Equation form expr-00ab483201161488

$\cardfont{a}$

Read as: the cardinal a

Means: the cardinal a

13 occurrences in this chapter

• Occurrence projected-formula-0032030
• Occurrence projected-formula-0032078
• Occurrence projected-formula-0032144
• Occurrence projected-formula-0032145
• Occurrence projected-formula-0032148
• Occurrence projected-formula-0032188
• Occurrence projected-formula-0032199
• Occurrence projected-formula-0032202
• Occurrence projected-formula-0032203
• Occurrence projected-formula-0032204
• Occurrence projected-formula-0032205
• Occurrence projected-formula-0032219
• Occurrence projected-formula-0032227

Equation form expr-01d96bf4bb5c1830

$\cardle{\Pow{\omega}}{\Real}$

Read as: the power set of omega has cardinality at most that of the real numbers

Means: the power set of omega has cardinality at most that of the real numbers

2 occurrences in this chapter

• Occurrence projected-formula-0032083
• Occurrence projected-formula-0032089

Equation form expr-026e768bdc4e4d23

$\cardtimes$

Read as: cardinal times

Means: cardinal times

2 occurrences in this chapter

• Occurrence projected-formula-0032053
• Occurrence projected-formula-0032057

Equation form expr-02ec3ef223c1d7a7

$\card{V_\alpha} = \beth_\alpha$

Read as: the cardinality of capital V subscript alpha equals beth subscript alpha

Means: the cardinality of capital V subscript alpha equals beth subscript alpha

1 occurrence in this chapter

• Occurrence projected-formula-0032288

Equation form expr-055de4ec3d8e4886

$\kappa$

Read as: kappa

Means: kappa

12 occurrences in this chapter

• Occurrence projected-formula-0032260
• Occurrence projected-formula-0032263
• Occurrence projected-formula-0032267
• Occurrence projected-formula-0032270
• Occurrence projected-formula-0032271
• Occurrence projected-formula-0032272
• Occurrence projected-formula-0032273
• Occurrence projected-formula-0032278
• Occurrence projected-formula-0032282
• Occurrence projected-formula-0032296
• Occurrence projected-formula-0032298
• Occurrence projected-formula-0032305

Equation form expr-06809210b9691700

$\card{V_{\omega+\alpha}} = \beth_{\alpha}$

Read as: the cardinality of capital V subscript omega plus alpha equals beth over alpha

Means: the cardinality of capital V subscript omega plus alpha equals beth over alpha

1 occurrence in this chapter

• Occurrence projected-formula-0032286

Equation form expr-0a31e077da6cbc01

$\cardfont{a}\leq \cardexpo{\cardfont{a}}{\cardfont{b}} \leq \cardsucc{\cardfont{a}}$

Read as: the cardinal a is less than or equal to cardinal exponentiation of the cardinal a to the power the cardinal b

Means: the cardinal a is less than or equal to cardinal exponentiation of the cardinal a to the power the cardinal b

1 occurrence in this chapter

• Occurrence projected-formula-0032237

Equation form expr-0aecbac1fa8cd841

$\card{A} \cardplus 1 = 1 \cardplus \card{A} = \card{A}$

Read as: the cardinality of capital A cardinal plus one equals one cardinal plus the cardinality of capital A equals the cardinality of capital A

Means: the cardinality of capital A cardinal plus one equals one cardinal plus the cardinality of capital A equals the cardinality of capital A

1 occurrence in this chapter

• Occurrence projected-formula-0032059

Equation form expr-0c632ffc56556321

$\funfromto{X}{Y}$

Read as: functions from capital X to capital Y

Means: functions from capital X to capital Y

2 occurrences in this chapter

• Occurrence projected-formula-0032034
• Occurrence projected-formula-0032038

Equation form expr-0d9ce8fbea95e426

$\tuple{\alpha_1, \alpha_2}$

Read as: the ordered pair alpha subscript one, then alpha subscript two

Means: the ordered pair alpha subscript one, then alpha subscript two

1 occurrence in this chapter

• Occurrence projected-formula-0032103

Equation form expr-0da775f751dd03e3

$\omega = \ordexpo{2}{\omega}$

Read as: omega equals ordinal exponentiation of two to the power omega

Means: omega equals ordinal exponentiation of two to the power omega

1 occurrence in this chapter

• Occurrence projected-formula-0032080

Equation form expr-0e663e0f57f57a74

$n \in \omega$

Read as: n belongs to omega

Means: n belongs to omega

1 occurrence in this chapter

• Occurrence projected-formula-0032189

Equation form expr-113ff1db723874c4

$\cardfont{a} < \cardexpo{2}{\cardfont{a}}$

Read as: the cardinal a is less than cardinal exponentiation of two to the power the cardinal a

Means: the cardinal a is less than cardinal exponentiation of two to the power the cardinal a

1 occurrence in this chapter

• Occurrence projected-formula-0032077

Equation form expr-11baa595827a4e0f

$\omega$

Read as: omega

Means: omega

1 occurrence in this chapter

• Occurrence projected-formula-0032076

Equation form expr-128583a37676f90d

$\alpha_2 = \beta_2$

Read as: alpha subscript two equals beta subscript two

Means: alpha subscript two equals beta subscript two

1 occurrence in this chapter

• Occurrence projected-formula-0032108

Equation form expr-16be23d752cb0d6e

$\funfromto{X}{Y} = \Setabs{f}{f \text{ is a function } X \to Y}$

Read as: functions from capital X to capital Y equals the set of f such that f, is a function, capital X maps to capital Y

Means: functions from capital X to capital Y equals the set of f such that f, is a function, capital X maps to capital Y

1 occurrence in this chapter

• Occurrence projected-formula-0032033

Equation form expr-18f5384d58bcb1bb

$Y$

Read as: capital Y

Means: capital Y

2 occurrences in this chapter

• Occurrence projected-formula-0032036
• Occurrence projected-formula-0032040

Equation form expr-1b16b1df538ba12d

$n$

Read as: n

Means: n

3 occurrences in this chapter

• Occurrence projected-formula-0032047
• Occurrence projected-formula-0032274
• Occurrence projected-formula-0032275

Equation form expr-1b7a9d3ef5f96fc3

$\cardeq{\omega}{\omega\times\omega}$

Read as: omega is equinumerous with omega times omega

Means: omega is equinumerous with omega times omega

1 occurrence in this chapter

• Occurrence projected-formula-0032124

Equation form expr-1c3fc2dbedca59df

$V_{W_\alpha}$

Read as: capital V subscript capital W subscript alpha

Means: capital V subscript capital W subscript alpha

1 occurrence in this chapter

• Occurrence projected-formula-0032311

Equation form expr-1d44b623d7327ce7

$\cardle{\alpha}{\alpha \times \alpha}$

Read as: alpha has cardinality at most that of alpha times alpha

Means: alpha has cardinality at most that of alpha times alpha

1 occurrence in this chapter

• Occurrence projected-formula-0032139

Equation form expr-1e756bbccd42b487

$\alpha \geq \omega$

Read as: alpha is greater than or equal to omega

Means: alpha is greater than or equal to omega

1 occurrence in this chapter

• Occurrence projected-formula-0032209

Equation form expr-208dd57aae84c8ca

$\ordtype{\alpha\times \alpha, \canonord} \leq \alpha$

Read as: the order type of alpha times alpha precedes in the canonical order is less than or equal to alpha

Means: the order type of alpha times alpha precedes in the canonical order is less than or equal to alpha

1 occurrence in this chapter

• Occurrence projected-formula-0032137

Equation form expr-20bb4634e6c63f72

$\beta \in \cardfont{a}$

Read as: beta belongs to the cardinal a

Means: beta belongs to the cardinal a

2 occurrences in this chapter

• Occurrence projected-formula-0032149
• Occurrence projected-formula-0032153

Equation form expr-22cf6d399835876b

$f(\gamma) \colon \cardfont{b} \to \cardfont{a}$

Read as: f of gamma colon the cardinal b maps to the cardinal a

Means: f of gamma colon the cardinal b maps to the cardinal a

1 occurrence in this chapter

• Occurrence projected-formula-0032174

Equation form expr-26d4eed27a4e196b

$\cardfont{b} = \aleph_{\gamma_\cardfont{b}}$

Read as: the cardinal b equals aleph over gamma subscript the cardinal b

Means: the cardinal b equals aleph over gamma subscript the cardinal b

1 occurrence in this chapter

• Occurrence projected-formula-0032223

Equation form expr-284c47be48fb0c66

$\lnot$

Read as: negation

Means: negation

1 occurrence in this chapter

• Occurrence projected-formula-0032242

Equation form expr-2addffea893434b6

$\cardfont{a} = \cardsucc{(\aleph_{\gamma_\cardfont{b}})}= \aleph_{\gamma_\cardfont{b}+1}$

Read as: the cardinal a equals the cardinal successor of open scope, aleph over gamma subscript the cardinal b, close scope

Means: the cardinal a equals the cardinal successor of open scope, aleph over gamma subscript the cardinal b, close scope

1 occurrence in this chapter

• Occurrence projected-formula-0032226

Equation form expr-316bf1f6e4be7245

$\cardle{\Real}{\Pow{\omega}}$

Read as: the real numbers has cardinality at most that of the power set of omega

Means: the real numbers has cardinality at most that of the power set of omega

2 occurrences in this chapter

• Occurrence projected-formula-0032084
• Occurrence projected-formula-0032090

Equation form expr-321563324f37c038

$\card{\Pow{A}} = \cardexpo{2}{\card{A}}$

Read as: the cardinality of the power set of capital A equals cardinal exponentiation of two to the power the cardinality of capital A

Means: the cardinality of the power set of capital A equals cardinal exponentiation of two to the power the cardinality of capital A

1 occurrence in this chapter

• Occurrence projected-formula-0032061

Equation form expr-33d71f01e44e606a

$\beta \in \cardfont{b}$

Read as: beta belongs to the cardinal b

Means: beta belongs to the cardinal b

1 occurrence in this chapter

• Occurrence projected-formula-0032167

Equation form expr-34d6291ebb0c22b3

$\cardeq{\Pow{A}}{\funfromto{\card{A}}{2}}$

Read as: the power set of capital A is equinumerous with functions from the cardinality of capital A to two

Means: the power set of capital A is equinumerous with functions from the cardinality of capital A to two

1 occurrence in this chapter

• Occurrence projected-formula-0032069

Equation form expr-37071ed0c173c7c5

$\card{\Pow{A}} = \card{\funfromto{\card{A}}{2}} = 2^{\card{A}}$

Read as: the cardinality of the power set of capital A equals the cardinality of functions from the cardinality of capital A to two

Means: the cardinality of the power set of capital A equals the cardinality of functions from the cardinality of capital A to two

