Set Theory

Replacement

Equation form expr-00905b5c3d249fd9

(∃x∈M)(\lexists[x \in M])

Read as: open scope, there exists x belongs to capital M, close scope

Means: open scope, there exists x belongs to capital M, close scope

1 occurrence in this chapter

Equation form expr-01208e159d2aec8c

∧\land

Read as: conjunction

Means: conjunction

1 occurrence in this chapter

Equation form expr-034ba937d3009ac4

T⊢θ\Th{T} \Proves \theta

Read as: the theory of capital T theta is derivable

Means: the theory of capital T theta is derivable

1 occurrence in this chapter

Equation form expr-04b23cb0c98d2f7d

ω+ω\omega + \omega

Read as: omega plus omega

Means: omega plus omega

1 occurrence in this chapter

Equation form expr-0758bba7c8612719

(∀x∈A)∃!yϕ(x,y)(\forall x \in A)\lexists![y][\phi(x,y)]

Read as: for every x in capital A, there exists exactly one y, phi of x and y

Means: for every x in capital A, there exists exactly one y, phi of x and y

3 occurrences in this chapter

Equation form expr-08dcd457412b4090

Vω+ωV_{\omega+\omega}

Read as: capital V subscript omega plus omega

Means: capital V subscript omega plus omega

4 occurrences in this chapter

Equation form expr-08f271887ce94707

MM

Read as: capital M

Means: capital M

2 occurrences in this chapter

Equation form expr-09ea12f2f72a055c

ϕ(v,w)\phi(v,w)

Read as: phi of v and w

Means: phi of v and w

1 occurrence in this chapter

Equation form expr-0bf9bc170ad957d9

VβV_\beta

Read as: capital V subscript beta

Means: capital V subscript beta

2 occurrences in this chapter

Equation form expr-10ce0200b450ea95

ϕ\phi

Read as: phi

Means: phi

13 occurrences in this chapter

Equation form expr-13c3d5d46363a18c

a¯1,…,a¯k\overline{a}_1, \ldots, \overline{a}_k

Read as: the tuple bar over a subscript one the tuple bar over a subscript k, and so on

Means: the tuple bar over a subscript one the tuple bar over a subscript k, and so on

1 occurrence in this chapter

Equation form expr-13dee6995bdcdb61

∀α∃β>α(∀x1…,xn∈Vβ)(ϕ(x1,…,xn)↔ϕVβ(x1,…,xn))\forall \alpha \exists \beta > \alpha (\forall x_1 \ldots, x_n \in V_\beta)(\phi(x_1, \ldots, x_n) \liff \phi^{V_\beta}(x_1, \ldots, x_n))

Read as: for every ordinal alpha there exists an ordinal beta greater than alpha such that, for all x subscript one through x subscript n in capital V subscript beta, phi holds of those objects if and only if its relativization to capital V subscript beta holds of them

Means: for every ordinal alpha there exists an ordinal beta greater than alpha such that, for all x subscript one through x subscript n in capital V subscript beta, phi holds of those objects if and only if its relativization to capital V subscript beta holds of them

1 occurrence in this chapter

Equation form expr-14bd1008238d85fd

∃xϕi(a¯i,x)→(∃x∈V)ϕi(a¯i,x))\exists x\phi_i(\overline{a}_i, x) \rightarrow (\exists x \in V) \phi_i(\overline{a}_i, x))

Read as: there exists x, phi subscript i of the tuple bar over a subscript i and x implies there exists x in capital V, phi subscript i of the tuple bar over a subscript i and x

Means: there exists x, phi subscript i of the tuple bar over a subscript i and x implies there exists x in capital V, phi subscript i of the tuple bar over a subscript i and x

1 occurrence in this chapter

Equation form expr-1b16b1df538ba12d

nn

Read as: n

Means: n

1 occurrence in this chapter

Equation form expr-21631083f64fc430

(∃x∈Sn+1)ϕi(a¯i,x)(\exists x \in S_{n+1})\phi_i(\overline{a}_i, x)

Read as: there exists x in capital S subscript n plus one, phi subscript i of the tuple bar over a subscript i and x

Means: there exists x in capital S subscript n plus one, phi subscript i of the tuple bar over a subscript i and x

1 occurrence in this chapter

Equation form expr-2225b5a8bdecda32

x∈Ax \in A

Read as: x belongs to capital A

Means: x belongs to capital A

4 occurrences in this chapter

Equation form expr-233b36fb0a3cde66

ψk=ϕ\psi_k = \phi

Read as: psi subscript k equals phi

Means: psi subscript k equals phi

2 occurrences in this chapter

Equation form expr-245c2c3b3d0e0c4d

τ[A]\funimage{\tau}{A}

Read as: the image of capital A under tau

Means: the image of capital A under tau

1 occurrence in this chapter

Equation form expr-250de4976f4a4b6c

(∀x∈A)(∃yϕ(x,y)↔(∃y∈S)ϕ(x,y))(\forall x \in A)(&\exists y\phi(x, y) \liff (\exists y \in S)\phi(x, y))

Read as: for every x in capital A, there exists a y such that phi holds of x and y if and only if there exists such a y in capital S

Means: for every x in capital A, there exists a y such that phi holds of x and y if and only if there exists such a y in capital S

1 occurrence in this chapter

Equation form expr-27272ef2b08a7dc8

(∃x∈μ(a¯1,…,a¯k))ϕi(a¯i,x)(\exists x \in \mu(\overline{a}_1, \ldots, \overline{a}_k))\phi_i(\overline{a}_i, x)

