Equation form expr-017759a03aa49f8c

$\emph{Regularity} \Rightarrow \emph{Foundation}$

Read as: Regularity implies Foundation

Means: Regularity implies Foundation

1 occurrence in this chapter

• Occurrence projected-formula-0031330

Equation form expr-028be7e8c27d107f

$A \subseteq \trcl{A} \subseteq V_\alpha$

Read as: capital A is a subset of the transitive closure of capital A is a subset of capital V subscript alpha

Means: capital A is a subset of the transitive closure of capital A is a subset of capital V subscript alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031249

Equation form expr-02fc33e5cd03ef78

$V_\beta \in V_\beta$

Read as: capital V subscript beta belongs to capital V subscript beta

Means: capital V subscript beta belongs to capital V subscript beta

2 occurrences in this chapter

• Occurrence projected-formula-0031183
• Occurrence projected-formula-0031194

Equation form expr-044d7a33e4e66515

$(\forall A \neq \emptyset)(\exists B \in A)A \cap B = \emptyset$

Read as: for every capital A not equal to the empty set, there exists capital B in capital A, capital A intersected with capital B equals the empty set

Means: for every capital A not equal to the empty set, there exists capital B in capital A, capital A intersected with capital B equals the empty set

1 occurrence in this chapter

• Occurrence projected-formula-0031209

Equation form expr-056b9ea6bbc0e648

$x \subseteq V_\delta$

Read as: x is a subset of capital V subscript delta

Means: x is a subset of capital V subscript delta

1 occurrence in this chapter

• Occurrence projected-formula-0031230

Equation form expr-05fe0e5b38f9e117

$c \in B$

Read as: c belongs to capital B

Means: c belongs to capital B

1 occurrence in this chapter

• Occurrence projected-formula-0031335

Equation form expr-06c9591f8f2986c1

$\delta \in \alpha$

Read as: delta belongs to alpha

Means: delta belongs to alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031232

Equation form expr-08ae3e69eba3f924

$V_\emptyset \notin V_\emptyset$

Read as: capital V subscript the empty set does not belong to capital V subscript the empty set

Means: capital V subscript the empty set does not belong to capital V subscript the empty set

1 occurrence in this chapter

• Occurrence projected-formula-0031171

Equation form expr-091eff496ce82fef

$B \in D$

Read as: capital B belongs to capital D

Means: capital B belongs to capital D

1 occurrence in this chapter

• Occurrence projected-formula-0031245

Equation form expr-097ececa9c41a75b

$\dom{f} = \alpha$

Read as: the domain of f equals alpha

Means: the domain of f equals alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031071

Equation form expr-0ae2a67c34d5e296

$x \in A \in V_\alpha$

Read as: x belongs to capital A belongs to capital V subscript alpha

Means: x belongs to capital A belongs to capital V subscript alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031158

Equation form expr-0bf9bc170ad957d9

$V_\beta$

Read as: capital V subscript beta

Means: capital V subscript beta

1 occurrence in this chapter

• Occurrence projected-formula-0031162

Equation form expr-0c355c9d7a4bd9c5

$V_\alpha \in V_\beta$

Read as: capital V subscript alpha belongs to capital V subscript beta

Means: capital V subscript alpha belongs to capital V subscript beta

3 occurrences in this chapter

• Occurrence projected-formula-0031180
• Occurrence projected-formula-0031189
• Occurrence projected-formula-0031190

Equation form expr-0c850994b22dd02d

$f_{\ordsucc{\alpha}}(\alpha)$

Read as: f subscript the ordinal successor of alpha of alpha

Means: f subscript the ordinal successor of alpha of alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031111

Equation form expr-0cac8b5c1fcece34

$\alpha \neq \beta$

Read as: alpha is not equal to beta

Means: alpha is not equal to beta

1 occurrence in this chapter

• Occurrence projected-formula-0031191

Equation form expr-0f0e1602c643459b

$\delta \in \gamma$

Read as: delta belongs to gamma

Means: delta belongs to gamma

1 occurrence in this chapter

• Occurrence projected-formula-0031097

Equation form expr-10ce0200b450ea95

$\phi$

Read as: phi

Means: phi

7 occurrences in this chapter

• Occurrence projected-formula-0031298
• Occurrence projected-formula-0031301
• Occurrence projected-formula-0031309
• Occurrence projected-formula-0031310
• Occurrence projected-formula-0031311
• Occurrence projected-formula-0031313
• Occurrence projected-formula-0031314

Equation form expr-1530582d946d5f25

$\forall A \exists \alpha\, A \subseteq V_\alpha$

Read as: for every capital A, there exists alpha, capital A is a subset of capital V subscript alpha

Means: for every capital A, there exists alpha, capital A is a subset of capital V subscript alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031208

Equation form expr-1539ef96e7fdc1b8

$B \in D \subseteq A$

Read as: capital B belongs to capital D is a subset of capital A

Means: capital B belongs to capital D is a subset of capital A

1 occurrence in this chapter

• Occurrence projected-formula-0031236

Equation form expr-181dc47425f7c51c

$A = \emptyset$

Read as: capital A equals the empty set

Means: capital A equals the empty set

1 occurrence in this chapter

• Occurrence projected-formula-0031135

Equation form expr-18ae661dbfa4d037

$\supstrict(X)$

Read as: the strict supremum of capital X

Means: the strict supremum of capital X

1 occurrence in this chapter

• Occurrence projected-formula-0031225

Equation form expr-19626854d007e4e5

$\lnot \forall A \phi(A)$

Read as: not for every capital A, phi of capital A

Means: not for every capital A, phi of capital A

1 occurrence in this chapter

• Occurrence projected-formula-0031300

Equation form expr-1afd1ce1b18f3739

$\setrank{\alpha} = \alpha$

Read as: the rank of alpha equals alpha

Means: the rank of alpha equals alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031325

Equation form expr-1b16b1df538ba12d

$n$

Read as: n

Means: n

1 occurrence in this chapter

• Occurrence projected-formula-0031220

Equation form expr-2225b5a8bdecda32

$x \in A$

Read as: x belongs to capital A

Means: x belongs to capital A

4 occurrences in this chapter

• Occurrence projected-formula-0031228
• Occurrence projected-formula-0031239
• Occurrence projected-formula-0031305
• Occurrence projected-formula-0031318

Equation form expr-238879a3746d5874

$\setrank{x} \in \alpha$

Read as: the rank of x belongs to alpha

Means: the rank of x belongs to alpha

2 occurrences in this chapter

• Occurrence projected-formula-0031286
• Occurrence projected-formula-0031319

Equation form expr-252f10c83610ebca

$f$

Read as: f

Means: f

2 occurrences in this chapter

• Occurrence projected-formula-0031066
• Occurrence projected-formula-0031068

Equation form expr-2782a1afcd1bccf4

$\exists \delta\ x \subseteq V_\delta$

Read as: there exists delta, x is a subset of capital V subscript delta

Means: there exists delta, x is a subset of capital V subscript delta

1 occurrence in this chapter

• Occurrence projected-formula-0031242

Equation form expr-285d2c9fb675ff29

$V_\gamma \in \bigcup_{\beta \in \alpha} V_\beta = V_\alpha$

Read as: capital V subscript gamma belongs to the union over beta belongs to alpha of capital V subscript beta equals capital V subscript alpha

Means: capital V subscript gamma belongs to the union over beta belongs to alpha of capital V subscript beta equals capital V subscript alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031168