1 occurrence in this chapter

• Occurrence projected-formula-0032070

Equation form expr-394d1165889bcde9

$f_\beta \colon X_\beta \to \cardfont{a}$

Read as: f subscript beta colon capital X subscript beta maps to the cardinal a

Means: f subscript beta colon capital X subscript beta maps to the cardinal a

1 occurrence in this chapter

• Occurrence projected-formula-0032154

Equation form expr-3963833522702523

$\cardfont{a} = \bigcup_{\cardfont{b} < \cardfont{a}} \cardfont{b} = \bigcup_{\cardfont{b} < \cardfont{a}}{\aleph_{\gamma_\cardfont{b}}}$

Read as: the cardinal a equals the union over the cardinal b is less than the cardinal a of the cardinal b

Means: the cardinal a equals the union over the cardinal b is less than the cardinal a of the cardinal b

1 occurrence in this chapter

• Occurrence projected-formula-0032228

Equation form expr-398949531980d0f4

$\tuple{\alpha_1, \alpha_2} = \tuple{\beta_1, \beta_2}$

Read as: the ordered pair alpha subscript one, then alpha subscript two equals the ordered pair beta subscript one, then beta subscript two

Means: the ordered pair alpha subscript one, then alpha subscript two equals the ordered pair beta subscript one, then beta subscript two

1 occurrence in this chapter

• Occurrence projected-formula-0032109

Equation form expr-40c08a2b7e20d725

$\kappa = \aleph_\kappa$

Read as: kappa equals aleph subscript kappa

Means: kappa equals aleph subscript kappa

1 occurrence in this chapter

• Occurrence projected-formula-0032279

Equation form expr-41cc6dcc2adab9ef

$\card{V_\kappa} = \beth_\kappa= \kappa$

Read as: the cardinality of capital V subscript kappa equals beth subscript kappa

Means: the cardinality of capital V subscript kappa equals beth subscript kappa

1 occurrence in this chapter

• Occurrence projected-formula-0032301

Equation form expr-441c7413926a416d

$v \in X_\beta$

Read as: v belongs to capital X subscript beta

Means: v belongs to capital X subscript beta

1 occurrence in this chapter

• Occurrence projected-formula-0032157

Equation form expr-448d6c0708a6613d

$v \notin X_\gamma$

Read as: v does not belong to capital X subscript gamma

Means: v does not belong to capital X subscript gamma

1 occurrence in this chapter

• Occurrence projected-formula-0032158

Equation form expr-4724bf76627efd8a

$\chi_{B}(x) &\defis \begin{cases} 1 & \text{if }x\in B\\ 0 & \text{otherwise.} \end{cases}$

Read as: chi subscript capital B of x is defined by cases. case one, one; if, x belongs to capital B. case two, zero; otherwise.. End cases

Means: chi subscript capital B of x is defined by cases. case one, one; if, x belongs to capital B. case two, zero; otherwise.. End cases

1 occurrence in this chapter

• Occurrence projected-formula-0032065

Equation form expr-476b9edd51844cb0

$\aleph_{0} &\defis \omega & \beth_{0} &\defis \omega\\ \aleph_{\alpha \ordplus 1} &\defis \cardsucc{(\aleph_{\alpha})} & \beth_{\alpha+1} &\defis \cardexpo{2}{\beth_{\alpha}}\\ \aleph_{\alpha} &\defis \bigcup_{\beta< \alpha} \aleph_{\beta} & \beth_{\alpha} &\defis \bigcup_{\beta < \alpha}\beth_{\beta} & \text{when $\alpha$ is a limit ordinal}.$

Read as: Source-ordered display. aleph over zero of is defined as omega applied to beth over zero. Then, aleph over alpha ordinal plus one of is defined as the cardinal successor of open scope, aleph over alpha, close scope beth over alpha plus one. Then, aleph over alpha of is defined as the union over beta is less than alpha of aleph over beta of beth over alpha. End display

Means: Source-ordered display. aleph over zero of is defined as omega applied to beth over zero. Then, aleph over alpha ordinal plus one of is defined as the cardinal successor of open scope, aleph over alpha, close scope beth over alpha plus one. Then, aleph over alpha of is defined as the union over beta is less than alpha of aleph over beta of beth over alpha. End display

1 occurrence in this chapter

• Occurrence projected-formula-0032200

Equation form expr-48647d63a534e3e1

$\card{V_\kappa} = \kappa$

Read as: the cardinality of capital V subscript kappa equals kappa

Means: the cardinality of capital V subscript kappa equals kappa

1 occurrence in this chapter

• Occurrence projected-formula-0032297

Equation form expr-4893e9df8b5496eb

$\alpha$

Read as: alpha

Means: alpha

9 occurrences in this chapter

• Occurrence projected-formula-0032100
• Occurrence projected-formula-0032111
• Occurrence projected-formula-0032122
• Occurrence projected-formula-0032123
• Occurrence projected-formula-0032126
• Occurrence projected-formula-0032212
• Occurrence projected-formula-0032234
• Occurrence projected-formula-0032276
• Occurrence projected-formula-0032312

Equation form expr-4b68ab3847feda7d

$X$

Read as: capital X

Means: capital X

3 occurrences in this chapter

• Occurrence projected-formula-0032035
• Occurrence projected-formula-0032039
• Occurrence projected-formula-0032116

Equation form expr-4bf5f5308abe549f

$\tuple{\alpha \times \alpha, \canonord}$

Read as: the ordered pair alpha times alpha, then the canonical order relation

Means: the ordered pair alpha times alpha, then the canonical order relation

1 occurrence in this chapter

• Occurrence projected-formula-0032099

Equation form expr-4d2eb930d95d1423

$\cardfont{b}$

Read as: the cardinal b

Means: the cardinal b

5 occurrences in this chapter

• Occurrence projected-formula-0032031
• Occurrence projected-formula-0032181
• Occurrence projected-formula-0032187
• Occurrence projected-formula-0032194
• Occurrence projected-formula-0032225

Equation form expr-4dc190c5ba3f4680

$2 \leq \cardfont{b} < \cardfont{a} \leq \cardexpo{2}{\cardfont{b}}$

Read as: two is less than or equal to the cardinal b is less than the cardinal a is less than or equal to cardinal exponentiation of two to the power the cardinal b

Means: two is less than or equal to the cardinal b is less than the cardinal a is less than or equal to cardinal exponentiation of two to the power the cardinal b

1 occurrence in this chapter

• Occurrence projected-formula-0032193

Equation form expr-4e3c7511020cb010

$\alpha \times \alpha$

Read as: alpha times alpha

Means: alpha times alpha

1 occurrence in this chapter

• Occurrence projected-formula-0032102

Equation form expr-4f1058ece857387f

$\Setabs{\max(\gamma_1, \gamma_2)}{\tuple{\gamma_1, \gamma_2} \in X}$

Read as: the set of the maximum of gamma subscript one, then gamma subscript two such that the ordered pair gamma subscript one, then gamma subscript two belongs to capital X

Means: the set of the maximum of gamma subscript one, then gamma subscript two such that the ordered pair gamma subscript one, then gamma subscript two belongs to capital X

1 occurrence in this chapter

• Occurrence projected-formula-0032113

Equation form expr-50232bdef8c40263

$\card{X_\beta} \leq \cardfont{a}$

Read as: the cardinality of capital X subscript beta is less than or equal to the cardinal a

Means: the cardinality of capital X subscript beta is less than or equal to the cardinal a

1 occurrence in this chapter

• Occurrence projected-formula-0032151

Equation form expr-512a911f16f7706a

$\cardeq{\card{\alpha}}{\alpha}$

Read as: the cardinality of alpha is equinumerous with alpha

Means: the cardinality of alpha is equinumerous with alpha

1 occurrence in this chapter

• Occurrence projected-formula-0032129

Equation form expr-5157c866e2f2bd7d

$\aleph_1 = \beth_1$

Read as: aleph subscript one of equals beth subscript one

Means: aleph subscript one of equals beth subscript one

1 occurrence in this chapter

• Occurrence projected-formula-0032244

Equation form expr-522df6ad7d1bb5ca

$\Zminus$

Read as: set theory Z minus

Means: set theory Z minus

1 occurrence in this chapter

• Occurrence projected-formula-0032037

Equation form expr-528b86f8c38f9730

$\cardexpo{\cardfont{a}}{\cardfont{b}}$

Read as: cardinal exponentiation of the cardinal a to the power the cardinal b

Means: cardinal exponentiation of the cardinal a to the power the cardinal b

4 occurrences in this chapter

• Occurrence projected-formula-0032179
• Occurrence projected-formula-0032185
• Occurrence projected-formula-0032192
• Occurrence projected-formula-0032236

Equation form expr-5408911f9f73eddf

$\omega \ordtimes \omega < \kappa$

Read as: omega ordinal times omega is less than kappa

Means: omega ordinal times omega is less than kappa

1 occurrence in this chapter

• Occurrence projected-formula-0032300

Equation form expr-54ce300aa2b3e6d6

$\setrank{\funfromto{X}{Y}}$

Read as: the rank of functions from capital X to capital Y

Means: the rank of functions from capital X to capital Y

1 occurrence in this chapter

• Occurrence projected-formula-0032042

Equation form expr-559aead08264d579

$A$

Read as: capital A

Means: capital A

5 occurrences in this chapter

• Occurrence projected-formula-0032046
• Occurrence projected-formula-0032058
• Occurrence projected-formula-0032062
• Occurrence projected-formula-0032075
• Occurrence projected-formula-0032307

Equation form expr-559c63bd803bef5b

$\gamma = \max(\gamma_1, \gamma_2)$

Read as: gamma equals the maximum of gamma subscript one, then gamma subscript two

Means: gamma equals the maximum of gamma subscript one, then gamma subscript two

1 occurrence in this chapter

• Occurrence projected-formula-0032133

Equation form expr-55d1823545b5b4c3

$\tau$

Read as: tau

Means: tau

1 occurrence in this chapter

• Occurrence projected-formula-0032306

Equation form expr-565e5f4eb5e72669

$\card{\bigcup_{\beta \in \cardfont{a}} X_\beta} \leq \cardfont{a}$

Read as: the cardinality of the union over beta belongs to the cardinal a of capital X subscript beta is less than or equal to the cardinal a

Means: the cardinality of the union over beta belongs to the cardinal a of capital X subscript beta is less than or equal to the cardinal a

1 occurrence in this chapter

• Occurrence projected-formula-0032152

Equation form expr-576dd5406577b02d

$f \colon (\cardfont{b}\disjointsum\cardfont{c}) \to \cardfont{a}$

Read as: f colon open scope, the cardinal b disjoint sum the cardinal c, close scope maps to the cardinal a

Means: f colon open scope, the cardinal b disjoint sum the cardinal c, close scope maps to the cardinal a

1 occurrence in this chapter

• Occurrence projected-formula-0032165

Equation form expr-5828eda51f773ec9

$f \colon \cardfont{c} \to (\funfromto{\cardfont{b}}{\cardfont{a}})$

Read as: f colon the cardinal c maps to open scope, functions from the cardinal b to the cardinal a, close scope