Read as: there exists x in mu of the tuple bar over a subscript one and , and so on and the tuple bar over a subscript k, phi subscript i of the tuple bar over a subscript i and x

Means: there exists x in mu of the tuple bar over a subscript one and , and so on and the tuple bar over a subscript k, phi subscript i of the tuple bar over a subscript i and x

1 occurrence in this chapter

Equation form expr-284c47be48fb0c66

¬\lnot

Read as: negation

Means: negation

1 occurrence in this chapter

Equation form expr-2aaa8b243d69250f

0,1,A∈S0, 1, A \in S

Read as: zero, then one, then capital A belongs to capital S

Means: zero, then one, then capital A belongs to capital S

1 occurrence in this chapter

Equation form expr-2b18c3f42bbf8aae

(∃x∈S)ϕi(a¯i,x)(\exists x \in S)\phi_i(\overline{a}_i, x)

Read as: there exists x in capital S, phi subscript i of the tuple bar over a subscript i and x

Means: there exists x in capital S, phi subscript i of the tuple bar over a subscript i and x

1 occurrence in this chapter

Equation form expr-2b4af44cb17349bb

θ\theta

Read as: theta

Means: theta

5 occurrences in this chapter

Equation form expr-30bda50e670619f7

(∀x,z∈S)(ϕ(x,z)↔ϕS(x,z))(∀x∈S)(∃yϕ(x,y)↔(∃y∈S)ϕS(x,y))(\forall x, z \in S)(&\phi(x, z) \liff \phi^S(x, z))\\ (\forall x \in S)(&\exists y\phi(x, y) \liff (\exists y \in S)\phi^S(x, y))

Read as: two absoluteness conditions. First, for all x and z in capital S, phi of x and z holds if and only if its relativization to capital S holds. Second, for every x in capital S, there exists a y satisfying phi of x and y if and only if there exists such a y in capital S satisfying the relativized formula. End conditions

Means: two absoluteness conditions. First, for all x and z in capital S, phi of x and z holds if and only if its relativization to capital S holds. Second, for every x in capital S, there exists a y satisfying phi of x and y if and only if there exists such a y in capital S satisfying the relativized formula. End conditions

1 occurrence in this chapter

Equation form expr-345a78129a0e73d5

Z+Weak-Reflection\Z + \text{Weak-Reflection}

Read as: set theory Z plus Weak-Reflection

Means: set theory Z plus Weak-Reflection

1 occurrence in this chapter

Equation form expr-35e291e0608087e1

(∀x,z∈S)((ψ↔ψS)∧(χ↔χS))(\forall x,z \in S)((\psi \liff \psi^S) \land (\chi \liff \chi^S))

Read as: for every x, then z in capital S, open scope, open scope, psi if and only if psi superscript capital S, close scope and open scope, chi if and only if chi superscript capital S, close scope, close scope

Means: for every x, then z in capital S, open scope, open scope, psi if and only if psi superscript capital S, close scope and open scope, chi if and only if chi superscript capital S, close scope, close scope

1 occurrence in this chapter

Equation form expr-3a6ae063cd1692e7

∃x\lexists[x]

Read as: there exists x

Means: there exists x

1 occurrence in this chapter

Equation form expr-3b989dcd8f42e1a9

a¯k\overline{a}_k

Read as: the tuple bar over a subscript k

Means: the tuple bar over a subscript k

1 occurrence in this chapter

Equation form expr-3cec2c066344d465

σ(x)\sigma(x)

Read as: sigma of x

Means: sigma of x

2 occurrences in this chapter

Equation form expr-42a918b2bd1a5a77

T⊢∃Mψ(M)\Th{T} \Proves \exists M \psi(M)

Read as: the theory of capital T there exists capital M, psi of capital M is derivable

Means: the theory of capital T there exists capital M, psi of capital M is derivable

1 occurrence in this chapter

Equation form expr-434c2f415bd78857

A⊆SA \subseteq S

Read as: capital A is a subset of capital S

Means: capital A is a subset of capital S

1 occurrence in this chapter

Equation form expr-46a1d65536c50157

∃xϕi(a¯i,x)\exists x \phi_i(\overline{a}_i,x)

Read as: there exists x, phi subscript i of the tuple bar over a subscript i and x

Means: there exists x, phi subscript i of the tuple bar over a subscript i and x

1 occurrence in this chapter

Equation form expr-46cf68e4bf624750

a¯1,…,a¯k∈Vβ\overline{a}_1, \ldots, \overline{a}_k \in V_\beta

Read as: the tuple bar over a subscript one the tuple bar over a subscript k, and so on belongs to capital V subscript beta

Means: the tuple bar over a subscript one the tuple bar over a subscript k, and so on belongs to capital V subscript beta

1 occurrence in this chapter

Equation form expr-47e4e158fdde441d

ZF⊢ϕVω+ω\ZF \vdash \phi^{V_{\omega+\omega}}

Read as: set theory Z F derives phi superscript capital V subscript omega plus omega

Means: set theory Z F derives phi superscript capital V subscript omega plus omega

1 occurrence in this chapter

Equation form expr-4893e9df8b5496eb

α\alpha

Read as: alpha

Means: alpha

3 occurrences in this chapter

Equation form expr-492247e1bab15272

μ\mu

Read as: mu

Means: mu

1 occurrence in this chapter

Equation form expr-4960b1d181b1e6f0

ϕM\phi^M

Read as: phi superscript capital M

Means: phi superscript capital M

1 occurrence in this chapter

Equation form expr-4b2c9ecd3e3cdcda

0≠10 \neq 1

Read as: zero is not equal to one

Means: zero is not equal to one

1 occurrence in this chapter

Equation form expr-4b68ab3847feda7d

XX

Read as: capital X

Means: capital X

1 occurrence in this chapter

Equation form expr-4be37f0622fedda1

ξ(z)↔ξS(z)\xi(z) \liff \xi^S(z)