Equation form expr-2bca212e11266c55

$\text{trcl}(A)$

Read as: transitive closure of capital A

Means: transitive closure of capital A

1 occurrence in this chapter

• Occurrence projected-formula-0031213

Equation form expr-2c25be572859426d

$\setrank{\beta} = \beta$

Read as: the rank of beta equals beta

Means: the rank of beta equals beta

1 occurrence in this chapter

• Occurrence projected-formula-0031326

Equation form expr-2e90174d015f7610

$A \subseteq V_{\alpha}$

Read as: capital A is a subset of capital V subscript alpha

Means: capital A is a subset of capital V subscript alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031234

Equation form expr-32113fc77ac33f34

$x \subseteq V_\alpha$

Read as: x is a subset of capital V subscript alpha

Means: x is a subset of capital V subscript alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031291

Equation form expr-339701fcc026a664

$V_\beta \notin V_\beta$

Read as: capital V subscript beta does not belong to capital V subscript beta

Means: capital V subscript beta does not belong to capital V subscript beta

1 occurrence in this chapter

• Occurrence projected-formula-0031184

Equation form expr-382afaf44f72f6f5

$V_{\ordsucc{\alpha}}$

Read as: capital V subscript the ordinal successor of alpha

Means: capital V subscript the ordinal successor of alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031202

Equation form expr-3950069bf33f5337

$\sigma(\alpha)$

Read as: sigma of alpha

Means: sigma of alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031110

Equation form expr-39f098dc6b231850

$\setrank{B} \in \setrank{A}$

Read as: the rank of capital B belongs to the rank of capital A

Means: the rank of capital B belongs to the rank of capital A

1 occurrence in this chapter

• Occurrence projected-formula-0031295

Equation form expr-3b2591dd01df745a

$x \in B$

Read as: x belongs to capital B

Means: x belongs to capital B

1 occurrence in this chapter

• Occurrence projected-formula-0031238

Equation form expr-3c9a98971489e386

$c \notin A$

Read as: c does not belong to capital A

Means: c does not belong to capital A

1 occurrence in this chapter

• Occurrence projected-formula-0031337

Equation form expr-3cec2c066344d465

$\sigma(x)$

Read as: sigma of x

Means: sigma of x

3 occurrences in this chapter

• Occurrence projected-formula-0031103
• Occurrence projected-formula-0031121
• Occurrence projected-formula-0031125

Equation form expr-3d676fd43723ee8f

$g \cup \{\tuple{\gamma, \tau(g)}\}$

Read as: g union the set containing the ordered pair gamma, then tau of g

Means: g union the set containing the ordered pair gamma, then tau of g

1 occurrence in this chapter

• Occurrence projected-formula-0031089

Equation form expr-3df09d9845fb9be2

$x \subseteq V_{\setrank{x}} \in V_\alpha$

Read as: x is a subset of capital V subscript the rank of x belongs to capital V subscript alpha

Means: x is a subset of capital V subscript the rank of x belongs to capital V subscript alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031287

Equation form expr-3e246acc80933105

$\funrestrictionto{\sigma}{\alpha} = \Setabs{\tuple{\beta, \sigma(\beta)}}{\beta \in \alpha}$

Read as: the restriction of sigma to alpha equals the set of the ordered pair beta, then sigma of beta such that beta belongs to alpha

Means: the restriction of sigma to alpha equals the set of the ordered pair beta, then sigma of beta such that beta belongs to alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031101

Equation form expr-3e4674b20e7e0af7

$A \subseteq V_{\setrank{A}} = \Setabs{x}{\setrank{x} \in \setrank{A}}$

Read as: capital A is a subset of capital V subscript the rank of capital A equals the set of x such that the rank of x belongs to the rank of capital A

Means: capital A is a subset of capital V subscript the rank of capital A equals the set of x such that the rank of x belongs to the rank of capital A

1 occurrence in this chapter

• Occurrence projected-formula-0031296

Equation form expr-41ad13a713bf1669

$\setrank{A}$

Read as: the rank of capital A

Means: the rank of capital A

2 occurrences in this chapter

• Occurrence projected-formula-0031279
• Occurrence projected-formula-0031282

Equation form expr-43fb79f4401dac77

$\alpha = \supstrict_{x \in A}\setrank{x}$

Read as: alpha equals the strict supremum over x belongs to capital A of the rank of x

Means: alpha equals the strict supremum over x belongs to capital A of the rank of x

1 occurrence in this chapter

• Occurrence projected-formula-0031316

Equation form expr-440c9fc4b6198138

$\delta \leq \alpha$

Read as: delta is less than or equal to alpha

Means: delta is less than or equal to alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031085

Equation form expr-463e2c0941f14a02

$\ordsucc{\gamma}$

Read as: the ordinal successor of gamma

Means: the ordinal successor of gamma

1 occurrence in this chapter

• Occurrence projected-formula-0031090

Equation form expr-46b7ba72e8e2dce7

$\setrank{\alpha} = \supstrict_{\beta \in \alpha}\setrank{\beta} = \supstrict_{\beta \in \alpha}\beta = \alpha$

Read as: the rank of alpha equals the strict supremum over beta belongs to alpha of the rank of beta

Means: the rank of alpha equals the strict supremum over beta belongs to alpha of the rank of beta

1 occurrence in this chapter

• Occurrence projected-formula-0031328

Equation form expr-4893e9df8b5496eb

$\alpha$

Read as: alpha

Means: alpha

21 occurrences in this chapter

• Occurrence projected-formula-0031067
• Occurrence projected-formula-0031069
• Occurrence projected-formula-0031074
• Occurrence projected-formula-0031075
• Occurrence projected-formula-0031099
• Occurrence projected-formula-0031105
• Occurrence projected-formula-0031106
• Occurrence projected-formula-0031107
• Occurrence projected-formula-0031127
• Occurrence projected-formula-0031129
• Occurrence projected-formula-0031134
• Occurrence projected-formula-0031142
• Occurrence projected-formula-0031164
• Occurrence projected-formula-0031169
• Occurrence projected-formula-0031178
• Occurrence projected-formula-0031197
• Occurrence projected-formula-0031223
• Occurrence projected-formula-0031248
• Occurrence projected-formula-0031280
• Occurrence projected-formula-0031284
• Occurrence projected-formula-0031324

Equation form expr-4b7ce6b75f0ec36f

$\phi(x)$

Read as: phi of x

Means: phi of x

1 occurrence in this chapter

• Occurrence projected-formula-0031307

Equation form expr-522df6ad7d1bb5ca

$\Zminus$

Read as: set theory Z minus

Means: set theory Z minus

4 occurrences in this chapter

• Occurrence projected-formula-0031258
• Occurrence projected-formula-0031261
• Occurrence projected-formula-0031269
• Occurrence projected-formula-0031270

Equation form expr-559aead08264d579

$A$

Read as: capital A

Means: capital A

12 occurrences in this chapter

• Occurrence projected-formula-0031120
• Occurrence projected-formula-0031140
• Occurrence projected-formula-0031211
• Occurrence projected-formula-0031214
• Occurrence projected-formula-0031222
• Occurrence projected-formula-0031240
• Occurrence projected-formula-0031246
• Occurrence projected-formula-0031276
• Occurrence projected-formula-0031277
• Occurrence projected-formula-0031278
• Occurrence projected-formula-0031283
• Occurrence projected-formula-0031302

Equation form expr-55d1823545b5b4c3

$\tau$

Read as: tau

Means: tau

3 occurrences in this chapter

• Occurrence projected-formula-0031070
• Occurrence projected-formula-0031076
• Occurrence projected-formula-0031109