Means: f colon the cardinal c maps to open scope, functions from the cardinal b to the cardinal a, close scope

1 occurrence in this chapter

• Occurrence projected-formula-0032172

Equation form expr-583e90eb7865276c

$B \subseteq A$

Read as: capital B is a subset of capital A

Means: capital B is a subset of capital A

1 occurrence in this chapter

• Occurrence projected-formula-0032063

Equation form expr-598ca71698daef6a

$\omega + \alpha = \alpha$

Read as: omega plus alpha equals alpha

Means: omega plus alpha equals alpha

1 occurrence in this chapter

• Occurrence projected-formula-0032290

Equation form expr-5c17ab74e47c9426

$\card{\alpha} \in \alpha$

Read as: the cardinality of alpha belongs to alpha

Means: the cardinality of alpha belongs to alpha

1 occurrence in this chapter

• Occurrence projected-formula-0032127

Equation form expr-5d63b44cd144534c

$\max(\alpha_1, \alpha_2) < \max(\beta_1, \beta_2)$

Read as: the maximum of open scope, alpha subscript one, then alpha subscript two, close scope is less than the maximum of beta subscript one, then beta subscript two

Means: the maximum of open scope, alpha subscript one, then alpha subscript two, close scope is less than the maximum of beta subscript one, then beta subscript two

1 occurrence in this chapter

• Occurrence projected-formula-0032093

Equation form expr-5e4eb86e776d9e86

$\cardeq{\alpha}{\beta}$

Read as: alpha is equinumerous with beta

Means: alpha is equinumerous with beta

1 occurrence in this chapter

• Occurrence projected-formula-0032117

Equation form expr-5ed93b3aeb41aeb8

$\cardle{\alpha \times \alpha}{\alpha}$

Read as: alpha times alpha has cardinality at most that of alpha

Means: alpha times alpha has cardinality at most that of alpha

1 occurrence in this chapter

• Occurrence projected-formula-0032138

Equation form expr-612a23d3904379cf

$\aleph_\alpha$

Read as: aleph subscript alpha

Means: aleph subscript alpha

3 occurrences in this chapter

• Occurrence projected-formula-0032210
• Occurrence projected-formula-0032214
• Occurrence projected-formula-0032277

Equation form expr-618d687533fbb236

$\cardexpo{2}{\cardfont{b}} &\leq \cardexpo{\cardfont{a}}{\cardfont{b}}\text{, as $2 \leq \cardfont{a}$}\\ &\leq \cardexpo{(2^\cardfont{a})}{\cardfont{b}} \text{, by \olref[opps]{lem:SizePowerset2Exp}}\\ &= \cardexpo{2}{\cardfont{a} \cardtimes \cardfont{b}} \text{, by \olref{simplecardexpo}} \\ &= \cardexpo{2}{\cardfont{b}} \text{, by \olref[simp]{cardplustimesmax}}$

Read as: Source-ordered display. cardinal exponentiation of two to the power the cardinal b is less than or equal to cardinal exponentiation of the cardinal a to the power the cardinal b , as two is less than or equal to the cardinal a. Then, is less than or equal to cardinal exponentiation of open scope, two superscript the cardinal a, close scope to the power the cardinal b , by the lemma on Size Powersettwo Exp. Then, equals cardinal exponentiation of two to the power the cardinal a cardinal times the cardinal b , by proposition five in chapter Cardinal Arithmetic. Then, equals cardinal exponentiation of two to the power the cardinal b , by theorem two in chapter Cardinal Arithmetic. End display

Means: Source-ordered display. cardinal exponentiation of two to the power the cardinal b is less than or equal to cardinal exponentiation of the cardinal a to the power the cardinal b , as two is less than or equal to the cardinal a. Then, is less than or equal to cardinal exponentiation of open scope, two superscript the cardinal a, close scope to the power the cardinal b , by the lemma on Size Powersettwo Exp. Then, equals cardinal exponentiation of two to the power the cardinal a cardinal times the cardinal b , by proposition five in chapter Cardinal Arithmetic. Then, equals cardinal exponentiation of two to the power the cardinal b , by theorem two in chapter Cardinal Arithmetic. End display

1 occurrence in this chapter

• Occurrence projected-formula-0032183

Equation form expr-62c66a7a5dd70c31

$m$

Read as: m

Means: m

1 occurrence in this chapter

• Occurrence projected-formula-0032049

Equation form expr-64f775ac6ae0544e

$\aleph_{\alpha+1}$

Read as: aleph over alpha plus one

Means: aleph over alpha plus one

1 occurrence in this chapter

• Occurrence projected-formula-0032216

Equation form expr-66d13378c8816533

$\cardeq{(\cardfont{a} \disjointsum \cardfont{b})}{(\cardfont{b} \disjointsum \cardfont{a})}$

Read as: open scope, the cardinal a disjoint sum the cardinal b, close scope is equinumerous with open scope, the cardinal b disjoint sum the cardinal a, close scope

Means: open scope, the cardinal a disjoint sum the cardinal b, close scope is equinumerous with open scope, the cardinal b disjoint sum the cardinal a, close scope

1 occurrence in this chapter

• Occurrence projected-formula-0032054

Equation form expr-67610b0632683369

$\gamma$

Read as: gamma

Means: gamma

2 occurrences in this chapter

• Occurrence projected-formula-0032135
• Occurrence projected-formula-0032221

Equation form expr-67c0a43903195983

$\tuple{\beta_1, \beta_2}$

Read as: the ordered pair beta subscript one, then beta subscript two

Means: the ordered pair beta subscript one, then beta subscript two

1 occurrence in this chapter

• Occurrence projected-formula-0032104

Equation form expr-6c31368de4379e34

$\tau_0(A) & \defis \card{A}\\ \tau_{n+1}(A) & \defis \beth_{\tau_n(A)}\\ \tau(A) & \defis \bigcup_{n < \omega}\tau_n(A) \intertext{As in \olref{bethfixed}, $\tau(A)$ is a $\beth$-fixed point for any $A$, and trivially $\card{A} < \tau(A)$. So now consider this recursive definition:} W_0 &\defis 0\\ W_{\alpha + 1} & \defis \tau(W_\alpha)\\ W_\alpha & \defis \bigcup_{\beta < \alpha} W_\beta \text{, when $\alpha$ is a limit}$

Read as: Source-ordered display. tau subscript zero of capital A is defined as the cardinality of capital A. Then, tau subscript n plus one of capital A is defined as beth over tau subscript n of capital A. Then, tau of capital A is defined as the union over n is less than omega of tau subscript n of capital A. Then, As in proposition twelve in chapter Cardinal Arithmetic, tau of capital A is a beth-fixed point for any capital A, and trivially the cardinality of capital A is less than tau of capital A. So now consider this recursive definition:. Then, capital W subscript zero is defined as zero. Then, capital W subscript alpha plus one is defined as tau of capital W subscript alpha. Then, capital W subscript alpha is defined as the union over beta is less than alpha of capital W subscript beta, , when alpha is a limit. End display

Means: Source-ordered display. tau subscript zero of capital A is defined as the cardinality of capital A. Then, tau subscript n plus one of capital A is defined as beth over tau subscript n of capital A. Then, tau of capital A is defined as the union over n is less than omega of tau subscript n of capital A. Then, As in proposition twelve in chapter Cardinal Arithmetic, tau of capital A is a beth-fixed point for any capital A, and trivially the cardinality of capital A is less than tau of capital A. So now consider this recursive definition:. Then, capital W subscript zero is defined as zero. Then, capital W subscript alpha plus one is defined as tau of capital W subscript alpha. Then, capital W subscript alpha is defined as the union over beta is less than alpha of capital W subscript beta, , when alpha is a limit. End display

1 occurrence in this chapter

• Occurrence projected-formula-0032308

Equation form expr-6e82ac27792a9ac6

$\Real$

Read as: the real numbers

Means: the real numbers

2 occurrences in this chapter

• Occurrence projected-formula-0032081
• Occurrence projected-formula-0032248

Equation form expr-6eb2cb875aed951f

$\cardexpo{\cardfont{a}}{n} = \cardfont{a}$

Read as: cardinal exponentiation of the cardinal a to the power n equals the cardinal a

Means: cardinal exponentiation of the cardinal a to the power n equals the cardinal a

1 occurrence in this chapter

• Occurrence projected-formula-0032190

Equation form expr-703c57c85df5fba6

$\beth_\alpha$

Read as: beth subscript alpha

Means: beth subscript alpha

3 occurrences in this chapter

• Occurrence projected-formula-0032208
• Occurrence projected-formula-0032211
• Occurrence projected-formula-0032215

Equation form expr-7046857daf500937

$\ZF$

Read as: set theory Z F

Means: set theory Z F

1 occurrence in this chapter

• Occurrence projected-formula-0032041

Equation form expr-705e78cac6bb8e8b

$\cardeq{\Real}{\Pow{\omega}}$

Read as: the real numbers is equinumerous with the power set of omega

Means: the real numbers is equinumerous with the power set of omega

1 occurrence in this chapter

• Occurrence projected-formula-0032085

Equation form expr-723699f6b0319bc1

$\alpha = (\omega\ordtimes\omega) \ordplus \beta$

Read as: alpha equals open scope, omega ordinal times omega, close scope ordinal plus beta

Means: alpha equals open scope, omega ordinal times omega, close scope ordinal plus beta

1 occurrence in this chapter

• Occurrence projected-formula-0032293

Equation form expr-7532402fed19c870

$\ZFC + \lnot\text{CH}$

Read as: set theory Z F C plus not CH

Means: set theory Z F C plus not CH

1 occurrence in this chapter

• Occurrence projected-formula-0032246

Equation form expr-76208888c14b5db6

$\Bin^{\omega}$

Read as: the set of binary sequences superscript omega

Means: the set of binary sequences superscript omega

1 occurrence in this chapter

• Occurrence projected-formula-0032073

Equation form expr-7701c56f6c7ea9fe

$\alpha_2 < \beta_2$

Read as: alpha subscript two is less than beta subscript two

Means: alpha subscript two is less than beta subscript two

1 occurrence in this chapter

• Occurrence projected-formula-0032098

Equation form expr-7864d74e178141e7

$\funfromto{\omega}{2}$

Read as: functions from omega to two

Means: functions from omega to two

1 occurrence in this chapter

• Occurrence projected-formula-0032074

Equation form expr-786c3d188fec6489

$\cardfont{a} = \cardsucc{\cardfont{b}}$

Read as: the cardinal a equals the cardinal successor of the cardinal b

Means: the cardinal a equals the cardinal successor of the cardinal b

1 occurrence in this chapter

• Occurrence projected-formula-0032224

Equation form expr-7a045c1ef97d790c

$A \times B$

Read as: capital A times capital B

Means: capital A times capital B

1 occurrence in this chapter

• Occurrence projected-formula-0032050

Equation form expr-7c582c2b017c6fbf

$\kappa< \aleph_\kappa$

Read as: kappa is less than aleph subscript kappa

Means: kappa is less than aleph subscript kappa

1 occurrence in this chapter

• Occurrence projected-formula-0032259

Equation form expr-7dd13371e57203fa

$\cardfont{a}\cardtimes\cardfont{a} = \cardfont{a} \leq \cardfont{a} \cardplus \cardfont{b} \leq \cardfont{a} \cardplus \cardfont{a} \leq \cardfont{a} \cardtimes \cardfont{a}$