Read as: xi of z if and only if xi superscript capital S of z

Means: xi of z if and only if xi superscript capital S of z

1 occurrence in this chapter

Equation form expr-4daa2ed1b05a7c05

n⋖m iff either n<m and m−n is even,or n is even and m is odd.n \lessdot m \text{ iff }&\text{either }n < m\text{ and }m-n\text{ is even,}\\ & \text{or $n$ is even and $m$ is odd.}

Read as: Source-ordered display. n is strictly smaller than m if and only if either, n is less than m and m minus n, is even,. Then, or n is even and m is odd.. End display

Means: Source-ordered display. n is strictly smaller than m if and only if either, n is less than m and m minus n, is even,. Then, or n is even and m is odd.. End display

1 occurrence in this chapter

Equation form expr-4ec6bb9152619c3f

T⊢ZF\Th{T} \Proves \ZF

Read as: the theory of capital T set theory Z F is derivable

Means: the theory of capital T set theory Z F is derivable

4 occurrences in this chapter

Equation form expr-507030fd57976f63

T⊢∀X(ψ(X)→(∃Nψ(N))X)Using this, and equation star in chapter Replacement again:T⊢∃M(ψ(M)∧(∃Nψ(N))M)In particular, then:T⊢∃M(ψ(M)∧(∃N∈M)((N is transitive)N∧(θN)M))So, by elementary reasoning concerning transitivity:T⊢∃M(ψ(M)∧(∃N∈M)(N is transitive∧θN))\Th{T} &\Proves \forall X(\psi(X) \lif (\exists N \psi(N))^X)\\ \intertext{Using this, and \eqref{Mpsi} again:} \Th{T} &\Proves \exists M(\psi(M) \land (\exists N \psi(N))^M) \intertext{In particular, then:} \Th{T} &\Proves \exists M(\psi(M) \land (\exists N \in M)((N\text{ is transitive})^N \land (\theta^N)^M)) \intertext{So, by elementary reasoning concerning transitivity:} \Th{T} &\Proves \exists M(\psi(M) \land (\exists N \in M)(N\text{ is transitive} \land \theta^N))

Read as: source-ordered derivation in theory capital T. First, capital T derives that every capital X satisfying psi also satisfies, inside capital X, that some capital N satisfies psi. Using this and equation star in chapter Replacement, capital T derives a capital M satisfying psi in which some capital N satisfies psi. In particular, capital T derives such a capital M containing a capital N that is internally transitive and satisfies theta through the displayed relativizations. Finally, by transitivity, capital T derives a capital M satisfying psi that contains a transitive capital N satisfying theta. End derivation

Means: source-ordered derivation in theory capital T. First, capital T derives that every capital X satisfying psi also satisfies, inside capital X, that some capital N satisfies psi. Using this and equation star in chapter Replacement, capital T derives a capital M satisfying psi in which some capital N satisfies psi. In particular, capital T derives such a capital M containing a capital N that is internally transitive and satisfies theta through the displayed relativizations. Finally, by transitivity, capital T derives a capital M satisfying psi that contains a transitive capital N satisfying theta. End derivation

1 occurrence in this chapter

Equation form expr-53ac347c35744f3d

{y∈S:(∃x∈A)ϕ(x,y)}={y:(∃x∈A)ϕ(x,y)}\Setabs{y \in S}{(\exists x \in A) \phi(x, y)} = \Setabs{y}{(\exists x \in A) \phi(x, y)}

Read as: the set of y belongs to capital S such that there exists x in capital A, phi of x and y, equals the set of y such that there exists x in capital A, phi of x and y

Means: the set of y belongs to capital S such that there exists x in capital A, phi of x and y, equals the set of y such that there exists x in capital A, phi of x and y

1 occurrence in this chapter

Equation form expr-559aead08264d579

AA

Read as: capital A

Means: capital A

8 occurrences in this chapter

Equation form expr-55d1823545b5b4c3

τ\tau

Read as: tau

Means: tau

4 occurrences in this chapter

Equation form expr-561d7c65461b8c7f

T⊢∀X(X is transitive→σX)\Th{T} \Proves \forall X(X\text{ is transitive} \lif \sigma^X)

Read as: the theory of capital T for every capital X, open scope, capital X, is transitive implies sigma superscript capital X, close scope is derivable

Means: the theory of capital T for every capital X, open scope, capital X, is transitive implies sigma superscript capital X, close scope is derivable

1 occurrence in this chapter

Equation form expr-57ccfab4b7e58601

T⊢∃βθVβ\Th{T} \Proves \exists \beta\ \theta^{V_\beta}

Read as: the theory of capital T there exists beta, theta superscript capital V subscript beta is derivable

Means: the theory of capital T there exists beta, theta superscript capital V subscript beta is derivable

2 occurrences in this chapter

Equation form expr-5b4436327a30249e

z=1z =1

Read as: z equals one

Means: z equals one

1 occurrence in this chapter

Equation form expr-5c2b11856e58eaec

σ(x)⊆τ(x)\sigma(x) \subseteq \tau(x)

Read as: sigma of x is a subset of tau of x

Means: sigma of x is a subset of tau of x

1 occurrence in this chapter

Equation form expr-5e96dc9bfb0a8fda

ψ(X)\psi(X)

