbrack {x = {F}^{c}\alpha }\right\rbrack .\n\]\n\nSince \( \mathrm{{... | Yes |
(1) \( \;\left( {\forall x \in L}\right) \left\lbrack {x \neq 0 \rightarrow {\mathrm{{As}}}^{i}x \in x}\right\rbrack \) . | Proof. (1) Since \( L \) is transitive \( L - \{ 0\} \subseteq \mathcal{D} \) (As). Furthermore As is single valued. Therefore\n\n\[ x \in L \land x \neq 0 \rightarrow {\operatorname{As}}^{i}x \in x. \] | Yes |
Theorem 15.33. (1) \( V = L \rightarrow \mathrm{AC} \) . (2) \( V = {L}_{a} \rightarrow \mathrm{AC} \) . | Proof. Obvious from Proposition 15.32. | No |
(1) \( L \) is a model of \( \mathrm{{AC}} \) .\n\n(2) \( {L}_{a} \) is a model of \( \mathrm{{AC}} \) . | Proof. Propositions 15.30 and 15.33. | No |
(1) \( {C}^{2}\alpha \leqq \alpha \) . | Proof. (1) If \( {F}^{c}\alpha = 0 \) then \( {\mathrm{{As}}}^{c}{F}^{c}\alpha = 0 \) and \( {\mathrm{{Od}}}^{c}{\mathrm{{As}}}^{c}{F}^{c}\alpha = 0 \), i.e., \( {C}^{c}\alpha = 0 \leqq \alpha \) . If \( {F}^{c}\alpha \neq 0 \) then \( {\mathrm{{As}}}^{c}{F}^{c}\alpha \in {F}^{c}\alpha \) . Therefore\n\n\[
{C}^{2}\alph... | Yes |
(1) \( \overline{\overline{{F}^{\prime \prime }{\aleph }_{\alpha }}} = {\aleph }_{\alpha } \) . | Proof. (1) Since \( F \) is a function it follows that\n\n\[ \overline{\overline{{F}^{\alpha }{\aleph }_{\alpha }}} \leqq {\aleph }_{\alpha }.\]\n\nFurthermore\n\n\[ {F}^{\iota }{J}^{\iota }\langle 0,\beta ,0\rangle = {F}^{u}{J}^{\iota }\langle 0,\beta ,0\rangle .\]\n\nTherefore since \( \gamma < \beta \) implies \( {J... | Yes |
(1) \( \left( {\forall \alpha }\right) \left\lbrack {\mathcal{P}\left( {{F}^{\alpha }{\aleph }_{\alpha }}\right) \subseteq {F}^{\alpha }{\aleph }_{\alpha + 1} \rightarrow \underline{\overline{{2}^{{\aleph }_{\alpha }}}} = {\aleph }_{\alpha + 1}}\right\rbrack \) . | (1) If \( \mathcal{P}\left( {{F}^{\alpha }{\aleph }_{\alpha }}\right) \subseteq {F}^{\alpha }{\aleph }_{\alpha + 1} \) then from Proposition 15.37\n\n\[ \overline{\overline{{2}^{{\aleph }_{\alpha }}}} = \overline{\overline{\mathcal{P}\left( {\aleph }_{\alpha }\right) }} = \overline{\overline{\mathcal{P}\left( {{F}^{\al... | Yes |
Proposition 15.39. (1) If \( V = L \) and \( \left( {\forall x}\right) \left( {\forall \gamma }\right) \left( {\forall f}\right) \lbrack 9 \subseteq x \subseteq \) On \( \land {C}^{\alpha }x \subseteq \) \( x \land {K}_{1}^{\alpha }x \subseteqq x \land {K}_{2}^{\alpha }x \subseteqq x \land {J}^{\alpha }{x}^{3} \subsete... | Proof. (1) If \( x \subseteq {F}^{\alpha }{\aleph }_{\alpha } \) then from the Axiom of Constructibility\n\n\[ \left( {\exists \delta }\right) \left\lbrack {x = {F}^{c}\delta }\right\rbrack \text{.} \]\n\nSince \( C,{K}_{1},{K}_{2} \) and \( J \) are each single valued and \( {\aleph }_{\alpha } \cup \{ \delta \} \) is... | Yes |
Proposition 15.41. If \( 9 \subseteqq b \subseteqq \mathrm{{On}} \land 9 \subseteqq c \subseteqq \mathrm{{On}} \land {K}_{1}^{\alpha }b \subseteqq b \land {K}_{2}^{\alpha }b \subseteqq b \land \n\n{J}^{\alpha }{b}^{3} \subseteq b \land {K}_{1}^{\alpha }c \subseteq c \land {K}_{2}^{\alpha }c \subseteq c \land {J}^{\alph... | Proof. (1) Since \( f{\operatorname{Isom}}_{E, E}\left( {b, c}\right) \) it follows that for some \( \eta ,{f}_{1} \), and \( {f}_{2} \) we have\n\n\[ \n{f}_{1}{\operatorname{Isom}}_{E, E}\left( {b,\eta }\right) \land {f}_{2}{\operatorname{Isom}}_{E, E}\left( {c,\eta }\right) \land {f}_{2} \circ f = {f}_{1}.\n\]\n\nSin... | Yes |
Lemma 1. If \( 9 \subseteq b \subseteq \) On and \( b \) is closed w.r.t. \( J \), then \( {F}^{\alpha }b \) is closed w.r.t. the fundamental operations. | Proof. If \( x, y \in {F}^{u}b \) then\n\n\[ \left( {\exists \alpha ,\beta \in b}\right) \left\lbrack {x = {F}^{c}\alpha \land y = {F}^{c}\beta }\right\rbrack \]\n\nand hence\n\n\[ {\mathcal{F}}_{n}\left( {x, y}\right) = {\mathcal{F}}_{n}\left( {{F}^{c}\alpha ,{F}^{c}\beta }\right) = {F}^{c}{J}^{c}\langle \alpha ,\beta... | Yes |