Read as: the cardinal a cardinal times the cardinal a equals the cardinal a is less than or equal to the cardinal a cardinal plus the cardinal b is less than or equal to the cardinal a cardinal plus the cardinal a is less than or equal to the cardinal a cardinal times the cardinal a

Means: the cardinal a cardinal times the cardinal a equals the cardinal a is less than or equal to the cardinal a cardinal plus the cardinal b is less than or equal to the cardinal a cardinal plus the cardinal a is less than or equal to the cardinal a cardinal times the cardinal a

1 occurrence in this chapter

• Occurrence projected-formula-0032143

Equation form expr-80dd722deb2674b6

$\cardfont{b} < \cardexpo{2}{\cardfont{b}}$

Read as: the cardinal b is less than cardinal exponentiation of two to the power the cardinal b

Means: the cardinal b is less than cardinal exponentiation of two to the power the cardinal b

1 occurrence in this chapter

• Occurrence projected-formula-0032184

Equation form expr-81a1c5037f99dacd

$\kappa= \bigcup_{n < \omega} \kappa_{n+1} = \bigcup_{n < \omega}\aleph_{\kappa_n} = \bigcup_{\alpha < \kappa}\aleph_\alpha = \aleph_\kappa$

Read as: kappa equals the union over n is less than omega of kappa subscript n plus one equals the union over n is less than omega of aleph over kappa subscript n equals the union over alpha is less than kappa of aleph subscript alpha equals aleph subscript kappa

Means: kappa equals the union over n is less than omega of kappa subscript n plus one equals the union over n is less than omega of aleph over kappa subscript n equals the union over alpha is less than kappa of aleph subscript alpha equals aleph subscript kappa

1 occurrence in this chapter

• Occurrence projected-formula-0032268

Equation form expr-81cbb13c1f4a5561

$\beth_{\alpha+1}$

Read as: beth over alpha plus one

Means: beth over alpha plus one

1 occurrence in this chapter

• Occurrence projected-formula-0032217

Equation form expr-84c1bc4c3222ea4b

$g \colon \Real \to \Pow{\omega}$

Read as: g colon the real numbers maps to the power set of omega

Means: g colon the real numbers maps to the power set of omega

1 occurrence in this chapter

• Occurrence projected-formula-0032088

Equation form expr-84e8c34537916246

$g \colon \bigcup_{\beta \in \cardfont{a}} X_\beta \to \cardfont{a} \times \cardfont{a}$

Read as: g colon the union over beta belongs to the cardinal a of capital X subscript beta maps to the cardinal a times the cardinal a

Means: g colon the union over beta belongs to the cardinal a of capital X subscript beta maps to the cardinal a times the cardinal a

1 occurrence in this chapter

• Occurrence projected-formula-0032155

Equation form expr-868fad332b60fd0a

$\aleph_0 = \beth_0$

Read as: aleph subscript zero of equals beth subscript zero

Means: aleph subscript zero of equals beth subscript zero

1 occurrence in this chapter

• Occurrence projected-formula-0032250

Equation form expr-869382de818e47fe

$\omega \ordplus \alpha = \omega \ordplus ((\omega \ordtimes \omega) \ordplus \beta) = (\omega \ordplus (\omega \ordtimes \omega)) \ordplus \beta = (\omega \ordtimes \omega) \ordplus \beta = \alpha$

Read as: omega ordinal plus alpha equals omega ordinal plus open scope, open scope, omega ordinal times omega, close scope ordinal plus beta, close scope equals open scope, omega ordinal plus open scope, omega ordinal times omega, close scope, close scope ordinal plus beta equals open scope, omega ordinal times omega, close scope ordinal plus beta equals alpha

Means: omega ordinal plus alpha equals omega ordinal plus open scope, open scope, omega ordinal times omega, close scope ordinal plus beta, close scope equals open scope, omega ordinal plus open scope, omega ordinal times omega, close scope, close scope ordinal plus beta equals open scope, omega ordinal times omega, close scope ordinal plus beta equals alpha

1 occurrence in this chapter

• Occurrence projected-formula-0032295

Equation form expr-8846f8f26404e16f

$\cardfont{b} < \cardfont{a}$

Read as: the cardinal b is less than the cardinal a

Means: the cardinal b is less than the cardinal a

3 occurrences in this chapter

• Occurrence projected-formula-0032186
• Occurrence projected-formula-0032222
• Occurrence projected-formula-0032235

Equation form expr-8bc9f3d0df963525

$\leq\cardfont{a}$

Read as: is less than or equal to the cardinal a

Means: is less than or equal to the cardinal a

2 occurrences in this chapter

• Occurrence projected-formula-0032146
• Occurrence projected-formula-0032147

Equation form expr-8dd91f8c4b09023d

$X \subseteq \alpha\times\alpha$

Read as: capital X is a subset of alpha times alpha

Means: capital X is a subset of alpha times alpha

1 occurrence in this chapter

• Occurrence projected-formula-0032110

Equation form expr-913cf3b881945151

$\tuple{\beta, \gamma} \in \cardfont{b} \times \cardfont{c}$

Read as: the ordered pair beta, then gamma belongs to the cardinal b times the cardinal c

Means: the ordered pair beta, then gamma belongs to the cardinal b times the cardinal c

1 occurrence in this chapter

• Occurrence projected-formula-0032176

Equation form expr-92a21d47df981641

$\cardexpo{\cardfont{a}}{n} = \cardfont{a} \cardtimes \cardfont{a} \cardtimes \ldots \cardtimes \cardfont{a} = \cardfont{a}$

Read as: cardinal exponentiation of the cardinal a to the power n equals the cardinal a cardinal times the cardinal a

Means: cardinal exponentiation of the cardinal a to the power n equals the cardinal a cardinal times the cardinal a

1 occurrence in this chapter

• Occurrence projected-formula-0032191

Equation form expr-937cdad7cd471979

$\cardexpo{(\cardexpo{\cardfont{a}}{\cardfont{b}})}{\cardfont{c}} = \cardexpo{\cardfont{a}}{\cardfont{b} \cardtimes \cardfont{c}}$

Read as: cardinal exponentiation of open scope, cardinal exponentiation of the cardinal a to the power the cardinal b, close scope to the power the cardinal c equals cardinal exponentiation of the cardinal a to the power the cardinal b cardinal times the cardinal c

Means: cardinal exponentiation of open scope, cardinal exponentiation of the cardinal a to the power the cardinal b, close scope to the power the cardinal c equals cardinal exponentiation of the cardinal a to the power the cardinal b cardinal times the cardinal c

1 occurrence in this chapter

• Occurrence projected-formula-0032163

Equation form expr-95a8355b06b66389

$\setrank{X}$

Read as: the rank of capital X

Means: the rank of capital X

1 occurrence in this chapter

• Occurrence projected-formula-0032043

Equation form expr-96b71b8ea23f3684

$\omega \ordtimes \omega = \omega \ordtimes (1 \ordplus \omega) = (\omega \ordtimes 1) \ordplus (\omega\ordtimes\omega) = \omega \ordplus (\omega \ordtimes \omega)$

Read as: omega ordinal times omega equals omega ordinal times open scope, one ordinal plus omega, close scope equals open scope, omega ordinal times one, close scope ordinal plus open scope, omega ordinal times omega, close scope equals omega ordinal plus open scope, omega ordinal times omega, close scope

Means: omega ordinal times omega equals omega ordinal times open scope, one ordinal plus omega, close scope equals open scope, omega ordinal times one, close scope ordinal plus open scope, omega ordinal times omega, close scope equals omega ordinal plus open scope, omega ordinal times omega, close scope

1 occurrence in this chapter

• Occurrence projected-formula-0032291

Equation form expr-96d0ac49aceb3fbf

$\cardsucc{\cardfont{a}}$

Read as: the cardinal successor of the cardinal a

Means: the cardinal successor of the cardinal a

2 occurrences in this chapter

• Occurrence projected-formula-0032198
• Occurrence projected-formula-0032201

Equation form expr-97320fa1c6111401

$f_\cardfont{c}(\gamma) = f(\gamma, 1)$

Read as: f subscript the cardinal c of gamma equals f of gamma and one

Means: f subscript the cardinal c of gamma equals f of gamma and one

1 occurrence in this chapter

• Occurrence projected-formula-0032168

Equation form expr-9836020725b6fff8

$0 < \aleph_0$

Read as: zero is less than aleph subscript zero

Means: zero is less than aleph subscript zero

1 occurrence in this chapter

• Occurrence projected-formula-0032256

Equation form expr-9881768c4b7b6085

$\cardeq{\alpha \times \alpha}{\beta \times \beta}$

Read as: alpha times alpha is equinumerous with beta times beta

Means: alpha times alpha is equinumerous with beta times beta

1 occurrence in this chapter

• Occurrence projected-formula-0032118

Equation form expr-9a89728b4bb49649

$\tuple{\gamma_1, \gamma_2} \in \alpha \times \alpha$

Read as: the ordered pair gamma subscript one, then gamma subscript two belongs to alpha times alpha

Means: the ordered pair gamma subscript one, then gamma subscript two belongs to alpha times alpha

1 occurrence in this chapter

• Occurrence projected-formula-0032131

Equation form expr-9b4aa9b2f0e1dbea

$2 \leq \cardfont{a} \leq \cardfont{b}$

Read as: two is less than or equal to the cardinal a is less than or equal to the cardinal b

Means: two is less than or equal to the cardinal a is less than or equal to the cardinal b

1 occurrence in this chapter

• Occurrence projected-formula-0032180

Equation form expr-9b5655188745a242

$\cardfont{a} = \max(\cardfont{a}, \cardfont{b})$

Read as: the cardinal a equals the maximum of the cardinal a the empty expression, then the cardinal b

Means: the cardinal a equals the maximum of the cardinal a the empty expression, then the cardinal b

1 occurrence in this chapter

• Occurrence projected-formula-0032142

Equation form expr-9e7f4a1822ad8447

$\beth$

Read as: beth

Means: beth

6 occurrences in this chapter

• Occurrence projected-formula-0032207
• Occurrence projected-formula-0032232
• Occurrence projected-formula-0032281
• Occurrence projected-formula-0032284
• Occurrence projected-formula-0032299
• Occurrence projected-formula-0032310