Read as: psi of capital X

Means: psi of capital X

2 occurrences in this chapter

Equation form expr-5feceb66ffc86f38

00

Read as: zero

Means: zero

2 occurrences in this chapter

Equation form expr-61a51a9ec738424b

T⊢∃Mψ(M).row label *\Th{T} \Proves \exists M \psi(M). \tag{*}\label{Mpsi}

Read as: the theory of capital T there exists capital M, psi of capital M applied to tag applied to star is derivable

Means: the theory of capital T there exists capital M, psi of capital M applied to tag applied to star is derivable

1 occurrence in this chapter

Equation form expr-636a20528ffea8ef

(∀x∈A)τ(x)∈V(\forall x \in A)\tau(x)\in V

Read as: for every x in capital A, tau of x belongs to capital V

Means: for every x in capital A, tau of x belongs to capital V

1 occurrence in this chapter

Equation form expr-63984962c330f4f1

n<ωn < \omega

Read as: n is less than omega

Means: n is less than omega

1 occurrence in this chapter

Equation form expr-692f336168c77b12

ψ is (ϕ(x,z)∧A=A)χ is ∃yϕ(x,y)\psi &\text{ is } (\phi(x, z) \land A = A)\\ \chi &\text{ is } \lexists[y][\phi(x, y)]

Read as: Source-ordered display. psi, is, open scope, phi of x and z and capital A equals capital A, close scope. Then, chi, is, there exists y, phi of x and y. End display

Means: Source-ordered display. psi, is, open scope, phi of x and z and capital A equals capital A, close scope. Then, chi, is, there exists y, phi of x and y. End display

1 occurrence in this chapter

Equation form expr-6a89e12b1d33d53f

∃xψi(a¯i,x)→(∃x∈Vβ)ψi(a¯i,x)row label *\label{reflectionnicelybehaved} \exists x\psi_i(\overline{a}_i, x) \rightarrow (\exists x \in V_\beta) \psi_i(\overline{a}_i, x)\tag{*}

Read as: there exists x, psi subscript i of the tuple bar over a subscript i and x implies there exists x in capital V subscript beta, psi subscript i of the tuple bar over a subscript i and x applied to tag applied to star

Means: there exists x, psi subscript i of the tuple bar over a subscript i and x implies there exists x in capital V subscript beta, psi subscript i of the tuple bar over a subscript i and x applied to tag applied to star

1 occurrence in this chapter

Equation form expr-6b768e624a7c412d

0,1∈S0, 1 \in S

Read as: zero, then one belongs to capital S

Means: zero, then one belongs to capital S

1 occurrence in this chapter

Equation form expr-6b86b273ff34fce1

11

Read as: one

Means: one

2 occurrences in this chapter

Equation form expr-6cd84ba3a876b175

(∀x∈A)∃!yϕ(x,y)(\forall x \in A)\lexists![y][\phi(x, y)]

Read as: for every x in capital A, there exists exactly one y, phi of x and y

Means: for every x in capital A, there exists exactly one y, phi of x and y

1 occurrence in this chapter

Equation form expr-6db479813e7c12fd

ψi\psi_i

Read as: psi subscript i

Means: psi subscript i

4 occurrences in this chapter

Equation form expr-7046857daf500937

ZF\ZF

Read as: set theory Z F

Means: set theory Z F

20 occurrences in this chapter

Equation form expr-70c52efae8978345

ξ\xi

Read as: xi

Means: xi

1 occurrence in this chapter

Equation form expr-729f2cd8398e9960

∈\in

Read as: the membership relation

Means: the membership relation

4 occurrences in this chapter

Equation form expr-78557b9a6d29fa42

ψi(a¯i)↔ψiVβ(a¯i)\psi_i(\overline{a}_i) \leftrightarrow \psi_i^{V_\beta}(\overline{a}_i)

Read as: psi subscript i of the tuple bar over a subscript i if and only if psi subscript i superscript capital V subscript beta of the tuple bar over a subscript i

Means: psi subscript i of the tuple bar over a subscript i if and only if psi subscript i superscript capital V subscript beta of the tuple bar over a subscript i

1 occurrence in this chapter

Equation form expr-7b2cffc10049e068

a¯i∈Vβ\overline{a}_i \in V_\beta

Read as: the tuple bar over a subscript i belongs to capital V subscript beta

Means: the tuple bar over a subscript i belongs to capital V subscript beta

3 occurrences in this chapter

Equation form expr-7f2315ed2fa0ec08

Z\Z

Read as: set theory Z

Means: set theory Z

19 occurrences in this chapter

Equation form expr-7facc807815fed40

a¯1\overline{a}_1

Read as: the tuple bar over a subscript one

Means: the tuple bar over a subscript one

2 occurrences in this chapter

Equation form expr-80c7dbc4dcbb466a

(∀x∈A)(∃!y∈S)ϕ(x,y)(\forall x \in A)(\lexists![y \in S])\phi(x, y)

Read as: for every x in capital A, there exists exactly one y belongs to capital S, phi of x and y

Means: for every x in capital A, there exists exactly one y belongs to capital S, phi of x and y

1 occurrence in this chapter

Equation form expr-829732fc395bcda1

(∀x∈M)(\lforall[x \in M])