Equation form expr-56f75af85a0d9936

$V_\alpha \notin V_\alpha$

Read as: capital V subscript alpha does not belong to capital V subscript alpha

Means: capital V subscript alpha does not belong to capital V subscript alpha

3 occurrences in this chapter

• Occurrence projected-formula-0031170
• Occurrence projected-formula-0031176
• Occurrence projected-formula-0031186

Equation form expr-5817f5cbb158cbe4

$x \in V_\beta$

Read as: x belongs to capital V subscript beta

Means: x belongs to capital V subscript beta

1 occurrence in this chapter

• Occurrence projected-formula-0031160

Equation form expr-5a38f036518cc2ab

$A \subseteq B \subseteq V_\beta$

Read as: capital A is a subset of capital B is a subset of capital V subscript beta

Means: capital A is a subset of capital B is a subset of capital V subscript beta

1 occurrence in this chapter

• Occurrence projected-formula-0031155

Equation form expr-5b07fb9b08e2bfe0

$A \subseteq B \in V_\alpha$

Read as: capital A is a subset of capital B belongs to capital V subscript alpha

Means: capital A is a subset of capital B belongs to capital V subscript alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031154

Equation form expr-5e8db33c3dc261a2

$V_\beta \subseteq V_\alpha$

Read as: capital V subscript beta is a subset of capital V subscript alpha

Means: capital V subscript beta is a subset of capital V subscript alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031201

Equation form expr-5f67a793a81728e3

$\alpha, \beta$

Read as: alpha, then beta

Means: alpha, then beta

1 occurrence in this chapter

• Occurrence projected-formula-0031187

Equation form expr-61b55354f7616dbb

$\beta \notin \alpha$

Read as: beta does not belong to alpha

Means: beta does not belong to alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031192

Equation form expr-61bb18adffaeba28

$\emph{Regularity}\Rightarrow\emph{Foundation}$

Read as: Regularity implies Foundation

Means: Regularity implies Foundation

1 occurrence in this chapter

• Occurrence projected-formula-0031253

Equation form expr-621717bfbaba6a20

$V_\gamma \subseteq V_\alpha$

Read as: capital V subscript gamma is a subset of capital V subscript alpha

Means: capital V subscript gamma is a subset of capital V subscript alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031147

Equation form expr-66d3594b2f760f46

$D \cap B = \emptyset$

Read as: capital D intersected with capital B equals the empty set

Means: capital D intersected with capital B equals the empty set

1 occurrence in this chapter

• Occurrence projected-formula-0031237

Equation form expr-67610b0632683369

$\gamma$

Read as: gamma

Means: gamma

4 occurrences in this chapter

• Occurrence projected-formula-0031078
• Occurrence projected-formula-0031081
• Occurrence projected-formula-0031088
• Occurrence projected-formula-0031091

Equation form expr-68f1ce892c3746fa

$D = \emptyset$

Read as: capital D equals the empty set

Means: capital D equals the empty set

2 occurrences in this chapter

• Occurrence projected-formula-0031227
• Occurrence projected-formula-0031235

Equation form expr-6a61148ce5d779b6

$\text{cl}_0(A) &= A,\\ \text{cl}_{n+1}(A) &= \bigcup \text{cl}_n(A),\\ \text{trcl}(A) &= \bigcup_{n < \omega} \text{cl}_{n}(A).$

Read as: Source-ordered display. closure stage zero of capital A equals capital A. Then, closure stage n plus one of capital A equals the union of closure stage n of capital A. Then, transitive closure of capital A equals the union over n is less than omega of closure stage n of capital A. End display

Means: Source-ordered display. closure stage zero of capital A equals capital A. Then, closure stage n plus one of capital A equals the union of closure stage n of capital A. Then, transitive closure of capital A equals the union over n is less than omega of closure stage n of capital A. End display

1 occurrence in this chapter

• Occurrence projected-formula-0031212

Equation form expr-6ca70d02253a917a

$x \in V_\alpha$

Read as: x belongs to capital V subscript alpha

Means: x belongs to capital V subscript alpha

5 occurrences in this chapter

• Occurrence projected-formula-0031163
• Occurrence projected-formula-0031233
• Occurrence projected-formula-0031288
• Occurrence projected-formula-0031290
• Occurrence projected-formula-0031320

Equation form expr-6ccaf2c1d932f2b5

$\beta \in \dom{g} \cap \dom{h} = \gamma \cap \delta = \min(\gamma, \delta)$

Read as: beta belongs to the domain of g intersected with the domain of h equals gamma intersected with delta equals the minimum of gamma, then delta

Means: beta belongs to the domain of g intersected with the domain of h equals gamma intersected with delta equals the minimum of gamma, then delta

1 occurrence in this chapter

• Occurrence projected-formula-0031084

Equation form expr-6e6f9b347c9e7892

$(\forall \gamma \in \delta)x \nsubseteq V_\gamma$

Read as: for every gamma in delta, x is not a subset of capital V subscript gamma

Means: for every gamma in delta, x is not a subset of capital V subscript gamma

1 occurrence in this chapter

• Occurrence projected-formula-0031231

Equation form expr-6ffc8155d28597c6

$\beta \in \alpha$

Read as: beta belongs to alpha

Means: beta belongs to alpha

4 occurrences in this chapter

• Occurrence projected-formula-0031181
• Occurrence projected-formula-0031185
• Occurrence projected-formula-0031200
• Occurrence projected-formula-0031327

Equation form expr-7046857daf500937

$\ZF$

Read as: set theory Z F

Means: set theory Z F

4 occurrences in this chapter

• Occurrence projected-formula-0031257
• Occurrence projected-formula-0031262
• Occurrence projected-formula-0031264
• Occurrence projected-formula-0031274

Equation form expr-7050bfbcf039c564

$V_\gamma \subseteq V_\beta$

Read as: capital V subscript gamma is a subset of capital V subscript beta

Means: capital V subscript gamma is a subset of capital V subscript beta

1 occurrence in this chapter

• Occurrence projected-formula-0031152

Equation form expr-729f2cd8398e9960

$\in$

Read as: the membership relation

Means: the membership relation

2 occurrences in this chapter

• Occurrence projected-formula-0031297
• Occurrence projected-formula-0031312

Equation form expr-72f486b87444c5e6

$g_\delta$

Read as: g subscript delta

Means: g subscript delta

2 occurrences in this chapter

• Occurrence projected-formula-0031092
• Occurrence projected-formula-0031096

Equation form expr-7677a4c84c312cb5

$V_\alpha \in V_{\ordsucc{\alpha}}$

Read as: capital V subscript alpha belongs to capital V subscript the ordinal successor of alpha

Means: capital V subscript alpha belongs to capital V subscript the ordinal successor of alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031174

Equation form expr-78827b0ef29849d2

$g(\delta) = g_{\ordsucc{\delta}}(\delta) = \tau(\funrestrictionto{g_{\ordsucc{\delta}}}{\delta}) = \tau(\funrestrictionto{g}{\delta})$

Read as: g of delta equals g subscript the ordinal successor of delta of delta equals tau of the restriction of g subscript the ordinal successor of delta to delta equals tau of the restriction of g to delta

Means: g of delta equals g subscript the ordinal successor of delta of delta equals tau of the restriction of g subscript the ordinal successor of delta to delta equals tau of the restriction of g to delta