Equation form expr-9ef714289deedcf5

$\gamma \in \beta$

Read as: gamma belongs to beta

Means: gamma belongs to beta

1 occurrence in this chapter

• Occurrence projected-formula-0032159

Equation form expr-a07d283fa971c66e

$\tuple{\gamma_1, \gamma_2} \mapsto \tuple{f(\gamma_1), f(\gamma_2)}$

Read as: the ordered pair gamma subscript one, then gamma subscript two maps to the ordered pair f of gamma subscript one, then f of gamma subscript two

Means: the ordered pair gamma subscript one, then gamma subscript two maps to the ordered pair f of gamma subscript one, then f of gamma subscript two

1 occurrence in this chapter

• Occurrence projected-formula-0032120

Equation form expr-a0e54f88d44846e3

$\gamma \in \cardfont{c}$

Read as: gamma belongs to the cardinal c

Means: gamma belongs to the cardinal c

2 occurrences in this chapter

• Occurrence projected-formula-0032169
• Occurrence projected-formula-0032173

Equation form expr-a45b71f0620cf4b1

$\cardfont{a}, \cardfont{b}, \cardfont{c}$

Read as: the cardinal a the empty expression, then the cardinal b, then the cardinal c

Means: the cardinal a the empty expression, then the cardinal b, then the cardinal c

1 occurrence in this chapter

• Occurrence projected-formula-0032164

Equation form expr-a4ef17a629d8bc6c

$\text{Seg}(\gamma_1, \gamma_2) & \precsim ((\gamma \ordplus 1)\ordtimes (\gamma \ordplus 1))\\ &\approx (\gamma \ordtimes \gamma) \text{, by \olref[ord-arithmetic][using-addition]{ordinfinitycharacter} and \olref{simplecardproduct}}\\ &\approx \gamma \text{, by the induction hypothesis}\\ & \prec \alpha\text{, since $\alpha$ is a cardinal}$

Read as: Source-ordered display. Seg, open scope, gamma subscript one, then gamma subscript two, close scope is dominated by open scope, open scope, gamma ordinal plus one, close scope ordinal times open scope, gamma ordinal plus one, close scope, close scope. Then, is approximately equal to open scope, gamma ordinal times gamma, close scope, , by lemma five in chapter Ordinal Arithmetic and proposition three in chapter Cardinal Arithmetic. Then, is approximately equal to gamma, , by the induction hypothesis. Then, is strictly less than alpha, , since alpha is a cardinal. End display

Means: Source-ordered display. Seg, open scope, gamma subscript one, then gamma subscript two, close scope is dominated by open scope, open scope, gamma ordinal plus one, close scope ordinal times open scope, gamma ordinal plus one, close scope, close scope. Then, is approximately equal to open scope, gamma ordinal times gamma, close scope, , by lemma five in chapter Ordinal Arithmetic and proposition three in chapter Cardinal Arithmetic. Then, is approximately equal to gamma, , by the induction hypothesis. Then, is strictly less than alpha, , since alpha is a cardinal. End display

1 occurrence in this chapter

• Occurrence projected-formula-0032136

Equation form expr-aac6d12ee6adf210

$\cardeq{\alpha}{\alpha \times \alpha}$

Read as: alpha is equinumerous with alpha times alpha

Means: alpha is equinumerous with alpha times alpha

1 occurrence in this chapter

• Occurrence projected-formula-0032121

Equation form expr-ab0c73e908963f59

$\setrank{Y}$

Read as: the rank of capital Y

Means: the rank of capital Y

1 occurrence in this chapter

• Occurrence projected-formula-0032044

Equation form expr-aba0ac8fc1b46a59

$\ZFC + \text{GCH}$

Read as: set theory Z F C plus GCH

Means: set theory Z F C plus GCH

1 occurrence in this chapter

• Occurrence projected-formula-0032241

Equation form expr-ad60575d91fc39d5

$\cardfont{a} = \aleph_\gamma$

Read as: the cardinal a equals aleph subscript gamma

Means: the cardinal a equals aleph subscript gamma

2 occurrences in this chapter

• Occurrence projected-formula-0032220
• Occurrence projected-formula-0032229

Equation form expr-b22e2621b9cb3013

$\aleph_\alpha = \beth_\alpha$

Read as: aleph subscript alpha of equals beth subscript alpha

Means: aleph subscript alpha of equals beth subscript alpha

1 occurrence in this chapter

• Occurrence projected-formula-0032233

Equation form expr-b35a570dd1397cc2

$\gamma = \bigcup_{\cardfont{b} < \cardfont{a}}\gamma_\cardfont{b}$

Read as: gamma equals the union over the cardinal b is less than the cardinal a of gamma subscript the cardinal b

Means: gamma equals the union over the cardinal b is less than the cardinal a of gamma subscript the cardinal b

1 occurrence in this chapter

• Occurrence projected-formula-0032230

Equation form expr-b4ba5eea812378b1

$\omega < \cardexpo{2}{\omega}$

Read as: omega is less than cardinal exponentiation of two to the power omega

Means: omega is less than cardinal exponentiation of two to the power omega

1 occurrence in this chapter

• Occurrence projected-formula-0032079

Equation form expr-b4c6291b64161960

$\aleph_0 = \beth_0 = \omega$

Read as: aleph subscript zero of equals beth subscript zero

Means: aleph subscript zero of equals beth subscript zero

1 occurrence in this chapter

• Occurrence projected-formula-0032213

Equation form expr-b4ee0f45f3b6961e

$1 < \aleph_1$

Read as: one is less than aleph subscript one

Means: one is less than aleph subscript one

1 occurrence in this chapter

• Occurrence projected-formula-0032257

Equation form expr-b5baefbbb1300da7

$\Bin^\omega$

Read as: the set of binary sequences superscript omega

Means: the set of binary sequences superscript omega

1 occurrence in this chapter

• Occurrence projected-formula-0032072

Equation form expr-b62fb38fa53c3be5

$\disjointsum$

Read as: disjoint sum

Means: disjoint sum

1 occurrence in this chapter

• Occurrence projected-formula-0032045

Equation form expr-b645d837ca596c1a

$\aleph$

Read as: aleph

Means: aleph

8 occurrences in this chapter

• Occurrence projected-formula-0032206
• Occurrence projected-formula-0032218
• Occurrence projected-formula-0032231
• Occurrence projected-formula-0032255
• Occurrence projected-formula-0032262
• Occurrence projected-formula-0032265
• Occurrence projected-formula-0032269
• Occurrence projected-formula-0032285

Equation form expr-b9e5414b3f040e8c

$\kappa_0 &= 0\\ \kappa_{n+1} &= \aleph_{\kappa_n}\\ \kappa&= \bigcup_{n < \omega}\kappa_n$

Read as: Source-ordered display. kappa subscript zero equals zero. Then, kappa subscript n plus one equals aleph over kappa subscript n. Then, kappa equals the union over n is less than omega of kappa subscript n. End display

Means: Source-ordered display. kappa subscript zero equals zero. Then, kappa subscript n plus one equals aleph over kappa subscript n. Then, kappa equals the union over n is less than omega of kappa subscript n. End display

1 occurrence in this chapter

• Occurrence projected-formula-0032266

Equation form expr-ba475e3fea4db18b

$V_\kappa$

Read as: capital V subscript kappa

Means: capital V subscript kappa

3 occurrences in this chapter

• Occurrence projected-formula-0032302
• Occurrence projected-formula-0032303
• Occurrence projected-formula-0032304

Equation form expr-bb19a1dc5b65c869

$\alpha_1 = \beta_1$

Read as: alpha subscript one equals beta subscript one

Means: alpha subscript one equals beta subscript one

2 occurrences in this chapter

• Occurrence projected-formula-0032097
• Occurrence projected-formula-0032107

Equation form expr-bdf15923f1863bda

$\cardexpo{\cardfont{a}}{\cardfont{b}} = \cardexpo{2}{\cardfont{b}}$

Read as: cardinal exponentiation of the cardinal a to the power the cardinal b equals cardinal exponentiation of two to the power the cardinal b

Means: cardinal exponentiation of the cardinal a to the power the cardinal b equals cardinal exponentiation of two to the power the cardinal b

2 occurrences in this chapter

• Occurrence projected-formula-0032182
• Occurrence projected-formula-0032195

Equation form expr-c04455bd1e7d7d88

$\text{Seg}(\gamma_1, \gamma_2) &= \Setabs{\tuple{\delta_1, \delta_2} \in \alpha \times \alpha}{\tuple{\delta_1, \delta_2} \canonord \tuple{\gamma_1, \gamma_2}}$

Read as: Seg, open scope, gamma subscript one, then gamma subscript two, close scope equals the set of the ordered pair delta subscript one, then delta subscript two belongs to alpha times alpha such that the ordered pair delta subscript one, then delta subscript two precedes in the canonical order the ordered pair gamma subscript one, then gamma subscript two

Means: Seg, open scope, gamma subscript one, then gamma subscript two, close scope equals the set of the ordered pair delta subscript one, then delta subscript two belongs to alpha times alpha such that the ordered pair delta subscript one, then delta subscript two precedes in the canonical order the ordered pair gamma subscript one, then gamma subscript two

1 occurrence in this chapter

• Occurrence projected-formula-0032132

Equation form expr-c1f84ae5ed05d268

$\cardexpo{\cardfont{a}}{\cardfont{b}} = \cardsucc{\cardfont{a}}$

Read as: cardinal exponentiation of the cardinal a to the power the cardinal b equals the cardinal successor of the cardinal a

Means: cardinal exponentiation of the cardinal a to the power the cardinal b equals the cardinal successor of the cardinal a

1 occurrence in this chapter

• Occurrence projected-formula-0032239

Equation form expr-c2048e02de4999a2

$f \colon \alpha \to \beta$

Read as: f colon alpha maps to beta

Means: f colon alpha maps to beta

1 occurrence in this chapter

• Occurrence projected-formula-0032119

Equation form expr-c22a834c5dd99eca

$\cardfont{a} \cardplus \cardfont{b} &\defis \card{\cardfont{a} \disjointsum \cardfont{b}}\\ \cardfont{a} \cardtimes \cardfont{b} &\defis \card{\cardfont{a} \times \cardfont{b}}\\ \cardexpo{\cardfont{a}}{\cardfont{b}} &\defis \card{\funfromto{\cardfont{b}}{\cardfont{a}}}$

Read as: Source-ordered display. the cardinal a cardinal plus the cardinal b is defined as the cardinality of the cardinal a disjoint sum the cardinal b. Then, the cardinal a cardinal times the cardinal b is defined as the cardinality of the cardinal a times the cardinal b. Then, cardinal exponentiation of the cardinal a to the power the cardinal b is defined as the cardinality of functions from the cardinal b to the cardinal a. End display