Read as: open scope, for every x belongs to capital M, close scope

Means: open scope, for every x belongs to capital M, close scope

1 occurrence in this chapter

Equation form expr-8423731058c96b5b

ϕVβ\phi^{V_\beta}

Read as: phi superscript capital V subscript beta

Means: phi superscript capital V subscript beta

1 occurrence in this chapter

Equation form expr-8428f44181dfe179

(∀x¯∈S)(ϕ↔ϕS)(\forall \overline{x} \in S)(\phi \liff \phi^S)

Read as: for every the tuple bar over x in capital S, open scope, phi if and only if phi superscript capital S, close scope

Means: for every the tuple bar over x in capital S, open scope, phi if and only if phi superscript capital S, close scope

1 occurrence in this chapter

Equation form expr-84ea45186761036f

β>α\beta > \alpha

Read as: beta is greater than alpha

Means: beta is greater than alpha

2 occurrences in this chapter

Equation form expr-88e8b56713ff57f6

A∈SA \in S

Read as: capital A belongs to capital S

Means: capital A belongs to capital S

1 occurrence in this chapter

Equation form expr-8951b18aca64248c

ω+ω\omega+\omega

Read as: omega plus omega

Means: omega plus omega

8 occurrences in this chapter

Equation form expr-8c0785670c74cfb3

SnS_n

Read as: capital S subscript n

Means: capital S subscript n

1 occurrence in this chapter

Equation form expr-8d19b772af5f0071

ϕi(v¯i,x)\phi_i(\overline{v}_i, x)

Read as: phi subscript i of the tuple bar over v subscript i and x

Means: phi subscript i of the tuple bar over v subscript i and x

1 occurrence in this chapter

Equation form expr-8de0b3c47f112c59

SS

Read as: capital S

Means: capital S

14 occurrences in this chapter

Equation form expr-9108a4bd94fbe6b5

ψ1,…,ψk\psi_1, \ldots, \psi_k

Read as: psi subscript one, then psi subscript k, and so on

Means: psi subscript one, then psi subscript k, and so on

1 occurrence in this chapter

Equation form expr-92becdaca2dd0d19

VαV_\alpha

Read as: capital V subscript alpha

Means: capital V subscript alpha

5 occurrences in this chapter

Equation form expr-94aec4edff8c2a34

{y:(∃x∈A)ϕ(x,y}\Setabs{y}{(\exists x \in A)\phi(x,y}

Read as: the set of y such that there exists x in capital A for which phi holds of x and y; the source is missing the final set-builder brace

Means: the set of y such that there exists x in capital A for which phi holds of x and y; the source is missing the final set-builder brace

1 occurrence in this chapter

Equation form expr-9a545057d452f075

z=1z=1

Read as: z equals one

Means: z equals one

1 occurrence in this chapter

Equation form expr-9add68a83ddacd04

ZF⊢ϕVω+ω\ZF \Proves \phi^{V_{\omega+\omega}}

Read as: set theory Z F derives phi superscript capital V subscript omega plus omega

Means: set theory Z F derives phi superscript capital V subscript omega plus omega

1 occurrence in this chapter

Equation form expr-9bdc2aeb25bc76b1

∃xϕi(a¯i,x)→(∃x∈Vβ)ϕi(a¯i,x)\exists x\phi_i(\overline{a}_i, x) \rightarrow (\exists x \in V_\beta) \phi_i(\overline{a}_i, x)

Read as: there exists x, phi subscript i of the tuple bar over a subscript i and x implies there exists x in capital V subscript beta, phi subscript i of the tuple bar over a subscript i and x

Means: there exists x, phi subscript i of the tuple bar over a subscript i and x implies there exists x in capital V subscript beta, phi subscript i of the tuple bar over a subscript i and x

1 occurrence in this chapter

Equation form expr-9c3245dfb4ac54c1

σ\sigma

Read as: sigma

Means: sigma

2 occurrences in this chapter

Equation form expr-a1fce4363854ff88

yy

Read as: y

Means: y

1 occurrence in this chapter

Equation form expr-a3466b8f618426a2

T\Th{T}

Read as: the theory of capital T

Means: the theory of capital T

11 occurrences in this chapter

Equation form expr-a4d1a4af3ae3c5f4

T⊢∃β(θ↔θVβ)\Th{T} \Proves \exists \beta(\theta \liff \theta^{V_\beta})

Read as: the theory of capital T there exists beta, open scope, theta if and only if theta superscript capital V subscript beta, close scope is derivable

Means: the theory of capital T there exists beta, open scope, theta if and only if theta superscript capital V subscript beta, close scope is derivable

1 occurrence in this chapter

Equation form expr-aad46a3031255dc3

(∀z,x¯∈S)(ϕ↔ϕS)i.e. (∀z,x¯∈S)(((z=0∧ψ)∨(z=1∧χ))↔(((z=0∧ψ)∨(z=1∧χ))S)i.e. (∀z,x¯∈S)(((z=0∧ψ)∨(z=1∧χ))↔(((z=0∧ψS)∨(z=1∧χS)))i.e. (∀x¯∈S)((ψ↔ψS)∧(χ↔χS))(\forall z, \overline{x} \in S)(&\phi \liff \phi^S)\\ \text{i.e. }(\forall z, \overline{x} \in S)(&((z = 0 \land \psi) \lor (z = 1 \land \chi)) \liff {}\\ &\phantom{(}((z = 0 \land \psi) \lor (z = 1 \land \chi))^S)\\ \text{i.e. }(\forall z, \overline{x} \in S)(&((z = 0 \land \psi) \lor (z = 1 \land \chi))\liff {}\\ &\phantom{(}((z = 0 \land \psi^S) \lor (z = 1 \land \chi^S)))\\ \text{i.e. }(\forall \overline{x} \in S)(&(\psi \liff \psi^S) \land (\chi \liff \chi^S))