Lemma 2. If \( 9 \subseteq b \subseteq \) On, if \( b \) is closed w.r.t. \( C \) and \( J \), and if \( x \in {F}^{\alpha }b \), then\n\n\[{\operatorname{Od}}^{c}x \in b\text{.}\] | Proof. From Lemma 1\n\n\[x \in {F}^{ii}b \rightarrow \{ x\} \in {F}^{ii}b\]\n\n\[\rightarrow \left( {\exists \alpha \in b}\right) \left\lbrack {\{ x\} = {F}^{c}\alpha }\right\rbrack \text{.}\]\n\nBut\n\n\[{\mathrm{{Od}}}^{4}x = {\mathrm{{Od}}}^{4}{\mathrm{{As}}}^{4}\{ x\} = {\mathrm{{Od}}}^{4}{\mathrm{{As}}}^{4}{F}^{2}... | No |
Lemma 3. If \( 9 \subseteq b \subseteq \) On and \( b \) is closed w.r.t. \( C \) then\n\n\[ x \in {F}^{a}b \land x \neq 0 \rightarrow x \cap {F}^{a}b \neq 0. \] | Proof. If \( x \in {F}^{\alpha }b \), then \( \left( {\exists \alpha \in b}\right) \left\lbrack {x = {F}^{i}\alpha }\right\rbrack \) . Since \( b \) is closed w.r.t. \( C \)\n\n\[ \alpha \in b \rightarrow {C}^{2}\alpha \in b \]\n\n\[ \rightarrow {F}^{4}{C}^{4}\alpha \in {F}^{44}b\text{.} \]\n\nBut\n\n\[ {F}^{2}{C}^{2}\... | Yes |
Lemma 4. If \( 9 \subseteq b \subseteq \) On and \( b \) is closed w.r.t. \( C \) and \( J \) then\n\n(1) \( \left( {\forall x, y}\right) \left\lbrack {\{ x, y\} \in {F}^{a}b \rightarrow x, y \in {F}^{a}b}\right\rbrack \) ,\n\n(2) \( \;\left( {\forall x, y}\right) \left\lbrack {\langle x, y\rangle \in {F}^{\omega }b \r... | Proof. (1) Since \( \langle x, y\rangle \neq 0 \) we have from Lemma 3 that\n\n\[ \{ x, y\} \cap {F}^{a}b \neq 0. \]\n\nTherefore\n\n\[ x \in {F}^{a}b \vee y \in {F}^{a}b. \]\n\nIf \( x \in {F}^{\alpha }b \) and \( x \neq y \) then from Lemma 1, \( \{ x\} \in {F}^{\alpha }b \) and\n\n\[ \{ y\} = \{ x, y\} - \{ x\} \in ... | No |
Lemma 5. If \( 9 \subseteq b \subseteq \) On and \( b \) is closed w.r.t. \( C \) and \( J \), then \[ \left( {\forall x, y}\right) \left\lbrack {\langle x, y\rangle \in {Q}_{n} \land y \in {F}^{\omega }b \rightarrow x \in {F}^{\omega }b}\right\rbrack ,\;n = 4,6,7,8. \] | Proof. If \( \langle x, y\rangle \in {Q}_{4} \), then \( \left( {\exists z}\right) \left\lbrack {y = \langle x, z\rangle }\right\rbrack \) . But by Lemma 4 \[ y \in {F}^{zz}b \rightarrow x \in {F}^{zz}b. \] If \( \langle x, y\rangle \in {Q}_{6} \), then \( \left( {\exists z, w}\right) \left\lbrack {x = \langle z, w\ran... | Yes |
Lemma 6. If \( 9 \subseteq b \subseteq \) On and if \( b \) is closed w.r.t. \( C \) and \( J \) then\n\n\[ \left( {\forall x, y}\right) \left\lbrack {x \in {F}^{ `` }\left( {b \cap \eta }\right) \land y \in x \cap {F}^{ `` }b \rightarrow y \in {F}^{ `` }\left( {b \cap \eta }\right) }\right\rbrack . | Proof. If \( x \in {F}^{u}\left( {b \cap \eta }\right) \), then \( x \in {F}^{u}b \) and \( x \in {F}^{u}\eta \) . Therefore\n\n\[ {\operatorname{Od}}^{2}x \in \eta \]\n\nand, by Lemma 2,\n\n\[ {\operatorname{Od}}^{c}x \in b\text{.} \]\n\nAlso\n\n\[ y \in x \cap {F}^{a}b \rightarrow y \in x \land y \in {F}^{a}b \]\n\n\... | Yes |
Lemma 7. If \( 9 \subseteq b \subseteq \) On and if \( b \) is closed w.r.t. \( C \) and \( J \), then \[ \left( {\forall x}\right) \left\lbrack {x \in {F}^{\iota }\eta \cap {F}^{u}b \rightarrow x \in {F}^{u}\left( {b \cap \eta }\right) }\right\rbrack . \] | Proof. If \( x \in {F}^{c}\eta \), then \( {\operatorname{Od}}^{c}x < {\operatorname{Od}}^{c}{F}^{c}\eta \leqq \eta \) . By Lemma 2 \[ x \in {F}^{u}b \rightarrow {\operatorname{Od}}^{c}x \in b. \] Then \[ x \in {F}^{a}\left( {b \cap \eta }\right) \] | No |
Lemma 8. If \( 9 \subseteq b \subseteq \) On and \( b \) is closed w.r.t. \( C \) and \( J \), then\n\n(1) \( \left( {\forall x, y}\right) \left\lbrack {\{ x, y\} \in {F}^{\alpha }\left( {b \cap \eta }\right) \rightarrow x \in {F}^{\alpha }\left( {b \cap \eta }\right) \land y \in {F}^{\alpha }\left( {b \cap \eta }\righ... | Proof. (1) Since \( {F}^{u}\left( {b \cap \eta }\right) \subseteq {F}^{u}b \), we have by Lemma 4.\n\n\[ \{ x, y\} \in {F}^{\omega }\left( {b \cap \eta }\right) \rightarrow \{ x, y\} \in {F}^{\omega }b \]\n\n\[ \rightarrow x \in {F}^{a}b \land y \in {F}^{a}b \]\n\nthen\n\n\[ \{ x, y\} \in {F}^{\omega }\left( {b \cap \e... | Yes |