1 occurrence in this chapter

• Occurrence projected-formula-0031098

Equation form expr-7956b6f7893bc248

$\setrank{c} \in \setrank{B}$

Read as: the rank of c belongs to the rank of capital B

Means: the rank of c belongs to the rank of capital B

1 occurrence in this chapter

• Occurrence projected-formula-0031336

Equation form expr-7a1da86849d4737b

$\alpha \leq \setrank{A}$

Read as: alpha is less than or equal to the rank of capital A

Means: alpha is less than or equal to the rank of capital A

1 occurrence in this chapter

• Occurrence projected-formula-0031317

Equation form expr-7ae0931189e7e6ed

$V_\beta \in V_\alpha$

Read as: capital V subscript beta belongs to capital V subscript alpha

Means: capital V subscript beta belongs to capital V subscript alpha

3 occurrences in this chapter

• Occurrence projected-formula-0031182
• Occurrence projected-formula-0031193
• Occurrence projected-formula-0031199

Equation form expr-7e3fb797657350ce

$\gamma \in \alpha$

Read as: gamma belongs to alpha

Means: gamma belongs to alpha

3 occurrences in this chapter

• Occurrence projected-formula-0031145
• Occurrence projected-formula-0031151
• Occurrence projected-formula-0031165

Equation form expr-7eb1cfc6df63a92c

$\setrank{x} \leq \alpha$

Read as: the rank of x is less than or equal to alpha

Means: the rank of x is less than or equal to alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031292

Equation form expr-7f2315ed2fa0ec08

$\Z$

Read as: set theory Z

Means: set theory Z

5 occurrences in this chapter

• Occurrence projected-formula-0031256
• Occurrence projected-formula-0031260
• Occurrence projected-formula-0031265
• Occurrence projected-formula-0031271
• Occurrence projected-formula-0031273

Equation form expr-8998cfb52f34aa19

$V_\alpha \in V_\alpha$

Read as: capital V subscript alpha belongs to capital V subscript alpha

Means: capital V subscript alpha belongs to capital V subscript alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031175

Equation form expr-89d413ba652b8a4a

$\gamma \in \ordsucc{\gamma} \in \alpha$

Read as: gamma belongs to the ordinal successor of gamma belongs to alpha

Means: gamma belongs to the ordinal successor of gamma belongs to alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031166

Equation form expr-8bd1f14468af7310

$\ZFminus + \text{Regularity}$

Read as: set theory Z F minus plus Regularity

Means: set theory Z F minus plus Regularity

2 occurrences in this chapter

• Occurrence projected-formula-0031331
• Occurrence projected-formula-0031332

Equation form expr-8c5d83da6d7806b7

$\setrank{x} \in \setrank{A}$

Read as: the rank of x belongs to the rank of capital A

Means: the rank of x belongs to the rank of capital A

1 occurrence in this chapter

• Occurrence projected-formula-0031306

Equation form expr-8cf6e1e176fca9e6

$A \subseteq V_\beta$

Read as: capital A is a subset of capital V subscript beta

Means: capital A is a subset of capital V subscript beta

2 occurrences in this chapter

• Occurrence projected-formula-0031156
• Occurrence projected-formula-0031159

Equation form expr-8d2cacefc75ba038

$\emptyset$

Read as: the empty set

Means: the empty set

1 occurrence in this chapter

• Occurrence projected-formula-0031086

Equation form expr-8d307401b31f0cb8

$\sigma(\emptyset) = \xi(\emptyset) = A$

Read as: sigma of the empty set equals xi of the empty set equals capital A

Means: sigma of the empty set equals xi of the empty set equals capital A

1 occurrence in this chapter

• Occurrence projected-formula-0031131

Equation form expr-8f8f3d76281a7200

$\forall x((\exists y \in A)x \subseteq y \lif x \in A)$

Read as: for every x, open scope, there exists y in capital A, x is a subset of y implies x belongs to capital A, close scope

Means: for every x, open scope, there exists y in capital A, x is a subset of y implies x belongs to capital A, close scope

1 occurrence in this chapter

• Occurrence projected-formula-0031141

Equation form expr-911f3dda1c556df1

$b \in \text{cl}_n(A)$

Read as: b belongs to closure stage n of capital A

Means: b belongs to closure stage n of capital A

1 occurrence in this chapter

• Occurrence projected-formula-0031219

Equation form expr-91c7df5e88e3759f

$\setrank{A} = \alpha$

Read as: the rank of capital A equals alpha

Means: the rank of capital A equals alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031323

Equation form expr-92becdaca2dd0d19

$V_\alpha$

Read as: capital V subscript alpha

Means: capital V subscript alpha

17 occurrences in this chapter

• Occurrence projected-formula-0031061
• Occurrence projected-formula-0031063
• Occurrence projected-formula-0031064
• Occurrence projected-formula-0031117
• Occurrence projected-formula-0031138
• Occurrence projected-formula-0031139
• Occurrence projected-formula-0031143
• Occurrence projected-formula-0031144
• Occurrence projected-formula-0031196
• Occurrence projected-formula-0031198
• Occurrence projected-formula-0031203
• Occurrence projected-formula-0031206
• Occurrence projected-formula-0031207
• Occurrence projected-formula-0031267
• Occurrence projected-formula-0031272
• Occurrence projected-formula-0031275
• Occurrence projected-formula-0031289

Equation form expr-9428f48a48ff1736

$\xi(x) = \begin{cases} A &\text{if $x$ is not a function whose}\\ &\hspace{1em}\text{domain is an ordinal; otherwise:}\\ \tau(x(\alpha)) & \text{if $\dom{x} = \ordsucc{\alpha}$}\\ \theta(\ran{x}) & \text{if $\dom{x}$ is a limit ordinal} \end{cases}$

Read as: xi of x is defined by cases. case one, capital A; if x is not a function whose. case two, domain is an ordinal; otherwise:. case three, tau of x of alpha; if the domain of x equals the ordinal successor of alpha. case four, theta of the range of x; if the domain of x is a limit ordinal. End cases

Means: xi of x is defined by cases. case one, capital A; if x is not a function whose. case two, domain is an ordinal; otherwise:. case three, tau of x of alpha; if the domain of x equals the ordinal successor of alpha. case four, theta of the range of x; if the domain of x is a limit ordinal. End cases

1 occurrence in this chapter

• Occurrence projected-formula-0031124

Equation form expr-974ac6749095664e

$\funrestrictionto{\sigma}{\alpha}$

Read as: the restriction of sigma to alpha

Means: the restriction of sigma to alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031128

Equation form expr-9a3afda901a5a2f0

$A = \text{cl}_0(A) \subseteq \text{trcl}(A)$

Read as: capital A equals closure stage zero of capital A is a subset of transitive closure of capital A

Means: capital A equals closure stage zero of capital A is a subset of transitive closure of capital A

1 occurrence in this chapter

• Occurrence projected-formula-0031217

Equation form expr-9acece532e677897

$\alpha = \ordsucc{\beta}$

Read as: alpha equals the ordinal successor of beta

Means: alpha equals the ordinal successor of beta

1 occurrence in this chapter

• Occurrence projected-formula-0031150

Equation form expr-9c3245dfb4ac54c1

$\sigma$

Read as: sigma

Means: sigma

3 occurrences in this chapter

• Occurrence projected-formula-0031100
• Occurrence projected-formula-0031115
• Occurrence projected-formula-0031130