Read as: source-ordered equivalence. For every z and tuple x in capital S, phi is equivalent to phi relativized to capital S. Expanding phi, the disjunction of z equals zero and psi with z equals one and chi is equivalent to its relativization. Relativizing the two cases yields the corresponding disjunction with psi and chi each relativized to capital S. Therefore, for every tuple x in capital S, psi is equivalent to relativized psi and chi is equivalent to relativized chi. End equivalence

Means: source-ordered equivalence. For every z and tuple x in capital S, phi is equivalent to phi relativized to capital S. Expanding phi, the disjunction of z equals zero and psi with z equals one and chi is equivalent to its relativization. Relativizing the two cases yields the corresponding disjunction with psi and chi each relativized to capital S. Therefore, for every tuple x in capital S, psi is equivalent to relativized psi and chi is equivalent to relativized chi. End equivalence

1 occurrence in this chapter

Equation form expr-b161bf60d6b2d19d

∃\exists

Read as: the existential quantifier

Means: the existential quantifier

1 occurrence in this chapter

Equation form expr-b3954b8b0b6abbe7

∀s(s∈z↔∀tt∉s)\forall s(s \in z \liff \forall t\, t \notin s)

Read as: for every s, open scope, s belongs to z if and only if for every t, t does not belong to s, close scope

Means: for every s, open scope, s belongs to z if and only if for every t, t does not belong to s, close scope

1 occurrence in this chapter

Equation form expr-b543b0f931ac4935

(z=0∧ψ)∨(z=1∧χ)(z = 0 \land \psi) \lor (z = 1 \land \chi)

Read as: open scope, z equals zero and psi, close scope or open scope, z equals one and chi, close scope

Means: open scope, z equals zero and psi, close scope or open scope, z equals one and chi, close scope

1 occurrence in this chapter

Equation form expr-ba4afe670821a805

Zr\Zr

Read as: the set theory Z sub r

Means: the set theory Z sub r

11 occurrences in this chapter

Equation form expr-bc57724b0c8fb691

τ(x)\tau(x)

Read as: tau of x

Means: tau of x

3 occurrences in this chapter

Equation form expr-bd402abbe0e586d1

ϕ∈ZF\phi \in \ZF

Read as: phi belongs to set theory Z F

Means: phi belongs to set theory Z F

1 occurrence in this chapter

Equation form expr-bf29875ede8f3e62

ψ,χ\psi, \chi

Read as: psi, then chi

Means: psi, then chi

1 occurrence in this chapter

Equation form expr-bf36a274460265c3

∀tt∉z\forall t\, t \notin z

Read as: for every t, t does not belong to z

Means: for every t, t does not belong to z

1 occurrence in this chapter

Equation form expr-c534d66803566493

a¯k∈Sn\overline{a}_k \in S_n

Read as: the tuple bar over a subscript k belongs to capital S subscript n

Means: the tuple bar over a subscript k belongs to capital S subscript n

1 occurrence in this chapter

Equation form expr-c6ae939512e1cac5

{y∈V:(∃x∈A)σ(x)=y}={y:(∃x∈A)ϕ(x,y)}\Setabs{y \in V}{(\exists x \in A)\sigma(x) = y} = \Setabs{y}{(\exists x \in A)\phi(x,y)}

Read as: the set of y belongs to capital V such that there exists x in capital A, sigma of x equals y, equals the set of y such that there exists x in capital A, phi of x and y

Means: the set of y belongs to capital V such that there exists x in capital A, sigma of x equals y, equals the set of y such that there exists x in capital A, phi of x and y

1 occurrence in this chapter

Equation form expr-cd541ce5e6d7d8bc

T⊢∀X(ψ(X)→σX)\Th{T} \Proves \forall X(\psi(X) \lif \sigma^X)

Read as: the theory of capital T for every capital X, open scope, psi of capital X implies sigma superscript capital X, close scope is derivable

Means: the theory of capital T for every capital X, open scope, psi of capital X implies sigma superscript capital X, close scope is derivable

2 occurrences in this chapter

Equation form expr-cd639ace2f2c87d0

τ[A]≼A\cardle{\funimage{\tau}{A}}{A}

Read as: the image of capital A under tau has cardinality at most that of capital A

Means: the image of capital A under tau has cardinality at most that of capital A

1 occurrence in this chapter

Equation form expr-cdb0f98000dfa727

LT\LT

Read as: the language of set theory

Means: the language of set theory

12 occurrences in this chapter

Equation form expr-cdf09e7d43eab2eb

S∈τ(x)S \in \tau(x)

Read as: capital S belongs to tau of x

Means: capital S belongs to tau of x

1 occurrence in this chapter

Equation form expr-d3639e0b107807d0

S0=Vα+1Sn+1=Sn∪⋃{μ(a¯1,…,a¯k):a¯1,…,a¯k∈Sn}S=⋃m<ωSn.S_0 & = V_{\alpha+1}\\ S_{n+1} & = S_n \cup \bigcup \Setabs{\mu(\overline{a}_1, \ldots, \overline{a}_k)} {\overline{a}_1, \ldots, \overline{a}_k \in S_n} \\ S &= \bigcup_{m < \omega} S_n.