Lemma 9. If \( 9 \subseteq b \subseteq \) On \( \land 9 \subseteq c \subseteq \) On, if \( b \) and \( c \) are each closed with respect to \( C \) and \( J \), iff \( {\operatorname{Isom}}_{E, E}\left( {b, c}\right) \) and if \( H{\operatorname{Isom}}_{E, E}\left( {{F}^{\alpha }\left( {b \cap \eta }\right) ,{F}^{\alph... | Proof. (1) From Lemma 8\n\n\[ \{ x, y\} \in {F}^{u}\left( {b \cap \eta }\right) \rightarrow x, y \in {F}^{u}\left( {b \cap \eta }\right) .\n\]\n\nTherefore since \( H{\operatorname{Isom}}_{E, E}\left( {{F}^{\alpha }\left( {b \cap \eta }\right) ,{F}^{\alpha }{f}^{\alpha }\left( {b \cap \eta }\right) }\right) \) and sinc... | No |
Proposition 15.42. (1) If \( 9 \subseteq b \subseteq \) On \( \land 9 \subseteq c \subseteq \) On, if \( b \) and \( c \) are each closed with respect to \( {K}_{1},{K}_{2}, C \), and \( J \), and iff \( {\operatorname{Isom}}_{E, E}\left( {b, c}\right) \), then\n\n\[ \left( {\forall \alpha ,\beta \in b}\right) \left\lb... | Proof. (1) By induction on \( \max \left( {\alpha ,\beta }\right) \) . If \( \eta = \max \left( {\alpha ,\beta }\right) \) and if \( \eta = \alpha = \beta \) then the result is true because\n\n\[ {F}^{c}\alpha = {F}^{c}\beta \land {F}^{c}{f}^{c}\alpha = {F}^{c}{f}^{c}\beta \land {F}^{c}\alpha \notin {F}^{c}\beta \land ... | Yes |
Proposition 15.43. (1) If \( 9 \subseteq b \subseteq \) On, if \( b \) is closed with respect to \( C,{K}_{1},{K}_{2} \) , and \( J \), and iff \( {\operatorname{Isom}}_{E, E}\left( {b,\eta }\right) \) then\n\n\[ \left( {\forall \alpha ,\beta \in b}\right) \left\lbrack {{F}^{\iota }\alpha \in {F}^{\iota }\beta \leftrig... | Proof. (1) From Proposition 15.40, \( \eta \) is closed with respect to \( J \) . Since\n\n\[ {K}_{1}^{c}\alpha \leqq \alpha \land {K}_{2}^{c}\alpha \leqq \alpha \land {C}^{c}\alpha \leqq \alpha \]\n\n\( \eta \) is also closed with respect to \( C,{K}_{1} \), and \( {K}_{2} \) . Therefore by Proposition 15.42\n\n\[ \le... | Yes |
(1) \( L \) is a model of \( \mathrm{{GCH}} \) .\n\n(2) \( {L}_{a} \) is a model of \( \mathrm{{GCH}} \) . | Proof. Theorems 15.44 and 15.30. | No |
Proposition 15.46. If \( M \) is a standard transitive model of \( \mathrm{{ZF}} \) then\n\n(1) \( \left( {\forall \alpha \in M}\right) \left\lbrack {{F}^{ \cdot }\alpha \in M}\right\rbrack \) ,\n\n(2) \( a \subseteq \omega \land a \in M \rightarrow \left( {\forall \alpha \in M}\right) \left\lbrack {{F}_{a}^{c}\alpha \... | Proof. (1) (By induction). Since \( M \) is a model of the Axiom Schema of Replacement it follows from the induction hypothesis that if \( {K}_{3}^{c}\alpha = 0 \), then\n\n\[ \n{F}^{c}\alpha = {F}^{\alpha }\alpha = \left\{ {x \mid \left( {\exists \beta \in \alpha }\right) \left\lbrack {x = {F}^{c}\beta }\right\rbrack ... | No |
Proposition 15.47. If \( M \) is a standard transitive model of \( \mathrm{{ZF}} \) then\n\n(1) \( {L}^{M} = \left\{ {x \mid \left( {\exists \alpha \in M}\right) \left\lbrack {x = {F}^{2}\alpha }\right\rbrack }\right\} \),\n\n(2) \( a \subseteq \omega \land a \in M \rightarrow {L}_{a}^{M} = \left\{ {x \mid \left( {\exi... | Proof. Obvious from Proposition 15.46. | No |
Theorem 15.48. If \( M \) is a standard transitive model of \( \mathrm{{ZF}} \) and if \( \varphi \) is a wff of \( \mathrm{{ZF}} \) then\n\n(1) \( \;{\left( {\varphi }^{L}\right) }^{M} \leftrightarrow {\varphi }^{{L}^{M}} \)\n\n(2) \( {\left( {\varphi }^{{L}_{a}}\right) }^{M} \leftrightarrow {\varphi }^{{L}_{a}^{M}} \... | Proof. (1) (By induction on the number of logical symbols in \( \varphi \) .) The formula \( \varphi \) must be of the form (1) \( a \in b,\left( 2\right) \neg \psi ,\left( 3\right) \psi \land \eta \), or \( \left( 4\right) \left( {\forall x}\right) \psi \) . The arguments for cases (1)-(3) we leave to the reader. If \... | No |