Equation form expr-9ecd9e4a3964eb4b

$f_\alpha$

Read as: f subscript alpha

Means: f subscript alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031108

Equation form expr-a28ddfdabdf3b3b3

$g = \bigcup_{\delta \in \gamma} g_\delta$

Read as: g equals the union over delta belongs to gamma of g subscript delta

Means: g equals the union over delta belongs to gamma of g subscript delta

1 occurrence in this chapter

• Occurrence projected-formula-0031095

Equation form expr-a37c86ceab9f4506

$\lnot \phi(A)$

Read as: not phi of capital A

Means: not phi of capital A

1 occurrence in this chapter

• Occurrence projected-formula-0031303

Equation form expr-a6bdabf308890856

$\beta < \alpha$

Read as: beta is less than alpha

Means: beta is less than alpha

2 occurrences in this chapter

• Occurrence projected-formula-0031114
• Occurrence projected-formula-0031148

Equation form expr-a9b9a05aefaee72b

$A \neq \emptyset$

Read as: capital A is not equal to the empty set

Means: capital A is not equal to the empty set

1 occurrence in this chapter

• Occurrence projected-formula-0031333

Equation form expr-aa7f0963df3719c9

$V_\gamma \in V_\alpha$

Read as: capital V subscript gamma belongs to capital V subscript alpha

Means: capital V subscript gamma belongs to capital V subscript alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031146

Equation form expr-aaa9402664f1a41f

$h$

Read as: h

Means: h

1 occurrence in this chapter

• Occurrence projected-formula-0031080

Equation form expr-aaaa5c61dc932a3a

$\sigma(\alpha) &= f_{\ordsucc{\alpha}}(\alpha) \\&= \tau(\funrestrictionto{f_{\ordsucc{\alpha}}}{\alpha}) \\&= \tau(\Setabs{\tuple{\beta, f_{\ordsucc{\alpha}}(\beta)}}{\beta \in \alpha}) \\&= \tau(\Setabs{\tuple{\beta, f_{\ordsucc{\beta}}(\beta)}}{\beta \in \alpha})\\&= \tau(\funrestrictionto{\sigma}{\alpha})$

Read as: Source-ordered display. sigma of alpha equals f subscript the ordinal successor of alpha of alpha. Then, equals tau of the restriction of f subscript the ordinal successor of alpha to alpha. Then, equals tau of the set of the ordered pair beta, then f subscript the ordinal successor of alpha of beta such that beta belongs to alpha. Then, equals tau of the set of the ordered pair beta, then f subscript the ordinal successor of beta of beta such that beta belongs to alpha. Then, equals tau of the restriction of sigma to alpha. End display

Means: Source-ordered display. sigma of alpha equals f subscript the ordinal successor of alpha of alpha. Then, equals tau of the restriction of f subscript the ordinal successor of alpha to alpha. Then, equals tau of the set of the ordered pair beta, then f subscript the ordinal successor of alpha of beta such that beta belongs to alpha. Then, equals tau of the set of the ordered pair beta, then f subscript the ordinal successor of beta of beta such that beta belongs to alpha. Then, equals tau of the restriction of sigma to alpha. End display

1 occurrence in this chapter

• Occurrence projected-formula-0031112

Equation form expr-af484f20df9e7413

$\trcl{A}$

Read as: the transitive closure of capital A

Means: the transitive closure of capital A

1 occurrence in this chapter

• Occurrence projected-formula-0031216

Equation form expr-afc2bb2dd9cceae5

$\theta(x)$

Read as: theta of x

Means: theta of x

1 occurrence in this chapter

• Occurrence projected-formula-0031119

Equation form expr-b01e91e82909a746

$V_\alpha \in V_\alpha = \bigcup_{\beta \in \alpha}V_\beta$

Read as: capital V subscript alpha belongs to capital V subscript alpha equals the union over beta belongs to alpha of capital V subscript beta

Means: capital V subscript alpha belongs to capital V subscript alpha equals the union over beta belongs to alpha of capital V subscript beta

1 occurrence in this chapter

• Occurrence projected-formula-0031179

Equation form expr-b1ecf61129494d07

$A \subseteq V_\alpha$

Read as: capital A is a subset of capital V subscript alpha

Means: capital A is a subset of capital V subscript alpha

3 occurrences in this chapter

• Occurrence projected-formula-0031224
• Occurrence projected-formula-0031281
• Occurrence projected-formula-0031321

Equation form expr-b29757dd371052eb

$\xi(x)$

Read as: xi of x

Means: xi of x

1 occurrence in this chapter

• Occurrence projected-formula-0031123

Equation form expr-b34cb15f81d627aa

$\gamma \leq \alpha$

Read as: gamma is less than or equal to alpha

Means: gamma is less than or equal to alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031077

Equation form expr-b3dc00852705587a

$\tau(x) = \Pow{x}$

Read as: tau of x equals the power set of x

Means: tau of x equals the power set of x

1 occurrence in this chapter

• Occurrence projected-formula-0031136

Equation form expr-b5466bfc959b86a0

$(\forall x \in A)\phi(x) \land \lnot \phi(A)$

Read as: for every x in capital A, phi of x and not phi of capital A

Means: for every x in capital A, phi of x and not phi of capital A

1 occurrence in this chapter

• Occurrence projected-formula-0031308

Equation form expr-b86d44e8a1ce10eb

$g(\beta) = h(\beta)$

Read as: g of beta equals h of beta

Means: g of beta equals h of beta

1 occurrence in this chapter

• Occurrence projected-formula-0031083

Equation form expr-b92a5760ea24224d

$(\forall \beta \in \alpha)f(\beta) = \tau(\funrestrictionto{f}{\beta})$

Read as: for every beta in alpha, f of beta equals tau of the restriction of f to beta

Means: for every beta in alpha, f of beta equals tau of the restriction of f to beta

1 occurrence in this chapter

• Occurrence projected-formula-0031072

Equation form expr-bc57724b0c8fb691

$\tau(x)$

Read as: tau of x

Means: tau of x

4 occurrences in this chapter

• Occurrence projected-formula-0031065
• Occurrence projected-formula-0031073
• Occurrence projected-formula-0031102
• Occurrence projected-formula-0031118

Equation form expr-bd0319fbfb5886ed

$\sigma(\alpha) = \xi(\funrestrictionto{\sigma}{\alpha}) = \theta(\ran{\funrestrictionto{\sigma}{\alpha}})$

Read as: sigma of alpha equals xi of the restriction of sigma to alpha equals theta of the range of the restriction of sigma to alpha

Means: sigma of alpha equals xi of the restriction of sigma to alpha equals theta of the range of the restriction of sigma to alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031133

Equation form expr-bd59a72cd6b7cbea

$V_\gamma \in V_{\ordsucc{\gamma}}$

Read as: capital V subscript gamma belongs to capital V subscript the ordinal successor of gamma

Means: capital V subscript gamma belongs to capital V subscript the ordinal successor of gamma

1 occurrence in this chapter

• Occurrence projected-formula-0031167

Equation form expr-bf58d4049a3e6dd3

$\theta(x) = \bigcup x$

Read as: theta of x equals the union of x

Means: theta of x equals the union of x

1 occurrence in this chapter

• Occurrence projected-formula-0031137

Equation form expr-c10b550fc997555d

$\beta = \supstrict\Setabs{\delta}{(\exists x \in b)(x \subseteq V_\delta \land (\forall \gamma < \delta)x \nsubseteq V_\gamma)}.$

Read as: beta equals the strict supremum of the set of delta such that there exists x in b, open scope, x is a subset of capital V subscript delta and for every gamma less than delta, x is not a subset of capital V subscript gamma, close scope

Means: beta equals the strict supremum of the set of delta such that there exists x in b, open scope, x is a subset of capital V subscript delta and for every gamma less than delta, x is not a subset of capital V subscript gamma, close scope