Read as: Source-ordered display. capital S subscript zero equals capital V subscript alpha plus one. Then, capital S subscript n plus one equals capital S subscript n union the union of the set of mu of the tuple bar over a subscript one and , and so on and the tuple bar over a subscript k such that the tuple bar over a subscript one the tuple bar over a subscript k, and so on belongs to capital S subscript n. Then, capital S equals the union over m is less than omega of capital S subscript n. End display

Means: Source-ordered display. capital S subscript zero equals capital V subscript alpha plus one. Then, capital S subscript n plus one equals capital S subscript n union the union of the set of mu of the tuple bar over a subscript one and , and so on and the tuple bar over a subscript k such that the tuple bar over a subscript one the tuple bar over a subscript k, and so on belongs to capital S subscript n. Then, capital S equals the union over m is less than omega of capital S subscript n. End display

1 occurrence in this chapter

Equation form expr-d4a0cfb83b24d498

∀x\lforall[x]

Read as: for every x

Means: for every x

1 occurrence in this chapter

Equation form expr-d903bee0b763c7d3

T⊢ϕ\Th{T} \Proves \phi

Read as: the theory of capital T phi is derivable

Means: the theory of capital T phi is derivable

1 occurrence in this chapter

Equation form expr-de5a6f78116eca62

VV

Read as: capital V

Means: capital V

2 occurrences in this chapter

Equation form expr-deb7ee55dd7f1328

ϕ(x,y)\phi(x,y)

Read as: phi of x and y

Means: phi of x and y

1 occurrence in this chapter

Equation form expr-def6d4dc9f9a068a

ϕ∈Z\phi \in \Z

Read as: phi belongs to set theory Z

Means: phi belongs to set theory Z

1 occurrence in this chapter

Equation form expr-e142bcfc7c7cbda9

akna_{k_n}

Read as: a subscript k subscript n

Means: a subscript k subscript n

1 occurrence in this chapter

Equation form expr-e256352fc3b41300

a¯k∈S\overline{a}_k \in S

Read as: the tuple bar over a subscript k belongs to capital S

Means: the tuple bar over a subscript k belongs to capital S

1 occurrence in this chapter

Equation form expr-e37f6f5735d00533

1≤i≤k1 \leq i \leq k

Read as: one is less than or equal to i is less than or equal to k

Means: one is less than or equal to i is less than or equal to k

5 occurrences in this chapter

Equation form expr-e58026a6b0118e3d

τ[A]={τ(x):x∈A}\funimage{\tau}{A} = \Setabs{\tau(x)}{x \in A}

Read as: the image of capital A under tau equals the set of tau of x such that x belongs to capital A

Means: the image of capital A under tau equals the set of tau of x such that x belongs to capital A

1 occurrence in this chapter

Equation form expr-e69001e7f79be850

θX∧X is transitive∧(∀Y∈X)(Y is transitive→¬θY)\theta^X \land X\text{ is transitive} \land (\forall Y \in X)(Y\text{ is transitive}\lif \lnot \theta^{Y})

Read as: theta superscript capital X and capital X, is transitive and for every capital Y in capital X, open scope, capital Y, is transitive implies not theta superscript capital Y, close scope

Means: theta superscript capital X and capital X, is transitive and for every capital Y in capital X, open scope, capital Y, is transitive implies not theta superscript capital Y, close scope

1 occurrence in this chapter

Equation form expr-e81e6bdb4f8cebeb

(∀x¯∈S)((ψ↔ψS)∧(χ↔χS))(\forall \overline{x} \in S)((\psi \liff \psi^S) \land (\chi \liff \chi^S))

Read as: for every the tuple bar over x in capital S, open scope, open scope, psi if and only if psi superscript capital S, close scope and open scope, chi if and only if chi superscript capital S, close scope, close scope

Means: for every the tuple bar over x in capital S, open scope, open scope, psi if and only if psi superscript capital S, close scope and open scope, chi if and only if chi superscript capital S, close scope, close scope

1 occurrence in this chapter

Equation form expr-ea427caa6a6950cf

z=0z = 0

Read as: z equals zero

Means: z equals zero

2 occurrences in this chapter

Equation form expr-ebc69219e42efcaf

Z⊢∀X(X is transitive→σX)\Z \Proves \forall X(X\text{ is transitive} \lif \sigma^X)

Read as: set theory Z derives for every capital X, open scope, capital X, is transitive implies sigma superscript capital X, close scope

Means: set theory Z derives for every capital X, open scope, capital X, is transitive implies sigma superscript capital X, close scope

1 occurrence in this chapter

Equation form expr-ef11f8fb38499d99

ak1a_{k_1}

Read as: a subscript k subscript one

Means: a subscript k subscript one

1 occurrence in this chapter

Equation form expr-f301f970eb09c336

ψ\psi

Read as: psi

Means: psi

2 occurrences in this chapter

Equation form expr-f79bf57332648c24

μ(a¯1,…,a¯k)\mu(\overline{a}_1, \ldots, \overline{a}_k)

Read as: mu of the tuple bar over a subscript one and , and so on and the tuple bar over a subscript k

Means: mu of the tuple bar over a subscript one and , and so on and the tuple bar over a subscript k

2 occurrences in this chapter

Equation form expr-f8b97ceb1fef4fc6

τ(x)∈V\tau(x) \in V

Read as: tau of x belongs to capital V

Means: tau of x belongs to capital V

1 occurrence in this chapter

Equation form expr-fcab22ac1563c5e2

(∃xψi(a¯i,x))Vβ iff (∃x∈Vβ)ψiVβ(a¯i,x)by definition iff (∃x∈Vβ)ψi(a¯i,x)by hypothesis iff ∃xψi(a¯i,x)by equation star in chapter Replacement(\exists x \psi_i(\overline{a}_i, x))^{V_\beta} &\text{ iff } (\exists x \in V_\beta)\psi_i^{V_\beta}(\overline{a}_i, x) &&\text{by definition}\\ &\text{ iff } (\exists x \in V_\beta)\psi_i(\overline{a}_i, x) &&\text{by hypothesis}\\ &\text{ iff } \exists x \psi_i(\overline{a}_i, x) &&\text{by \eqref{reflectionnicelybehaved}}