Theorem 15.49. If \( M \) is a standard transitive model of \( \mathrm{{ZF}} \) then\n\n(1) \( {L}^{M} \) is a standard transitive model of \( \mathrm{{ZF}} + \mathrm{{AC}} + \mathrm{{GCH}} + V = L \) and \( {\mathrm{{On}}}^{{L}^{M}} = {\mathrm{{On}}}^{M} \) . | Proof. (1) Since \( 0 \in M \) and \( {F}^{\mathfrak{c}}0 = 0 \) it follows that \( 0 \in {L}^{M} \) and hence \( {L}^{M} \) is not empty. Furthermore if \( y \in x \in {L}^{M} \) then\n\n\[ \left( {\exists \alpha \in M}\right) \left\lbrack {y \in x = {F}^{c}\alpha \subseteq {F}^{a}\alpha }\right\rbrack .\n\]\n\nTheref... | No |
Proposition 16.5. Let \( \left\{ {{\left\langle {A}_{b},{F}_{bx}\right\rangle }_{x \in I} \mid b \in B}\right\} \) be a collection of subalgebras of an algebra \( \mathfrak{A} \) . Then\n\n(1) \( {\left\langle \mathop{\bigcap }\limits_{{b \in B}}{A}_{b},\mathop{\bigcap }\limits_{{b \in B}}{F}_{bx}\right\rangle }_{x \in... | The proof is obvious. | No |
Proposition 16.14. If \( \pi : \xi \rightarrow \eta \) is a strong \( \mathfrak{A} \) -map and \( \delta \leqq \eta \) is an ordinal such that \( \mathcal{W}\left( \pi \right) \subseteq \delta \), then \( \pi : \xi \rightarrow \delta \) is a strong \( \mathfrak{A} \) -map. | Proof. For any \( \mathbf{\alpha } \in {\xi }^{\omega } \) ,\n\n\[ \n{F}_{i}^{\xi }\left( \mathbf{\alpha }\right) \simeq \beta \rightarrow {F}_{i}^{\eta }\left( {\pi \left( \mathbf{\alpha }\right) }\right) \simeq \pi \left( \mathbf{\beta }\right) \land \pi \left( \mathbf{\beta }\right) < \delta \n\]\n\n\[ \n\rightarrow... | Yes |
Proposition 16.15. If \( \pi : \xi \rightarrow \eta \) is a medium \( \mathfrak{A} \) -map and \( \gamma \leqq \xi \), then \( \pi \upharpoonright \gamma : \gamma \rightarrow \eta \) is a medium \( \mathfrak{A} \) -map. | Proof. We may assume that \( \pi \) is strong. Let \( \delta = \sup \{ \pi \left( \alpha \right) + 1 \mid \alpha < \gamma \} \) . We will show that \( \pi \upharpoonright \gamma : \gamma \rightarrow \delta \) is a strong \( \mathfrak{A} \) -map. For any \( \mathbf{\alpha } \in {\gamma }^{\omega } \) ,\n\n\[ \n{F}_{i}^{... | Yes |
Proposition 16.16. If \( {\pi }_{1} : {\xi }_{1} \rightarrow {\xi }_{2} \) and \( {\pi }_{2} : {\xi }_{2} \rightarrow {\xi }_{3} \) are strong QI-maps, then \( {\pi }_{2} \circ {\pi }_{1} : {\xi }_{1} \rightarrow {\xi }_{3} \) is a strong \( \mathfrak{A} \) -map. | The proof is left to the reader. | No |
Proposition 16.17. If \( {\pi }_{1} : {\xi }_{1} \rightarrow {\xi }_{2} \) and \( {\pi }_{2} : {\xi }_{2} \rightarrow {\xi }_{3} \) are medium \( \mathfrak{A} \) -maps, then \( {\pi }_{2} \circ {\pi }_{1} : {\xi }_{1} \rightarrow {\xi }_{3} \) is a medium \( \mathfrak{A} \) -map. | Proof. Let \( {\delta }_{2} = \sup \left\{ {{\pi }_{1}\left( \alpha \right) + 1 \mid \alpha < {\xi }_{1}}\right\} \) and let \( {\delta }_{3} = \sup \left\{ {{\pi }_{2}\left( \beta \right) + 1 \mid }\right. \) \( \left. {\beta < {\delta }_{2}}\right\} \) . Then \( {\pi }_{1} : {\xi }_{1} \rightarrow {\delta }_{2} \) an... | Yes |
Proposition 16.18. If \( {\pi }_{1} : {\xi }_{1} \rightarrow \eta \) and \( {\pi }_{2} : {\xi }_{2} \rightarrow \eta \) are medium \( \mathfrak{A} \) -maps such that \( \mathcal{W}\left( {\pi }_{1}\right) \subseteq \mathcal{W}\left( {\pi }_{2}\right) \), then \( {\pi }_{2}^{-1} \circ {\pi }_{1} : {\xi }_{1} \rightarrow... | Proof. Let \( {\delta }_{1} = \sup \left\{ {\pi \left( \alpha \right) + 1 \mid \alpha < {\xi }_{1}}\right\} \) and let \( {\delta }_{2} = \sup \left\{ {{\pi }_{2}^{-1}\left( {{\pi }_{1}\left( \alpha \right) }\right) + }\right. \) \( \left. {1 \mid \alpha < {\xi }_{1}}\right\} \) . Then \( {\pi }_{1} : {\xi }_{1} \right... | Yes |
Theorem 16.19. If \( \pi : \xi \rightarrow \eta \) is a strong \( \mathfrak{A} \) -map and \( X \subseteq \xi \), then \( {\pi }^{\mathfrak{``}}{\mathfrak{A}}^{\xi }\left( X\right) = \) \( {\mathfrak{A}}^{\eta }\left( {{\pi }^{a}X}\right) \) . | Proof. It is easy to see that \( {\pi }^{\alpha }{\mathfrak{A}}^{\xi }\left( X\right) \) is a subalgebra of \( {\mathfrak{A}}^{\eta } \) that contains \( {\pi }^{\alpha }X \) , and thus \( {\mathfrak{A}}^{\eta }\left( {{\pi }^{\mathfrak{u}}X}\right) \subseteq {\pi }^{\mathfrak{u}}{\mathfrak{A}}^{\xi }\left( X\right) \)... | Yes |