Means: Source-ordered display. the cardinal a cardinal plus the cardinal b is defined as the cardinality of the cardinal a disjoint sum the cardinal b. Then, the cardinal a cardinal times the cardinal b is defined as the cardinality of the cardinal a times the cardinal b. Then, cardinal exponentiation of the cardinal a to the power the cardinal b is defined as the cardinality of functions from the cardinal b to the cardinal a. End display

1 occurrence in this chapter

• Occurrence projected-formula-0032032

Equation form expr-c552f81100ba4b60

$f \mapsto f^*$

Read as: f maps to f superscript star

Means: f maps to f superscript star

1 occurrence in this chapter

• Occurrence projected-formula-0032177

Equation form expr-c817daca09969708

$\cardfont{a} \cardtimes \cardfont{b} = \cardfont{a} \cardplus \cardfont{b} = \text{max}(\cardfont{a}, \cardfont{b}).$

Read as: the cardinal a cardinal times the cardinal b equals the cardinal a cardinal plus the cardinal b equals max, open scope, the cardinal a the empty expression, then the cardinal b, close scope

Means: the cardinal a cardinal times the cardinal b equals the cardinal a cardinal plus the cardinal b equals max, open scope, the cardinal a the empty expression, then the cardinal b, close scope

1 occurrence in this chapter

• Occurrence projected-formula-0032141

Equation form expr-c81a6a0df60c1020

$\Setabs{\tuple{\gamma_1,\gamma_2} \in X}{\max(\gamma_1, \gamma_2) = \delta}$

Read as: the set of the ordered pair gamma subscript one, then gamma subscript two belongs to capital X such that the maximum of open scope, gamma subscript one, then gamma subscript two, close scope equals delta

Means: the set of the ordered pair gamma subscript one, then gamma subscript two belongs to capital X such that the maximum of open scope, gamma subscript one, then gamma subscript two, close scope equals delta

1 occurrence in this chapter

• Occurrence projected-formula-0032114

Equation form expr-c8949eab46ebfc77

$\cardfont{a} = \cardexpo{\cardfont{a}}{\cardfont{b}}$

Read as: the cardinal a equals cardinal exponentiation of the cardinal a to the power the cardinal b

Means: the cardinal a equals cardinal exponentiation of the cardinal a to the power the cardinal b

1 occurrence in this chapter

• Occurrence projected-formula-0032238

Equation form expr-c9626e05648546b9

$\cardexpo{\cardfont{a}}{\cardfont{b} \cardplus \cardfont{c}} = \cardexpo{\cardfont{a}}{\cardfont{b}} \cardtimes \cardexpo{\cardfont{a}}{\cardfont{c}}$

Read as: cardinal exponentiation of the cardinal a to the power the cardinal b cardinal plus the cardinal c equals cardinal exponentiation of the cardinal a to the power the cardinal b cardinal times cardinal exponentiation of the cardinal a to the power the cardinal c

Means: cardinal exponentiation of the cardinal a to the power the cardinal b cardinal plus the cardinal c equals cardinal exponentiation of the cardinal a to the power the cardinal b cardinal times cardinal exponentiation of the cardinal a to the power the cardinal c

1 occurrence in this chapter

• Occurrence projected-formula-0032162

Equation form expr-c9895046d036ca65

$\tuple{\gamma_1, \gamma_2} \canonord \tuple{\gamma+1, \gamma + 1}$

Read as: the ordered pair gamma subscript one, then gamma subscript two precedes in the canonical order the ordered pair gamma plus one, then gamma plus one

Means: the ordered pair gamma subscript one, then gamma subscript two precedes in the canonical order the ordered pair gamma plus one, then gamma plus one

1 occurrence in this chapter

• Occurrence projected-formula-0032134

Equation form expr-caf5d72a278bcfcd

$2 = \{0, 1\}$

Read as: two equals the set containing zero, then one

Means: two equals the set containing zero, then one

1 occurrence in this chapter

• Occurrence projected-formula-0032060

Equation form expr-cafaf2312799238d

$\kappa= \beth_\kappa$

Read as: kappa equals beth subscript kappa

Means: kappa equals beth subscript kappa

1 occurrence in this chapter

• Occurrence projected-formula-0032283

Equation form expr-cb95c4cb42bc2c4e

$\omega \in \alpha$

Read as: omega belongs to alpha

Means: omega belongs to alpha

1 occurrence in this chapter

• Occurrence projected-formula-0032125

Equation form expr-ce411d89d0f53d5b

$\cardeq{(\cardfont{a} \times \cardfont{b})}{(\cardfont{b} \times \cardfont{a})}$

Read as: open scope, the cardinal a times the cardinal b, close scope is equinumerous with open scope, the cardinal b times the cardinal a, close scope

Means: open scope, the cardinal a times the cardinal b, close scope is equinumerous with open scope, the cardinal b times the cardinal a, close scope

1 occurrence in this chapter

• Occurrence projected-formula-0032055

Equation form expr-d203ba01eef4198c

$\delta$

Read as: delta

Means: delta

1 occurrence in this chapter

• Occurrence projected-formula-0032112

Equation form expr-d318976757631908

$\cardeq{\alpha}{\alpha\times\alpha}$

Read as: alpha is equinumerous with alpha times alpha

Means: alpha is equinumerous with alpha times alpha

1 occurrence in this chapter

• Occurrence projected-formula-0032130

Equation form expr-d3d79d86ea776a48

$f \colon \Pow{A} \to \funfromto{A}{2}$

Read as: f colon the power set of capital A maps to functions from capital A to two

Means: f colon the power set of capital A maps to functions from capital A to two

1 occurrence in this chapter

• Occurrence projected-formula-0032067

Equation form expr-d4735e3a265e16ee

$2$

Read as: two

Means: two

1 occurrence in this chapter

• Occurrence projected-formula-0032197

Equation form expr-d7b2a6054576e967

$\alpha_1 < \beta_1$

Read as: alpha subscript one is less than beta subscript one

Means: alpha subscript one is less than beta subscript one

1 occurrence in this chapter

• Occurrence projected-formula-0032095

Equation form expr-d87cc01ef52fd93f

$\cardplus$

Read as: cardinal plus

Means: cardinal plus

2 occurrences in this chapter

• Occurrence projected-formula-0032052
• Occurrence projected-formula-0032056

Equation form expr-d8824e9378aad1f3

$\kappa=\aleph_\kappa$

Read as: kappa equals aleph subscript kappa

Means: kappa equals aleph subscript kappa

1 occurrence in this chapter

• Occurrence projected-formula-0032264

Equation form expr-de25e05e7f7c0144

$\card{\Real} = \cardexpo{2}{\omega}$

Read as: the cardinality of the real numbers equals cardinal exponentiation of two to the power omega

Means: the cardinality of the real numbers equals cardinal exponentiation of two to the power omega

2 occurrences in this chapter

• Occurrence projected-formula-0032082
• Occurrence projected-formula-0032086

Equation form expr-df264a722a1ae280

$\funfromto{\cardfont{b} \disjointsum \cardfont{c}}{\cardfont{a}} \to (\funfromto{\cardfont{b}}{\cardfont{a}} \times \funfromto{\cardfont{c}}{\cardfont{a}})$

Read as: functions from the cardinal b disjoint sum the cardinal c to the cardinal a maps to open scope, functions from the cardinal b to the cardinal a times functions from the cardinal c to the cardinal a, close scope

Means: functions from the cardinal b disjoint sum the cardinal c to the cardinal a maps to open scope, functions from the cardinal b to the cardinal a times functions from the cardinal c to the cardinal a, close scope

1 occurrence in this chapter

• Occurrence projected-formula-0032171

Equation form expr-df7e70e5021544f4

$B$

Read as: capital B

Means: capital B

1 occurrence in this chapter

• Occurrence projected-formula-0032048

Equation form expr-e349851823caf578

$\ZFC$

Read as: set theory Z F C

Means: set theory Z F C

9 occurrences in this chapter

• Occurrence projected-formula-0032240
• Occurrence projected-formula-0032243
• Occurrence projected-formula-0032245
• Occurrence projected-formula-0032247
• Occurrence projected-formula-0032252
• Occurrence projected-formula-0032253
• Occurrence projected-formula-0032254
• Occurrence projected-formula-0032261
• Occurrence projected-formula-0032280

Equation form expr-e3ebd12aa88a9541

$\cardfont{a}, \cardfont{b}$

Read as: the cardinal a the empty expression, then the cardinal b

Means: the cardinal a the empty expression, then the cardinal b

1 occurrence in this chapter

• Occurrence projected-formula-0032140

Equation form expr-e459021f2080582e

$\card{\Real} = \beth_1$

Read as: the cardinality of the real numbers equals beth subscript one

Means: the cardinality of the real numbers equals beth subscript one

1 occurrence in this chapter

• Occurrence projected-formula-0032249

Equation form expr-e4d28c0b3e89a680

$f_\cardfont{b}(\beta) = f(\beta, 0)$

Read as: f subscript the cardinal b of beta equals f of beta and zero

Means: f subscript the cardinal b of beta equals f of beta and zero

1 occurrence in this chapter

• Occurrence projected-formula-0032166

Equation form expr-e5a69bbb6bda64f3

$X_\beta$

Read as: capital X subscript beta

Means: capital X subscript beta

1 occurrence in this chapter

• Occurrence projected-formula-0032150

Equation form expr-e699a88f5fe9eec8

$\Pow{\omega}$

Read as: the power set of omega

Means: the power set of omega

1 occurrence in this chapter

• Occurrence projected-formula-0032071

Equation form expr-e722e620f9c5a72b

$\canonord$

Read as: the canonical order relation

Means: the canonical order relation

5 occurrences in this chapter

• Occurrence projected-formula-0032091
• Occurrence projected-formula-0032101
• Occurrence projected-formula-0032105
• Occurrence projected-formula-0032115
• Occurrence projected-formula-0032161

Equation form expr-e8e03daa6ead8005

$f^*(\beta, \gamma) = (f(\gamma))(\beta)$

Read as: f superscript star of beta and gamma equals f of gamma applied to beta

Means: f superscript star of beta and gamma equals f of gamma applied to beta

1 occurrence in this chapter

• Occurrence projected-formula-0032175

Equation form expr-e9ed9c39c76e6441

$f(B) = \chi_B$

Read as: f of capital B equals chi subscript capital B

Means: f of capital B equals chi subscript capital B

1 occurrence in this chapter

• Occurrence projected-formula-0032066

Equation form expr-ea0ba03ad68b6f72

$\cardeq{\Pow{A}}{\funfromto{A}{2}}$

Read as: the power set of capital A is equinumerous with functions from capital A to two

Means: the power set of capital A is equinumerous with functions from capital A to two

1 occurrence in this chapter

• Occurrence projected-formula-0032068

Equation form expr-ea13392c3c89c82e

$2 < \aleph_2$

Read as: two is less than aleph subscript two

Means: two is less than aleph subscript two

1 occurrence in this chapter

• Occurrence projected-formula-0032258

Equation form expr-ebc33acdb2456947

$\tuple{\alpha_1, \alpha_2} \canonord \tuple{\beta_1, \beta_2}$

Read as: the ordered pair alpha subscript one, then alpha subscript two precedes in the canonical order the ordered pair beta subscript one, then beta subscript two