1 occurrence in this chapter

• Occurrence projected-formula-0031243

Equation form expr-c73254e6088d1bc4

$\alpha \in \beta$

Read as: alpha belongs to beta

Means: alpha belongs to beta

2 occurrences in this chapter

• Occurrence projected-formula-0031188
• Occurrence projected-formula-0031195

Equation form expr-cc01e8febd312574

$V_{\ordsucc{\alpha}} \subseteq V_\alpha$

Read as: capital V subscript the ordinal successor of alpha is a subset of capital V subscript alpha

Means: capital V subscript the ordinal successor of alpha is a subset of capital V subscript alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031173

Equation form expr-cd0aa9856147b6c5

$g$

Read as: g

Means: g

2 occurrences in this chapter

• Occurrence projected-formula-0031079
• Occurrence projected-formula-0031087

Equation form expr-cd7f04f0c2bdba69

$x \subseteq V_\beta$

Read as: x is a subset of capital V subscript beta

Means: x is a subset of capital V subscript beta

1 occurrence in this chapter

• Occurrence projected-formula-0031161

Equation form expr-d203ba01eef4198c

$\delta$

Read as: delta

Means: delta

3 occurrences in this chapter

• Occurrence projected-formula-0031082
• Occurrence projected-formula-0031093
• Occurrence projected-formula-0031229

Equation form expr-d221f5671d9ec45b

$\ZFminus$

Read as: set theory Z F minus

Means: set theory Z F minus

12 occurrences in this chapter

• Occurrence projected-formula-0031204
• Occurrence projected-formula-0031205
• Occurrence projected-formula-0031210
• Occurrence projected-formula-0031250
• Occurrence projected-formula-0031252
• Occurrence projected-formula-0031254
• Occurrence projected-formula-0031255
• Occurrence projected-formula-0031259
• Occurrence projected-formula-0031263
• Occurrence projected-formula-0031266
• Occurrence projected-formula-0031268
• Occurrence projected-formula-0031329

Equation form expr-d3d4957312332192

$\sigma(\alpha) = \xi(\funrestrictionto{\sigma}{\alpha})$

Read as: sigma of alpha equals xi of the restriction of sigma to alpha

Means: sigma of alpha equals xi of the restriction of sigma to alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031126

Equation form expr-d45441d1afded8f6

$\setrank{A} = \supstrict_{x \in A}\setrank{x}$

Read as: the rank of capital A equals the strict supremum over x belongs to capital A of the rank of x

Means: the rank of capital A equals the strict supremum over x belongs to capital A of the rank of x

1 occurrence in this chapter

• Occurrence projected-formula-0031315

Equation form expr-d4d1413a7b6b3ff1

$V_\alpha = \Setabs{x}{\setrank{x} \in \alpha}$

Read as: capital V subscript alpha equals the set of x such that the rank of x belongs to alpha

Means: capital V subscript alpha equals the set of x such that the rank of x belongs to alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031285

Equation form expr-d5070db80f3e1b6e

$B \subseteq V_\beta$

Read as: capital B is a subset of capital V subscript beta

Means: capital B is a subset of capital V subscript beta

1 occurrence in this chapter

• Occurrence projected-formula-0031244

Equation form expr-d91609dd987768dc

$D &= \Setabs{x \in A}{\forall \delta\ x \nsubseteq V_\delta}\\ \alpha &= \supstrict\Setabs{\delta}{(\exists x \in A) (x \subseteq V_\delta \land (\forall \gamma \in \delta)x \nsubseteq V_\gamma)}$

Read as: Source-ordered display. capital D equals the set of x belongs to capital A such that for every delta, x is not a subset of capital V subscript delta. Then, alpha equals the strict supremum of the set of delta such that there exists x in capital A, open scope, x is a subset of capital V subscript delta and for every gamma in delta, x is not a subset of capital V subscript gamma, close scope. End display

Means: Source-ordered display. capital D equals the set of x belongs to capital A such that for every delta, x is not a subset of capital V subscript delta. Then, alpha equals the strict supremum of the set of delta such that there exists x in capital A, open scope, x is a subset of capital V subscript delta and for every gamma in delta, x is not a subset of capital V subscript gamma, close scope. End display

1 occurrence in this chapter

• Occurrence projected-formula-0031226

Equation form expr-dd2f1825648f9860

$V_\emptyset &\defis \emptyset\\ V_{\ordsucc{\alpha}} &\defis \Pow{V_\alpha} & & \text{for any ordinal }\alpha\\ V_{\alpha} &\defis \bigcup_{\gamma < \alpha} V_\gamma & & \text{when }\alpha\text{ is a limit ordinal}$

Read as: Source-ordered display. capital V subscript the empty set is defined as the empty set. Then, capital V subscript the ordinal successor of alpha is defined as the power set of capital V subscript alpha for any ordinal, alpha. Then, capital V subscript alpha is defined as the union over gamma is less than alpha of capital V subscript gamma, when, alpha, is a limit ordinal. End display

Means: Source-ordered display. capital V subscript the empty set is defined as the empty set. Then, capital V subscript the ordinal successor of alpha is defined as the power set of capital V subscript alpha for any ordinal, alpha. Then, capital V subscript alpha is defined as the union over gamma is less than alpha of capital V subscript gamma, when, alpha, is a limit ordinal. End display

1 occurrence in this chapter

• Occurrence projected-formula-0031062

Equation form expr-dd54d04e21d39ef5

$\setrank{A} \leq \alpha$

Read as: the rank of capital A is less than or equal to alpha

Means: the rank of capital A is less than or equal to alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031322

Equation form expr-ddf25ca65bdc64b0

$\alpha = \emptyset$

Read as: alpha equals the empty set

Means: alpha equals the empty set

1 occurrence in this chapter

• Occurrence projected-formula-0031149

Equation form expr-dee1674dd2a24314

$A \subseteq \trcl{A}$

Read as: capital A is a subset of the transitive closure of capital A

Means: capital A is a subset of the transitive closure of capital A

2 occurrences in this chapter

• Occurrence projected-formula-0031215
• Occurrence projected-formula-0031247

Equation form expr-df7e70e5021544f4

$B$

Read as: capital B

Means: capital B

1 occurrence in this chapter

• Occurrence projected-formula-0031338

Equation form expr-e1ec762a49137b2e

$A \in V_\alpha$

Read as: capital A belongs to capital V subscript alpha

Means: capital A belongs to capital V subscript alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031157

Equation form expr-e32100276452cf69

$\sigma(\ordsucc{\alpha}) = \xi(\funrestrictionto{\sigma}{\ordsucc{\alpha}}) = \tau(\funrestrictionto{\sigma}{\ordsucc{\alpha}}(\alpha)) = \tau(\sigma(\alpha))$

Read as: sigma of the ordinal successor of alpha equals xi of the restriction of sigma to the ordinal successor of alpha equals tau of the restriction of sigma to the ordinal successor of alpha open scope, alpha, close scope equals tau of sigma of alpha

Means: sigma of the ordinal successor of alpha equals xi of the restriction of sigma to the ordinal successor of alpha equals tau of the restriction of sigma to the ordinal successor of alpha open scope, alpha, close scope equals tau of sigma of alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031132

Equation form expr-e42959e147294a1d

$V_{\ordsucc{\alpha}} \notin V_{\ordsucc{\alpha}}$

Read as: capital V subscript the ordinal successor of alpha does not belong to capital V subscript the ordinal successor of alpha

Means: capital V subscript the ordinal successor of alpha does not belong to capital V subscript the ordinal successor of alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031177