Lemma 16.22. Let \( \kappa \) be an arbitrary limit ordinal and let \( \eta \) be the least ordinal greater than or equal to \( \kappa \) such that for some \( \alpha < \kappa \) and some finite \( P \subseteq \eta \), the order type of \( {\mathfrak{A}}^{n}\left( {\alpha \cup P}\right) \) is not less than \( \kappa \)... | Proof. Take \( \alpha < \kappa \) and let \( P \subseteq \eta \) be a finite set so chosen that the order type of \( {\mathfrak{A}}^{\eta }\left( {\alpha \cup P}\right) \) is greater than or equal to \( \kappa \) . Assume \( \eta = v + 1 \) for some \( v \) . Since \( \mathfrak{A} \) has the finiteness property, there ... | Yes |
Lemma 16.28. \( \Pi \) is well founded iff there are no sequences \( \left\langle {{i}_{n} \mid n < \omega }\right\rangle \) and \( \left\langle {{\sigma }_{n} \mid n < \omega }\right\rangle \) such that \( {i}_{0} \leqq {i}_{1} \leqq \cdots \leqq {i}_{n} \leqq \cdots \) and \( {\pi }_{{i}_{n}{i}_{n + 1}}\left( {\sigma... | Proof. If such sequences exist, then \( \left\langle {{\pi }_{{i}_{n}\infty }\left( {\sigma }_{n}\right) \mid n < \omega }\right\rangle \) is an infinite descending sequence in \( M/ \equiv \) . Thus \( \Pi \) is not well founded. Suppose \( \Pi \) is not well founded and \( \left\langle {\left\lbrack {{j}_{n},{\tau }_... | Yes |
Theorem 16.34. The pairing machine \( {\mathcal{P}}_{ < } \) has the collapsing property. | Proof. Let \( \eta \) be an ordinal, let \( X \subseteq \eta \) be a subalgebra of \( {\mathcal{P}}_{ < }^{\eta } \) and let \( \pi : \xi \rightarrow X \) be the collapsing map of \( X \) . We want to show that \( \pi : \xi \rightarrow \eta \) is a strong \( {\mathcal{P}}_{ < } \) -map.\n\nLet \( Z = \left\{ {\mathbf{\... | Yes |
Theorem 16.35. \( {\mathcal{P}}_{ < } \) has the finiteness property. | Proof. Let \( \eta \) be an arbitrary ordinal and \( {\beta }_{0},\ldots ,{\beta }_{n - 1} \) be such that \( J\left( \left\langle {{\beta }_{0},\ldots ,{\beta }_{n - 1}}\right\rangle \right) = \eta \), then \( {\beta }_{i} \leqq \eta \) for all \( i < n \) . Let \( {H}_{\eta } = \left\{ {{\beta }_{i} \mid {\beta }_{i}... | No |
Theorem 17.2. \( V = L \rightarrow \mathrm{{GCH}} \). | Proof. We shall show that\n\n\[ \mathcal{P}\left( {\aleph }_{\alpha }\right) \subseteq {L}_{{\aleph }_{\alpha + 1}} \]\n\nSuppose that \( a \subseteq {\aleph }_{\alpha } \) and let \( t \) be a term such that \( D\left( t\right) = a \). Let \( \eta = \max \left( {{\aleph }_{\alpha + 1},}\right. \) \( {\bar{t}}^{ + } \)... | Yes |
Lemma 17.6. Let \( \kappa \) be a regular uncountable cardinal and let \( \lambda \) be an ordinal such that \( \kappa < \lambda \) . For each set \( X \) with \( X \subseteq \lambda \) and \( \overline{\bar{X}} < \kappa \), there exists a subalgebra \( Y \) of \( {M}^{\lambda } \) such that \( X \) is a subset of \( Y... | Proof. By induction on \( n \), we define \( {\alpha }_{n} \) and \( {Y}_{n} \) as follows:\n\n\[ \n{Y}_{0} = X \n\]\n\n\[ \n{\alpha }_{n} = \sup \left\{ {\alpha + 1 \mid \alpha \in {Y}_{n} \cap \kappa }\right\} \n\]\n\n\[ \n{Y}_{n + 1} = {M}^{\lambda }\left( {{Y}_{n} \cup {\alpha }_{n}}\right) \n\]\n\nFinally if \( Y ... | Yes |
Lemma17.11. Let \( M \) be a transitive set which is elementarily equivalent to \( {L}_{\kappa } \) for some uncountable cardinal in \( L \) . If \( \pi : \langle \delta ,\alpha, P\rangle \rightarrow \left\langle {{\delta }^{\prime },{\alpha }^{\prime },{P}^{\prime }}\right\rangle \) is an acceptable map such that \( \... | Proof. Note that each of \( {F}_{0}, J,{C}_{i}, T \) and \( K \) is absolute with respect to \( M \) . Let \( {F}_{1} = J,{F}_{2} = T,{F}_{3} = K \) and \( {F}_{i + 4} = {C}_{i}\left( {i < \omega }\right) \) . We define a sequence \( \left\langle {{X}_{n} : n < \omega }\right\rangle \in M \) as follows:\n\n\[ \n{X}_{0}... | Yes |