Means: the ordered pair alpha subscript one, then alpha subscript two precedes in the canonical order the ordered pair beta subscript one, then beta subscript two

1 occurrence in this chapter

• Occurrence projected-formula-0032092

Equation form expr-f0bb9b0835d4f4f6

$\chi_B \in \funfromto{A}{2}$

Read as: chi subscript capital B belongs to functions from capital A to two

Means: chi subscript capital B belongs to functions from capital A to two

1 occurrence in this chapter

• Occurrence projected-formula-0032064

Equation form expr-f1672d6cf895c128

$\max(\alpha_1, \alpha_2) = \max(\beta_1, \beta_2)$

Read as: the maximum of open scope, alpha subscript one, then alpha subscript two, close scope equals the maximum of beta subscript one, then beta subscript two

Means: the maximum of open scope, alpha subscript one, then alpha subscript two, close scope equals the maximum of beta subscript one, then beta subscript two

3 occurrences in this chapter

• Occurrence projected-formula-0032094
• Occurrence projected-formula-0032096
• Occurrence projected-formula-0032106

Equation form expr-f3f3804480e8551a

$\beta$

Read as: beta

Means: beta

1 occurrence in this chapter

• Occurrence projected-formula-0032294

Equation form expr-f55d458785a64e48

$g(v) = \tuple{\beta, f_\beta(v)}$

Read as: g of v equals the ordered pair beta, then f subscript beta of v

Means: g of v equals the ordered pair beta, then f subscript beta of v

1 occurrence in this chapter

• Occurrence projected-formula-0032156

Equation form expr-f6718113dab3a636

$\omega \ordtimes \omega \leq \alpha$

Read as: omega ordinal times omega is less than or equal to alpha

Means: omega ordinal times omega is less than or equal to alpha

3 occurrences in this chapter

• Occurrence projected-formula-0032287
• Occurrence projected-formula-0032289
• Occurrence projected-formula-0032292

Equation form expr-f72ebed11a3595ce

$\bigcup_{\beta \in \cardfont{a}} X_\beta \preceq \cardfont{a} \times \cardfont{a} \approx \cardfont{a}$

Read as: the union over beta belongs to the cardinal a of capital X subscript beta is less than or equivalent to the cardinal a times the cardinal a is approximately equal to the cardinal a

Means: the union over beta belongs to the cardinal a of capital X subscript beta is less than or equivalent to the cardinal a times the cardinal a is approximately equal to the cardinal a

1 occurrence in this chapter

• Occurrence projected-formula-0032160

Equation form expr-f7acd2561ab2bd63

$f \mapsto (f_{\cardfont{b}} \times f_\cardfont{c})$

Read as: f maps to open scope, f subscript the cardinal b times f subscript the cardinal c, close scope

Means: f maps to open scope, f subscript the cardinal b times f subscript the cardinal c, close scope

1 occurrence in this chapter

• Occurrence projected-formula-0032170

Equation form expr-f85fa8064284ca5f

$\funfromto{\cardfont{c}}{(\funfromto{\cardfont{b}}{\cardfont{a}})} \to \funfromto{\cardfont{b} \cardtimes \cardfont{c}}{\cardfont{a}}$

Read as: functions from the cardinal c to open scope, functions from the cardinal b to the cardinal a, close scope maps to functions from the cardinal b cardinal times the cardinal c to the cardinal a

Means: functions from the cardinal c to open scope, functions from the cardinal b to the cardinal a, close scope maps to functions from the cardinal b cardinal times the cardinal c to the cardinal a

1 occurrence in this chapter

• Occurrence projected-formula-0032178

Equation form expr-f926f4e136aa55c9

$f \colon \Pow{\omega} \to \Real$

Read as: f colon the power set of omega maps to the real numbers

Means: f colon the power set of omega maps to the real numbers

1 occurrence in this chapter

• Occurrence projected-formula-0032087

Equation form expr-f972265d166ed1f4

$\cardeq{\card{\alpha}}{\card{\alpha} \times \card{\alpha}}$

Read as: the cardinality of alpha is equinumerous with the cardinality of alpha times the cardinality of alpha

Means: the cardinality of alpha is equinumerous with the cardinality of alpha times the cardinality of alpha

1 occurrence in this chapter

• Occurrence projected-formula-0032128

Equation form expr-fa764ecb38aaa05b

$\cardexpo{2}{\cardfont{b}}\leq \cardexpo{\cardfont{a}}{\cardfont{b}} \leq \cardexpo{(\cardexpo{2}{\cardfont{b}})}{\cardfont{b}} = \cardexpo{2}{\cardfont{b}\cardtimes\cardfont{b}} = \cardexpo{2}{\cardfont{b}}$

Read as: cardinal exponentiation of two to the power the cardinal b is less than or equal to cardinal exponentiation of the cardinal a to the power the cardinal b

Means: cardinal exponentiation of two to the power the cardinal b is less than or equal to cardinal exponentiation of the cardinal a to the power the cardinal b

1 occurrence in this chapter

• Occurrence projected-formula-0032196

Equation form expr-fcb5f40df9be6bae

$W$

Read as: capital W

Means: capital W

1 occurrence in this chapter

• Occurrence projected-formula-0032309

Equation form expr-fe3756fb5599cf1a

$n \cdot m$

Read as: n times m

Means: n times m

1 occurrence in this chapter

• Occurrence projected-formula-0032051

Equation form expr-ff4be5f76635a0aa

$\beth_1$

Read as: beth subscript one

Means: beth subscript one

1 occurrence in this chapter

• Occurrence projected-formula-0032251

Definition one in this chapter

This source definition contains, in source order: the cardinal a; then the cardinal b; then Source-ordered display. the cardinal a cardinal plus the cardinal b is defined as the cardinality of the cardinal a disjoint sum the cardinal b. Then, the cardinal a cardinal times the cardinal b is defined as the cardinality of the cardinal a times the cardinal b. Then, cardinal exponentiation of the cardinal a to the power the cardinal b is defined as the cardinality of functions from the cardinal b to the cardinal a. End display; then functions from capital X to capital Y equals the set of f such that f, is a function, capital X maps to capital Y; then functions from capital X to capital Y; then capital X; then capital Y. The complete surrounding source prose remains in the continuous listener stream.

Source

Display math one in this chapter

This source display math contains, in source order: Source-ordered display. the cardinal a cardinal plus the cardinal b is defined as the cardinality of the cardinal a disjoint sum the cardinal b. Then, the cardinal a cardinal times the cardinal b is defined as the cardinality of the cardinal a times the cardinal b. Then, cardinal exponentiation of the cardinal a to the power the cardinal b is defined as the cardinality of functions from the cardinal b to the cardinal a. End display. The complete surrounding source prose remains in the continuous listener stream.

Source

Exercise one in this chapter

This source exercise contains, in source order: set theory Z minus; then functions from capital X to capital Y; then capital X; then capital Y; then set theory Z F; then the rank of functions from capital X to capital Y; then the rank of capital X; then the rank of capital Y. The complete surrounding source prose remains in the continuous listener stream. The exercise is preserved as stated and no solution is supplied.

Source

Proposition one in this chapter

This source proposition contains, in source order: cardinal plus; then cardinal times. The complete surrounding source prose remains in the continuous listener stream.

Source

Exercise two in this chapter

This source exercise contains, in source order: cardinal plus; then cardinal times. The complete surrounding source prose remains in the continuous listener stream. The exercise is preserved as stated and no solution is supplied.

Source

Proposition two in this chapter

This source proposition contains, in source order: capital A; then the cardinality of capital A cardinal plus one equals one cardinal plus the cardinality of capital A equals the cardinality of capital A. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma one in this chapter

This source lemma contains, in source order: the cardinality of the power set of capital A equals cardinal exponentiation of two to the power the cardinality of capital A; then capital A. The complete surrounding source prose remains in the continuous listener stream.

Source

Display math two in this chapter

This source display math contains, in source order: chi subscript capital B of x is defined by cases. case one, one; if, x belongs to capital B. case two, zero; otherwise.. End cases. The complete surrounding source prose remains in the continuous listener stream.

Source

Corollary one in this chapter

This source corollary contains, in source order: the cardinal a is less than cardinal exponentiation of two to the power the cardinal a; then the cardinal a. The complete surrounding source prose remains in the continuous listener stream.

Source

Theorem one in this chapter

This source theorem contains, in source order: the cardinality of the real numbers equals cardinal exponentiation of two to the power omega. The complete surrounding source prose remains in the continuous listener stream.

Source

Exercise three in this chapter

This source exercise contains, in source order: the power set of omega has cardinality at most that of the real numbers; then the real numbers has cardinality at most that of the power set of omega. The complete surrounding source prose remains in the continuous listener stream. The exercise is preserved as stated and no solution is supplied.

Source

Definition two in this chapter

This source definition contains, in source order: the canonical order relation; then the ordered pair alpha subscript one, then alpha subscript two precedes in the canonical order the ordered pair beta subscript one, then beta subscript two; then the maximum of open scope, alpha subscript one, then alpha subscript two, close scope is less than the maximum of beta subscript one, then beta subscript two; then the maximum of open scope, alpha subscript one, then alpha subscript two, close scope equals the maximum of beta subscript one, then beta subscript two; then alpha subscript one is less than beta subscript one; then the maximum of open scope, alpha subscript one, then alpha subscript two, close scope equals the maximum of beta subscript one, then beta subscript two; then alpha subscript one equals beta subscript one; then alpha subscript two is less than beta subscript two. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma two in this chapter

This source lemma contains, in source order: the ordered pair alpha times alpha, then the canonical order relation; then alpha. The complete surrounding source prose remains in the continuous listener stream.

Source

Proposition three in this chapter

This source proposition contains, in source order: alpha is equinumerous with beta; then alpha times alpha is equinumerous with beta times beta. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma three in this chapter

This source lemma contains, in source order: alpha is equinumerous with alpha times alpha; then alpha. The complete surrounding source prose remains in the continuous listener stream.

Source

Display math three in this chapter

This source display math contains, in source order: Seg, open scope, gamma subscript one, then gamma subscript two, close scope equals the set of the ordered pair delta subscript one, then delta subscript two belongs to alpha times alpha such that the ordered pair delta subscript one, then delta subscript two precedes in the canonical order the ordered pair gamma subscript one, then gamma subscript two. The complete surrounding source prose remains in the continuous listener stream.