Equation form expr-e5478990b9696cbf

$\forall A((\forall x \in A)\phi(x) \lif \phi(A)) \lif \forall A \phi(A).$

Read as: for every capital A, open scope, for every x in capital A, phi of x implies phi of capital A, close scope implies for every capital A, phi of capital A

Means: for every capital A, open scope, for every x in capital A, phi of x implies phi of capital A, close scope implies for every capital A, phi of capital A

1 occurrence in this chapter

• Occurrence projected-formula-0031299

Equation form expr-e7cedb234c1f4e03

$V_\gamma \in \Pow{V_\beta} = V_\alpha$

Read as: capital V subscript gamma belongs to the power set of capital V subscript beta equals capital V subscript alpha

Means: capital V subscript gamma belongs to the power set of capital V subscript beta equals capital V subscript alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031153

Equation form expr-e97616e0e4fc3820

$\delta < \gamma$

Read as: delta is less than gamma

Means: delta is less than gamma

1 occurrence in this chapter

• Occurrence projected-formula-0031094

Equation form expr-ea6d0ccbca97a901

$x \in b \in \trcl{A}$

Read as: x belongs to b belongs to the transitive closure of capital A

Means: x belongs to b belongs to the transitive closure of capital A

1 occurrence in this chapter

• Occurrence projected-formula-0031218

Equation form expr-ec1aa1d84c1fae8c

$\dom{\sigma}$

Read as: the domain of sigma

Means: the domain of sigma

1 occurrence in this chapter

• Occurrence projected-formula-0031116

Equation form expr-ee32906a10914339

$B \in A$

Read as: capital B belongs to capital A

Means: capital B belongs to capital A

2 occurrences in this chapter

• Occurrence projected-formula-0031294
• Occurrence projected-formula-0031334

Equation form expr-ee441b458889ac08

$x \notin V_\alpha$

Read as: x does not belong to capital V subscript alpha

Means: x does not belong to capital V subscript alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031293

Equation form expr-f34172048104aad9

$\forall x(\setrank{x} \in \setrank{A} \lif \phi(x))$

Read as: for every x, open scope, the rank of x belongs to the rank of capital A implies phi of x, close scope

Means: for every x, open scope, the rank of x belongs to the rank of capital A implies phi of x, close scope

1 occurrence in this chapter

• Occurrence projected-formula-0031304

Equation form expr-f5caabd0d15f58f0

$V_{\ordsucc{\alpha}} \in V_{\ordsucc{\alpha}} = \Pow{V_\alpha}$

Read as: capital V subscript the ordinal successor of alpha belongs to capital V subscript the ordinal successor of alpha equals the power set of capital V subscript alpha

Means: capital V subscript the ordinal successor of alpha belongs to capital V subscript the ordinal successor of alpha equals the power set of capital V subscript alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031172

Equation form expr-f67e8d417cfe54d8

$x \in \text{cl}_{n+1}(A) \subseteq \text{trcl}(A)$

Read as: x belongs to closure stage n plus one of capital A is a subset of transitive closure of capital A

Means: x belongs to closure stage n plus one of capital A is a subset of transitive closure of capital A

1 occurrence in this chapter

• Occurrence projected-formula-0031221

Equation form expr-f6a0b89baba6d3fe

$\emph{Foundation}\Rightarrow\emph{Regularity}$

Read as: Foundation implies Regularity

Means: Foundation implies Regularity

1 occurrence in this chapter

• Occurrence projected-formula-0031251

Equation form expr-f97e6102d66e7336

$\sigma(\alpha) = \tau(\funrestrictionto{\sigma}{\alpha})$

Read as: sigma of alpha equals tau of the restriction of sigma to alpha

Means: sigma of alpha equals tau of the restriction of sigma to alpha

1 occurrence in this chapter

• Occurrence projected-formula-0031104

Equation form expr-fa3d72bba23e60fd

$\sigma(\emptyset) &= A\\ \sigma(\ordsucc{\alpha}) &= \tau(\sigma(\alpha)) && \text{for any ordinal }\alpha\\ \sigma(\alpha) &= \theta(\ran{\funrestrictionto{\sigma}{\alpha}})&& \text{when }\alpha\text{ is a limit ordinal}$

Read as: Source-ordered display. sigma of the empty set equals capital A. Then, sigma of the ordinal successor of alpha equals tau of sigma of alpha, for any ordinal, alpha. Then, sigma of alpha equals theta of the range of the restriction of sigma to alpha, when, alpha, is a limit ordinal. End display

Means: Source-ordered display. sigma of the empty set equals capital A. Then, sigma of the ordinal successor of alpha equals tau of sigma of alpha, for any ordinal, alpha. Then, sigma of alpha equals theta of the range of the restriction of sigma to alpha, when, alpha, is a limit ordinal. End display

1 occurrence in this chapter

• Occurrence projected-formula-0031122

Equation form expr-fadddc479eeedf12

$x \notin D$

Read as: x does not belong to capital D

Means: x does not belong to capital D

1 occurrence in this chapter

• Occurrence projected-formula-0031241

Equation form expr-fcfa50982ded070a

$f_{\ordsucc{\beta}}(\beta) = f_{\ordsucc{\alpha}}(\beta)$

Read as: f subscript the ordinal successor of beta of beta equals f subscript the ordinal successor of alpha of beta

Means: f subscript the ordinal successor of beta of beta equals f subscript the ordinal successor of alpha of beta

1 occurrence in this chapter

• Occurrence projected-formula-0031113

Definition one in this chapter

This source definition contains, in source order: Source-ordered display. capital V subscript the empty set is defined as the empty set. Then, capital V subscript the ordinal successor of alpha is defined as the power set of capital V subscript alpha for any ordinal, alpha. Then, capital V subscript alpha is defined as the union over gamma is less than alpha of capital V subscript gamma, when, alpha, is a limit ordinal. End display. 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. capital V subscript the empty set is defined as the empty set. Then, capital V subscript the ordinal successor of alpha is defined as the power set of capital V subscript alpha for any ordinal, alpha. Then, capital V subscript alpha is defined as the union over gamma is less than alpha of capital V subscript gamma, when, alpha, is a limit ordinal. End display. The complete surrounding source prose remains in the continuous listener stream.

Source

Definition two in this chapter

This source definition contains, in source order: tau of x; then f; then alpha; then f; then alpha; then tau; then the domain of f equals alpha; then for every beta in alpha, f of beta equals tau of the restriction of f to beta. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma: Bounded Recursion

This source lemma contains, in source order: tau of x; then alpha; then alpha; then tau. The complete surrounding source prose remains in the continuous listener stream.

Source

Theorem: General Recursion

This source theorem contains, in source order: tau of x; then sigma of x; then sigma of alpha equals tau of the restriction of sigma to alpha; then alpha. 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: Source-ordered display. sigma of alpha equals f subscript the ordinal successor of alpha of alpha. Then, equals tau of the restriction of f subscript the ordinal successor of alpha to alpha. Then, equals tau of the set of the ordered pair beta, then f subscript the ordinal successor of alpha of beta such that beta belongs to alpha. Then, equals tau of the set of the ordered pair beta, then f subscript the ordinal successor of beta of beta such that beta belongs to alpha. Then, equals tau of the restriction of sigma to alpha. End display. The complete surrounding source prose remains in the continuous listener stream.