Lemma 17.14. Let \( \kappa \) be a limit ordinal, and let \( \left\langle {{I}_{1},{ \leqq }_{1}}\right\rangle \) and \( \left\langle {{I}_{2},{ \leqq }_{2}}\right\rangle \) be directed sets such that \( {I}_{1} \cap {I}_{2} = 0 \) . If \( {\Pi }_{1} = \left\langle {\left\langle {{\xi }_{i},{\alpha }_{i},{P}_{i}}\right... | Proof. We need to define \( i < j \) and \( {\pi }_{ij} \) when \( i \in {I}_{1} \) and \( j \in {I}_{2} \) : If \( i \in {I}_{1} \) and \( j \in {I}_{2} \), then\n\n\[ i < j \Leftrightarrow {\xi }_{i} \leqq {\eta }_{j} \land {\alpha }_{i} \leqq {\beta }_{j} \land {\rho }_{i\infty }^{a}{P}_{i} \leqq \mathcal{W}\left( {... | Yes |
Lemma 17.16. Let \( \kappa \) be an infinite cardinal in \( L \) and \( h : {L}_{\bar{\kappa }} \rightarrow {L}_{\kappa } \) be an elementary embedding. Let \( \bar{\Pi } = {\left\langle \left\langle {\bar{\delta }}_{i},{\bar{\alpha }}_{i},{\bar{P}}_{i}\right\rangle ,{\bar{\pi }}_{ij}\right\rangle }_{i, j \in I} \) be ... | Proof. Let \( {\delta }_{i} = h\left( {\bar{\delta }}_{i}\right) ,{\alpha }_{i} = h\left( {\bar{\alpha }}_{i}\right) ,{P}_{i} = h\left( {\bar{P}}_{i}\right) \) and \( {\pi }_{ij} = h\left( {\bar{\pi }}_{ij}\right) \) . For each \( \bar{\sigma } < {\bar{\delta }}_{i} \), we let \( {h}^{ * }\left( {{\bar{\pi }}_{i\infty ... | Yes |
(1) If \( \lambda \) is a regular cardinal in \( L \) and \( \lambda \geqq {\aleph }_{2} \), then \( \operatorname{cf}\left( \lambda \right) = \overline{\bar{\lambda }} \) . | Proof. (1) Let \( x \subseteq \lambda \) be such that \( \bar{x} = \operatorname{cf}\left( \lambda \right) \) and \( \cup x = \lambda \) . Then by \( S \) there exists a \( y \in L \) such that \( x \subseteq y \subseteq \lambda \) and \( \bar{y} = \max \left( {\bar{x},{\aleph }_{1}}\right) \) . Since \( \cup y = \lamb... | Yes |
Theorem 17.21. \( \mathrm{I} \rightarrow S \) | Remark. To prove Theorem 17.21 we will prove a sequence of lemmas that will enable us to prove the contrapositive \( \neg S \rightarrow \neg \mathbf{I} \) . So from this point until we complete the proof of the contrapositive of Theorem 17.21 we assume \( \neg S \) . That is, we assume\n\n\[ \left( {\exists X \subseteq... | Yes |
(1) \( \left( {\forall Y \in L}\right) \left( {X \subseteq Y \rightarrow {\overline{\bar{Y}}}^{L} \geqq \kappa }\right) \) . | Proof. (1) Assume\n\n\[ \left( {\exists Y \in L}\right) \left( {X \subseteq Y \land {\overline{\bar{Y}}}^{L} < \kappa }\right) ,\]\n\nand let \( \lambda = {\overline{\bar{Y}}}^{L} \) . Then\n\n\[ \left( {\exists f \in L}\right) \left( {f : \lambda \xrightarrow[\text{onto }]{1 - 1}Y}\right) \]\n\nLet \( {X}^{\prime } = ... | Yes |
Lemma 17.23. There exists an elementary embedding \( h : {L}_{\bar{\kappa }} \rightarrow {L}_{\kappa } \) such that\n\n(1) \( X \subseteq \mathcal{W}\left( h\right) \) and \( \overline{\bar{X}} = \overline{\overline{\mathcal{W}\left( h\right) }} \) ;\n\n(2) if \( \bar{\Pi } = {\left\langle \left\langle {\bar{\delta }}_... | Remark. We shall prove this lemma later. | No |
Lemma 17.27. Let \( Z \) be an elementary submodel of \( {L}_{\kappa } \) such that \( X \subseteq Z \) and \( \overline{\bar{Z}} = \overline{\bar{X}} \) . Then there exists a \( {Z}^{\prime } \), an elementary submodel of \( {L}_{\kappa } \) such that\n\n(1) \( Z \subseteq {Z}^{\prime },{\overline{\bar{Z}}}^{\prime } ... | Proof. Let \( h : {L}_{\bar{\kappa }} \rightarrow {L}_{\bar{\kappa }} \) be an elementary embedding such that \( \mathcal{W}\left( h\right) = Z \) . For any \( \lambda \in S \), by the same proof as that of Lemma 17.26, we can find a countable set \( {A}_{\lambda } \) such that \( {A}_{\lambda } \subseteq \lambda \) an... | Yes |
Corollary 17.28. There exists a \( Z \), an elementary submodel of \( {L}_{\kappa } \) such that\n\n(1) \( X \subseteq Z,\overline{\bar{Z}} = \overline{\bar{X}} \) ; and\n\n(2) for any \( \lambda \in S \), if \( \Pi \subseteq Z \) is a countable \( \lambda \) -direct limit system which is not well founded, then \( \Pi ... | Proof. We can construct a sequence \( \left\langle {{Z}_{\alpha } : \alpha < {\aleph }_{1}}\right\rangle \) such that\n\n(a) \( {Z}_{\alpha } \prec {L}_{\kappa }, X \subseteq {Z}_{\alpha } \) and \( \overline{\overline{{Z}_{\alpha }}} = \overline{\bar{X}} \) ;\n\n(b) \( \alpha < \beta \rightarrow {Z}_{\alpha } \leqq {Z... | Yes |