Source

Display math four in this chapter

This source display math contains, in source order: Source-ordered display. Seg, open scope, gamma subscript one, then gamma subscript two, close scope is dominated by open scope, open scope, gamma ordinal plus one, close scope ordinal times open scope, gamma ordinal plus one, close scope, close scope. Then, is approximately equal to open scope, gamma ordinal times gamma, close scope, , by the referenced source passage and the referenced source passage. Then, is approximately equal to gamma, , by the induction hypothesis. Then, is strictly less than alpha, , since alpha is a cardinal. End display. The complete surrounding source prose remains in the continuous listener stream.

Source

Theorem two in this chapter

This source theorem contains, in source order: the cardinal a the empty expression, then the cardinal b; then the cardinal a cardinal times the cardinal b equals the cardinal a cardinal plus the cardinal b equals max, open scope, the cardinal a the empty expression, then the cardinal b, close scope. The complete surrounding source prose remains in the continuous listener stream.

Source

Proposition four in this chapter

This source proposition contains, in source order: the cardinal a; then beta belongs to the cardinal a; then capital X subscript beta; then the cardinality of capital X subscript beta is less than or equal to the cardinal a; then the cardinality of the union over beta belongs to the cardinal a of capital X subscript beta is less than or equal to the cardinal a. The complete surrounding source prose remains in the continuous listener stream.

Source

Proposition five in this chapter

This source proposition contains, in source order: cardinal exponentiation of the cardinal a to the power the cardinal b cardinal plus the cardinal c equals cardinal exponentiation of the cardinal a to the power the cardinal b cardinal times cardinal exponentiation of the cardinal a to the power the cardinal c; then cardinal exponentiation of open scope, cardinal exponentiation of the cardinal a to the power the cardinal b, close scope to the power the cardinal c equals cardinal exponentiation of the cardinal a to the power the cardinal b cardinal times the cardinal c; then the cardinal a the empty expression, then the cardinal b, then the cardinal c. The complete surrounding source prose remains in the continuous listener stream.

Source

Proposition six in this chapter

This source proposition contains, in source order: two is less than or equal to the cardinal a is less than or equal to the cardinal b; then the cardinal b; then cardinal exponentiation of the cardinal a to the power the cardinal b equals cardinal exponentiation of two to the power the cardinal b. The complete surrounding source prose remains in the continuous listener stream.

Source

Display math five in this chapter

This source display math contains, in source order: Source-ordered display. cardinal exponentiation of two to the power the cardinal b is less than or equal to cardinal exponentiation of the cardinal a to the power the cardinal b , as two is less than or equal to the cardinal a. Then, is less than or equal to cardinal exponentiation of open scope, two superscript the cardinal a, close scope to the power the cardinal b , by the referenced source passage. Then, equals cardinal exponentiation of two to the power the cardinal a cardinal times the cardinal b , by the referenced source passage. Then, equals cardinal exponentiation of two to the power the cardinal b , by the referenced source passage. End display. The complete surrounding source prose remains in the continuous listener stream.

Source

Proposition seven in this chapter

This source proposition contains, in source order: the cardinal a; then n belongs to omega; then cardinal exponentiation of the cardinal a to the power n equals the cardinal a. The complete surrounding source prose remains in the continuous listener stream.

Source

Proposition eight in this chapter

This source proposition contains, in source order: two is less than or equal to the cardinal b is less than the cardinal a is less than or equal to cardinal exponentiation of two to the power the cardinal b; then the cardinal b; then cardinal exponentiation of the cardinal a to the power the cardinal b equals cardinal exponentiation of two to the power the cardinal b. The complete surrounding source prose remains in the continuous listener stream.

Source

Definition three in this chapter

This source definition contains, in source order: the cardinal successor of the cardinal a; then the cardinal a; then Source-ordered display. aleph over zero of is defined as omega applied to beth over zero. Then, aleph over alpha ordinal plus one of is defined as the cardinal successor of open scope, aleph over alpha, close scope beth over alpha plus one. Then, aleph over alpha of is defined as the union over beta is less than alpha of aleph over beta of beth over alpha. End display. The complete surrounding source prose remains in the continuous listener stream.

Source

Display math six in this chapter

This source display math contains, in source order: Source-ordered display. aleph over zero of is defined as omega applied to beth over zero. Then, aleph over alpha ordinal plus one of is defined as the cardinal successor of open scope, aleph over alpha, close scope beth over alpha plus one. Then, aleph over alpha of is defined as the union over beta is less than alpha of aleph over beta of beth over alpha. End display. The complete surrounding source prose remains in the continuous listener stream.

Source

Proposition nine in this chapter

This source proposition contains, in source order: aleph subscript alpha; then beth subscript alpha; then alpha. The complete surrounding source prose remains in the continuous listener stream.

Source

Proposition ten in this chapter

This source proposition contains, in source order: the cardinal a; then the cardinal a equals aleph subscript gamma; then gamma. The complete surrounding source prose remains in the continuous listener stream.

Source

Definition four in this chapter

This source definition contains, in source order: aleph subscript alpha of equals beth subscript alpha; then alpha. The complete surrounding source prose remains in the continuous listener stream.

Source

Definition five in this chapter

This source definition contains, in source order: aleph subscript one of equals beth subscript one. The complete surrounding source prose remains in the continuous listener stream.

Source

Proposition eleven in this chapter

This source proposition contains, in source order: aleph. The complete surrounding source prose remains in the continuous listener stream.

Source

Display math seven in this chapter

This source display math contains, in source order: Source-ordered display. kappa subscript zero equals zero. Then, kappa subscript n plus one equals aleph over kappa subscript n. Then, kappa equals the union over n is less than omega of kappa subscript n. End display. The complete surrounding source prose remains in the continuous listener stream.

Source

Proposition twelve in this chapter

This source proposition contains, in source order: beth; then kappa; then kappa equals beth subscript kappa. The complete surrounding source prose remains in the continuous listener stream.

Source

Proposition thirteen in this chapter

This source proposition contains, in source order: the cardinality of capital V subscript omega plus alpha equals beth over alpha; then omega ordinal times omega is less than or equal to alpha; then the cardinality of capital V subscript alpha equals beth subscript alpha. The complete surrounding source prose remains in the continuous listener stream.

Source

Corollary two in this chapter

This source corollary contains, in source order: kappa; then the cardinality of capital V subscript kappa equals kappa. The complete surrounding source prose remains in the continuous listener stream.

Source

Display math eight in this chapter

This source display math contains, in source order: Source-ordered display. tau subscript zero of capital A is defined as the cardinality of capital A. Then, tau subscript n plus one of capital A is defined as beth over tau subscript n of capital A. Then, tau of capital A is defined as the union over n is less than omega of tau subscript n of capital A. Then, As in the referenced source passage, tau of capital A is a beth-fixed point for any capital A, and trivially the cardinality of capital A is less than tau of capital A. So now consider this recursive definition:. Then, capital W subscript zero is defined as zero. Then, capital W subscript alpha plus one is defined as tau of capital W subscript alpha. Then, capital W subscript alpha is defined as the union over beta is less than alpha of capital W subscript beta, , when alpha is a limit. End display. The complete surrounding source prose remains in the continuous listener stream.

Source

Cross-reference reference-001513

lemma four in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001514

definition one in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001515

proposition in chapter “Sets”

Source occurrence

Cross-reference reference-001516

the lemma on Cardinals Behave Right

Source occurrence

Cross-reference reference-001517

theorem one in chapter “Cardinals”

Source occurrence

Cross-reference reference-001518

lemma five in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001519

the lemma on Cardinals Behave Right

Source occurrence

Cross-reference reference-001520

definition of the natural numbers and omega in chapter “Steps towards Z”

Source occurrence

Cross-reference reference-001521

section “Reduction” in chapter “The Size of Sets”

Source occurrence

Cross-reference reference-001522

section “Reduction” in chapter “The Size of Sets”

Source occurrence

Cross-reference reference-001523

the theorem on cantor

Source occurrence

Cross-reference reference-001524

the lemma on Size Powersettwo Exp

Source occurrence

Cross-reference reference-001525

section “Ordinal Exponentiation” in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001526

the lemma on Size Powersettwo Exp

Source occurrence

Cross-reference reference-001527

theorem one in chapter “Cardinal Arithmetic”

Source occurrence

Cross-reference reference-001528

proposition “Enumerability of pairs of natural numbers” in chapter “The Size of Sets”

Source occurrence

Cross-reference reference-001529

proposition three in chapter “Cardinal Arithmetic”

Source occurrence

Cross-reference reference-001530

lemma five in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001531

proposition three in chapter “Cardinal Arithmetic”

Source occurrence

Cross-reference reference-001532

lemma three in chapter “Cardinal Arithmetic”

Source occurrence

Cross-reference reference-001533

section “Countable Choice” in chapter “Choice”

Source occurrence

Cross-reference reference-001534

theorem two in chapter “Cardinal Arithmetic”

Source occurrence

Cross-reference reference-001535

theorem two in chapter “Cardinal Arithmetic”

Source occurrence

Cross-reference reference-001536

the lemma on Size Powersettwo Exp

Source occurrence

Cross-reference reference-001537

proposition five in chapter “Cardinal Arithmetic”

Source occurrence

Cross-reference reference-001538

theorem two in chapter “Cardinal Arithmetic”

Source occurrence

Cross-reference reference-001539

the lemma on Size Powersettwo Exp

Source occurrence

Cross-reference reference-001540

theorem two in chapter “Cardinal Arithmetic”

Source occurrence

Cross-reference reference-001541

proposition six in chapter “Cardinal Arithmetic”

Source occurrence

Cross-reference reference-001542

the lemma on Size Powersettwo Exp

Source occurrence

Cross-reference reference-001543

the lemma on No Largest Cardinal

Source occurrence

Cross-reference reference-001544

corollary two in chapter “Cardinals”

Source occurrence

Cross-reference reference-001545

proposition three in chapter “Cardinals”

Source occurrence

Cross-reference reference-001546

Kurt Gödel (1938)

Source occurrence

Cross-reference reference-001547

Paul J. Cohen (1963)

Source occurrence

Cross-reference reference-001548

theorem one in chapter “Cardinal Arithmetic”

Source occurrence

Cross-reference reference-001549

Michael Potter (2004), §15.6

Source occurrence

Cross-reference reference-001550

chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001551

proposition three in chapter “Cardinals”

Source occurrence

Cross-reference reference-001552

(George Boolos, 2000, p. 257)

Source occurrence

Cross-reference reference-001553

(George Boolos, 2000, p. 268)

Source occurrence

Cross-reference reference-001554

section “Extrinsic Considerations about Replacement” in chapter “Replacement”

Source occurrence

Cross-reference reference-001555

proposition eleven in chapter “Cardinal Arithmetic”

Source occurrence

Cross-reference reference-001556

chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001557

proposition twelve in chapter “Cardinal Arithmetic”

Source occurrence

Cross-reference reference-001558

proposition thirteen in chapter “Cardinal Arithmetic”

Source occurrence

Cross-reference reference-001559

proposition twelve in chapter “Cardinal Arithmetic”

Source occurrence