Source

Theorem: Simple Recursion

This source theorem contains, in source order: tau of x; then theta of x; then capital A; then sigma of x; then Source-ordered display. sigma of the empty set equals capital A. Then, sigma of the ordinal successor of alpha equals tau of sigma of alpha, for any ordinal, alpha. Then, sigma of alpha equals theta of the range of the restriction of sigma to alpha, when, alpha, is a limit ordinal. End display. 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: Source-ordered display. sigma of the empty set equals capital A. Then, sigma of the ordinal successor of alpha equals tau of sigma of alpha, for any ordinal, alpha. Then, sigma of alpha equals theta of the range of the restriction of sigma to alpha, when, alpha, is a limit ordinal. End display. The complete surrounding source prose remains in the continuous listener stream.

Source

Definition three in this chapter

This source definition contains, in source order: capital A; then for every x, open scope, there exists y in capital A, x is a subset of y implies x belongs to capital A, close scope. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma two in this chapter

This source lemma contains, in source order: alpha; then capital V subscript alpha; then capital V subscript alpha; then gamma belongs to alpha; then capital V subscript gamma belongs to capital V subscript alpha; then capital V subscript gamma is a subset of capital V subscript alpha. 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; then capital V subscript alpha does not belong to capital V subscript alpha. The complete surrounding source prose remains in the continuous listener stream.

Source

Corollary one in this chapter

This source corollary contains, in source order: alpha, then beta; then alpha belongs to beta; then capital V subscript alpha belongs to capital V subscript beta. The complete surrounding source prose remains in the continuous listener stream.

Source

Definition four in this chapter

This source definition contains, in source order: for every capital A, there exists alpha, capital A is a subset of capital V subscript alpha. The complete surrounding source prose remains in the continuous listener stream.

Source

Axiom: Foundation

This source axiom contains, in source order: for every capital A not equal to the empty set, there exists capital B in capital A, capital A intersected with capital B equals the empty set. The complete surrounding source prose remains in the continuous listener stream.

Source

Definition five in this chapter

This source definition contains, in source order: capital A; then Source-ordered display. closure stage zero of capital A equals capital A. Then, closure stage n plus one of capital A equals the union of closure stage n of capital A. Then, transitive closure of capital A equals the union over n is less than omega of closure stage n of capital A. End display; then transitive closure of capital A; then capital A. 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. closure stage zero of capital A equals capital A. Then, closure stage n plus one of capital A equals the union of closure stage n of capital A. Then, transitive closure of capital A equals the union over n is less than omega of closure stage n of capital A. End display. The complete surrounding source prose remains in the continuous listener stream.

Source

Proposition one in this chapter

This source proposition contains, in source order: capital A is a subset of the transitive closure of capital A; then the transitive closure of capital A. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma four in this chapter

This source lemma contains, in source order: capital A; then alpha; then capital A is a subset of capital V subscript alpha. 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. capital D equals the set of x belongs to capital A such that for every delta, x is not a subset of capital V subscript delta. Then, alpha equals the strict supremum of the set of delta such that there exists x in capital A, open scope, x is a subset of capital V subscript delta and for every gamma in delta, x is not a subset of capital V subscript gamma, close scope. End display. The complete surrounding source prose remains in the continuous listener stream.

Source

Theorem three in this chapter

This source theorem contains no delimiter-counted formula. Its complete source wording remains in the continuous listener stream.

Source

Definition six in this chapter

This source definition contains, in source order: set theory Z; then set theory Z minus; then set theory Z F; then set theory Z F minus; then set theory Z F; then set theory Z. The complete surrounding source prose remains in the continuous listener stream.

Source

Definition seven in this chapter

This source definition contains, in source order: capital A; then the rank of capital A; then alpha; then capital A is a subset of capital V subscript alpha. The complete surrounding source prose remains in the continuous listener stream.

Source

Proposition two in this chapter

This source proposition contains, in source order: the rank of capital A; then capital A. The complete surrounding source prose remains in the continuous listener stream.

Source

Exercise one in this chapter

This source exercise contains no delimiter-counted formula. Its complete source wording remains in the continuous listener stream. The exercise is preserved as stated and no solution is supplied.

Source

Proposition three in this chapter

This source proposition contains, in source order: alpha; then capital V subscript alpha equals the set of x such that the rank of x belongs to alpha. The complete surrounding source prose remains in the continuous listener stream.

Source

Exercise two in this chapter

This source exercise contains no delimiter-counted formula. Its complete source wording remains in the continuous listener stream. The exercise is preserved as stated and no solution is supplied.

Source

Proposition four in this chapter

This source proposition contains, in source order: capital B belongs to capital A; then the rank of capital B belongs to the rank of capital A. The complete surrounding source prose remains in the continuous listener stream.

Source

Theorem: the membership relation-Induction Scheme

This source theorem contains, in source order: the membership relation; then phi; then for every capital A, open scope, for every x in capital A, phi of x implies phi of capital A, close scope implies for every capital A, phi of capital 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: the rank of capital A equals the strict supremum over x belongs to capital A of the rank of x. The complete surrounding source prose remains in the continuous listener stream.

Source

Corollary two in this chapter

This source corollary contains, in source order: alpha; then the rank of alpha equals alpha. The complete surrounding source prose remains in the continuous listener stream.

Source

Proposition: working in set theory Z F minus plus Regularity

This source proposition contains, in source order: set theory Z F minus plus Regularity. The complete surrounding source prose remains in the continuous listener stream.

Source

Cross-reference reference-001301

chapter “Ordinals”

Source occurrence

Cross-reference reference-001302

section “Successor and Limit Ordinals” in chapter “Ordinals”

Source occurrence

Cross-reference reference-001303

section “Dedekind Algebras” in chapter “Infinite Sets”

Source occurrence

Cross-reference reference-001304

definition one in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001305

theorem “Simple Transfinite Induction” in chapter “Ordinals”

Source occurrence

Cross-reference reference-001306

lemma “Bounded Recursion” in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001307

lemma “Bounded Recursion” in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001308

theorem “General Recursion” in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001309

theorem “Burali-Forti Paradox” in chapter “Ordinals”

Source occurrence

Cross-reference reference-001310

theorem “General Recursion” in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001311

theorem “General Recursion” in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001312

theorem “Simple Transfinite Induction” in chapter “Ordinals”

Source occurrence

Cross-reference reference-001313

definition one in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001314

Tim Button (2021)

Source occurrence

Cross-reference reference-001315

item 1 of lemma two in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001316

item 1 of lemma two in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001317

item 3 of lemma two in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001318

item 3 of lemma two in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001319

item 2 of lemma two in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001320

item 1 of lemma two in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001321

item 3 of lemma two in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001322

item 1 of lemma two in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001323

item 2 of lemma two in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001324

lemma two in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001325

lemma two in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001326

Lemma two in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001327

lemma three in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001328

lemma two in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001329

lemma three in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001330

section “The Story in More Detail” in chapter “Steps towards Z”

Source occurrence

Cross-reference reference-001331

definition nine in chapter “Ordinals”

Source occurrence

Cross-reference reference-001332

lemma two in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001333

proposition one in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001334

the lemma on Transitive Well Founded

Source occurrence

Cross-reference reference-001335

proposition “working in set theory Z F minus plus Regularity” in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001336

section “The Transfinite Recursion Theorem(s)” in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001337

proposition two in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001338

lemma two in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001339

proposition three in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001340

proposition three in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001341

theorem “Transfinite Induction” in chapter “Ordinals”

Source occurrence

Cross-reference reference-001342

proposition four in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001343

proposition four in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001344

proposition three in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001345

proposition five in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001346

section “Foundation” in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001347

proposition four in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001348

proposition four in chapter “Stages and Ranks”

Source occurrence