Theorem 18.1. If there exists a set that is a standard model of \( {ZF} \) then there exists one and only one set \( {M}_{0} \) such that\n\n(1) \( {M}_{0} \) is a countable standard transitive model of \( \mathrm{{ZF}} + V = L \) and\n\n(2) \( {M}_{0} \) is a submodel of every standard transitive model of \( \mathrm{{... | Proof. From Mostowski’s theorem (Theorem 12.8) every standard model is \( \in \) -isomorphic to a standard transitive model. Therefore the existence of a set that is a standard model of ZF implies the existence of a set that is a standard transitive model. For transitive models the property of being an ordinal is absol... | Yes |
Lemma 1. For every \( {p}_{0} \in P \), there exists a \( \mathcal{P} \) -generic set \( G \) such that \( {p}_{0} \in G \) . | Proof. Since \( M \) is countable, we can enumerate all dense subsets of \( P \) in \( M \) , say \( {D}_{1},{D}_{2},\ldots \) . We then define \( {p}_{n + 1} \) by introduction on \( n \) such that\n\n\[ \n{p}_{n + 1} \in {D}_{n + 1} \land {p}_{n + 1} \leqq {p}_{n}.\n\]\n\nLet \( G = \left\{ {q \in P \mid \left( {\exi... | Yes |
Proposition 18.11. \( \left( 1\right) \;t \in {T}_{\alpha } \rightarrow \mathrm{{Ord}}\left( {\varphi \left( t\right) }\right) < \mathrm{{Ord}}\left( {{\exists }^{\alpha }{x}_{i}\varphi \left( {x}_{i}\right) }\right) . \) | The proof is left to the reader. | No |
Proposition 18.13. \( M\left\lbrack G\right\rbrack \) is transitive and the ordinals in \( M\left\lbrack G\right\rbrack \) are the ordinals in \( M \) . | The proof is left to the reader. | No |
Lemma 18.19. Let \( \varphi \left( {{x}_{1},\ldots ,{x}_{n}}\right) \) be a first-order formula without any occurrence of \( {\exists }^{\alpha } \), or \( {\widehat{x}}^{\alpha } \) and let \( {t}_{1},\ldots ,{t}_{n} \in T \) . Then\n\n\[ \n{D}^{G}\left( {\varphi \left( {{t}_{1},\ldots ,{t}_{n}}\right) }\right) \;\tex... | The proof is left to the reader. | No |
Lemma 19.3. \( q \leqq p \land p \mid + \varphi \rightarrow q \mid + \varphi \) . | Proof. By transfinite induction on \( \operatorname{Ord}\left( \varphi \right) \) .\n\nThe details are left to the reader. | No |
Lemma 19.4. \( \left( {\exists p \in G}\right) \left( {p \mid +,\varphi \vee p\mid +\neg \varphi }\right) \) | Proof. Since \( p \mid + \varphi \) is \( M \) -definable, \( D = \{ p \in P \mid p \mid + \varphi \vee p \mid + \neg \varphi \} \) is a member of \( M \) . We claim that \( D \) is dense in \( P \) . Let \( p \in P \) . If \( p \mid + \neg \varphi \), then \( p \in D \) . If \( \neg \left( {p\rightarrow \neg \varphi }... | Yes |
Lemma 19.5. \( \neg \left( {p \mid + \varphi \land p\mid +\neg \varphi }\right) \) . | Proof. Immediate from the definition of \( p \mid + \neg \varphi \) . | No |
\[ \left( {\forall p \in G}\right) \left( {\exists q \leqq p}\right) \left( {q \mid + \varphi }\right) \leftrightarrow \left( {\exists p \in G}\right) \left( {p \mid + \varphi }\right) . \] | Proof. The \ | No |
Theorem 19.7. For a sentence \( \varphi ,\left( {\exists p \in G}\right) \left( {p \mid + \varphi }\right) \) iff \( M\left\lbrack G\right\rbrack \vDash \varphi \) . | Proof. First we prove this for a limited sentence \( \varphi \) by transfinite induction on \( \operatorname{Ord}\left( \varphi \right) \) then we prove it for an unlimited sentence \( \varphi \) by induction on the number of logical symbols in \( \varphi \) . Though there are many cases, almost all of the cases can be... | Yes |
Corollary 19.8. For a finite order sentence \( \varphi \left( {{t}_{1},\ldots ,{t}_{n}}\right) \) without any occurrence of \( {\widehat{x}}^{\alpha } \) or \( {\exists }^{\alpha } \), we have\n\n\[ \left( {\exists p \in G}\right) \left( {p \Vdash \varphi \left( {{t}_{1},\ldots ,{t}_{n}}\right) }\right) \leftrightarrow... | Proof. This is immediate from Theorem 19.7 and Lemma 18.19. | No |
Corollary 1 (Topological invarance of Whitehead torsion): If \( f : X \rightarrow Y \) is a homeomorphism (onto) then \( f \) is a simple-homotopy equivalence. | PROOF: \( f \times {1}_{Q} : X \times Q \rightarrow Y \times Q \) is a homeomorphism. | No |
Theorem 1.1 For integers \( n \geq 0,0 \leq l < m, a > 0 \) and \( b > 0 \), we have the following identities:\n\n\[ \mathop{\sum }\limits_{{k = 0}}^{n}\left( \begin{array}{l} n \\ k \end{array}\right) {a}^{n - k}{b}^{k + 1}{B}_{n - k}^{\left( m\right) }\left( {bx}\right) \mathop{\sum }\limits_{{i = 0}}^{k}\left( \begi... | Proof Let\n\n\[ f\left( t\right) = \frac{{t}^{{2m} - 1}{\mathrm{e}}^{abxt}\left( {{\mathrm{e}}^{abt} - 1}\right) {\mathrm{e}}^{abyt}}{{\left( {\mathrm{e}}^{at} - 1\right) }^{m}{\left( {\mathrm{e}}^{bt} - 1\right) }^{m}}. \]\n\nNote that this expression for \( f\left( t\right) \) is symmetric in \( a \) and \( b \) . Th... | Yes |
Theorem 2.1 For integers \( n \geq 0,0 \leq l < m, a > 0 \) and \( b > 0 \), if \( a \) and \( b \) have the same parity, then we have the following identities:\n\n\[ \mathop{\sum }\limits_{{k = 0}}^{n}\left( \begin{array}{l} n \\ k \end{array}\right) {a}^{n - k}{b}^{k}{E}_{n - k}^{\left( m\right) }\left( {bx}\right) \... | Proof Let\n\n\[ g\left( t\right) = \frac{{2}^{{2m} - 1}{\mathrm{e}}^{abxt}\left( {1 + {\left( -1\right) }^{a - 1}{\mathrm{e}}^{abt}}\right) {\mathrm{e}}^{abyt}}{{\left( {\mathrm{e}}^{at} + 1\right) }^{m}{\left( {\mathrm{e}}^{bt} + 1\right) }^{m}}. \]\n\nNote that this expression for \( g\left( t\right) \) is symmetric ... | Yes |
Theorem 2.2 For integers \( n \geq 0,0 \leq l < m, a > 0 \) and \( b > 0 \), if \( a \) is odd and \( b \) is even, then we have the following identities:\n\n\[ \mathop{\sum }\limits_{{k = 0}}^{n}\left( \begin{array}{l} n \\ k \end{array}\right) {a}^{n - k}{b}^{k}{E}_{n - k}^{\left( m\right) }\left( {bx}\right) \mathop... | Proof We expand \( g\left( t\right) \) into series in two different ways.\n\n\[ g\left( t\right) = \frac{{2}^{{2m} - 1}{\mathrm{e}}^{abxt}\left( {1 + {\left( -1\right) }^{a - 1}{\mathrm{e}}^{abt}}\right) {\mathrm{e}}^{abyt}}{{\left( {\mathrm{e}}^{at} + 1\right) }^{m}{\left( {\mathrm{e}}^{bt} + 1\right) }^{m}} \]\n\n\[ ... | Yes |
Theorem 2.3 For integers \( n \geq 0,0 \leq l < m, a > 0 \) and \( b > 0 \), if \( a \) is even and \( b \) is odd, then we have the following identities:\n\n\[ \mathop{\sum }\limits_{{k = 0}}^{n}\left( \begin{array}{l} n \\ k \end{array}\right) {a}^{n - k}{b}^{k}{E}_{n - k}^{\left( m\right) }\left( {bx}\right) \mathop... | Proof We expand \( g\left( t\right) \) into series in two different ways.\n\n\[ g\left( t\right) = \frac{{2}^{{2m} - 1}{\mathrm{e}}^{abxt}\left( {1 + {\left( -1\right) }^{a - 1}{\mathrm{e}}^{abt}}\right) {\mathrm{e}}^{abyt}}{{\left( {\mathrm{e}}^{at} + 1\right) }^{m}{\left( {\mathrm{e}}^{bt} + 1\right) }^{m}} \]\n\n\[ ... | Yes |
Corollary 2.3 For integers \( n \geq 1, a > 0 \) and \( b > 0 \), if \( a \) is even and \( b \) is odd, then we have the following identities: | \[ \mathop{\sum }\limits_{{k = 0}}^{n}\left( \begin{array}{l} n \\ k \end{array}\right) {a}^{n - k}{b}^{k}{E}_{n - k}^{\left( m\right) }\left( {bx}\right) {T}_{k}^{\left( m\right) }\left( {a - 1}\right) \] \[ = \mathop{\sum }\limits_{{j = 0}}^{{n - 1}}\left( \begin{array}{l} n \\ j \end{array}\right) {a}^{n - j}{E}_{n ... | Yes |
Theorem 3.1 For integers \( n \geq 0,0 \leq l < m, a > 0 \) and \( b > 0 \), if \( a \) is even, then we have the following identities:\n\n\[ \mathop{\sum }\limits_{{k = 0}}^{n}\left( \begin{array}{l} n \\ k \end{array}\right) {a}^{n - k}{b}^{k + 1}{B}_{n - k}^{\left( m\right) }\left( {bx}\right) \mathop{\sum }\limits_... | On the other hand, we have\n\n\[ h\left( t\right) = \frac{2}{{a}^{m}}\mathop{\sum }\limits_{{n = 0}}^{\infty }\left( {\mathop{\sum }\limits_{{k = 0}}^{n}\left( \begin{array}{l} n \\ k \end{array}\right) {a}^{n - k}{b}^{k}{B}_{n - k}^{\left( m\right) }\left( {bx}\right) \mathop{\sum }\limits_{{i = 0}}^{k}\left( \begin{a... | Yes |
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