Read as: Source-ordered display. there exists x applied to psi subscript i of the tuple bar over a subscript i and x, the empty expression superscript capital V subscript beta if and only if there exists x in capital V subscript beta, psi subscript i superscript capital V subscript beta of the tuple bar over a subscript i and x, by definition. Then, if and only if there exists x in capital V subscript beta, psi subscript i of the tuple bar over a subscript i and x, by hypothesis. Then, if and only if there exists x, psi subscript i of the tuple bar over a subscript i and x, by equation star in chapter Replacement. End display

Means: Source-ordered display. there exists x applied to psi subscript i of the tuple bar over a subscript i and x, the empty expression superscript capital V subscript beta if and only if there exists x in capital V subscript beta, psi subscript i superscript capital V subscript beta of the tuple bar over a subscript i and x, by definition. Then, if and only if there exists x in capital V subscript beta, psi subscript i of the tuple bar over a subscript i and x, by hypothesis. Then, if and only if there exists x, psi subscript i of the tuple bar over a subscript i and x, by equation star in chapter Replacement. End display

1 occurrence in this chapter

Equation form expr-ffb2d087802a8b1c

T⊢σ\Th{T} \Proves \sigma

Read as: the theory of capital T sigma is derivable

Means: the theory of capital T sigma is derivable

2 occurrences in this chapter

Definition one in this chapter

This source definition contains, in source order: capital M; then phi; then phi superscript capital M; then phi; then capital M; then there exists x; then open scope, there exists x belongs to capital M, close scope; then for every x; then open scope, for every x belongs to capital M, close scope. 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. n is strictly smaller than m if and only if either, n is less than m and m minus n, is even,. Then, or n is even and m is odd.. 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 F. The complete surrounding source prose remains in the continuous listener stream. The exercise is preserved as stated and no solution is supplied.

Source

Theorem: Reflection Schema

This source theorem contains, in source order: phi; then for every ordinal alpha there exists an ordinal beta greater than alpha such that, for all x subscript one through x subscript n in capital V subscript beta, phi holds of those objects if and only if its relativization to capital V subscript beta holds of them. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma one in this chapter

This source lemma contains, in source order: one is less than or equal to i is less than or equal to k; then phi subscript i of the tuple bar over v subscript i and x; then alpha; then beta is greater than alpha; then the tuple bar over a subscript one the tuple bar over a subscript k, and so on belongs to capital V subscript beta; then one is less than or equal to i is less than or equal to k; then there exists x, phi subscript i of the tuple bar over a subscript i and x implies there exists x in capital V subscript beta, phi subscript i of the tuple bar over a subscript i and x. 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. capital S subscript zero equals capital V subscript alpha plus one. Then, capital S subscript n plus one equals capital S subscript n union the union of the set of mu of the tuple bar over a subscript one and , and so on and the tuple bar over a subscript k such that the tuple bar over a subscript one the tuple bar over a subscript k, and so on belongs to capital S subscript n. Then, capital S equals the union over m is less than omega of capital S subscript n. 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: there exists x, psi subscript i of the tuple bar over a subscript i and x implies there exists x in capital V subscript beta, psi subscript i of the tuple bar over a subscript i and x applied to tag applied to star. 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. there exists x applied to psi subscript i of the tuple bar over a subscript i and x, the empty expression superscript capital V subscript beta if and only if there exists x in capital V subscript beta, psi subscript i superscript capital V subscript beta of the tuple bar over a subscript i and x, by definition. Then, if and only if there exists x in capital V subscript beta, psi subscript i of the tuple bar over a subscript i and x, by hypothesis. Then, if and only if there exists x, psi subscript i of the tuple bar over a subscript i and x, by the referenced equation. 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: phi; then capital S; then zero; then one; then phi; then capital S; then for every the tuple bar over x in capital S, open scope, phi if and only if phi superscript capital S, close scope. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma: in set theory Z plus Weak-Reflection.

This source lemma contains, in source order: set theory Z plus Weak-Reflection; then psi, then chi; then capital S; then zero; then one; then capital S; then for every the tuple bar over x in capital S, open scope, open scope, psi if and only if psi superscript capital S, close scope and open scope, chi if and only if chi superscript capital S, close scope, close scope. 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 equivalence. For every z and tuple x in capital S, phi is equivalent to phi relativized to capital S. Expanding phi, the disjunction of z equals zero and psi with z equals one and chi is equivalent to its relativization. Relativizing the two cases yields the corresponding disjunction with psi and chi each relativized to capital S. Therefore, for every tuple x in capital S, psi is equivalent to relativized psi and chi is equivalent to relativized chi. End equivalence. The complete surrounding source prose remains in the continuous listener stream.

Source

Theorem: in set theory Z + Weak-Reflection

This source theorem contains, in source order: set theory Z; then phi of v and w; then capital A; then for every x in capital A, there exists exactly one y, phi of x and y; then the set of y such that there exists x in capital A for which phi holds of x and y; the source is missing the final set-builder brace. 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. psi, is, open scope, phi of x and z and capital A equals capital A, close scope. Then, chi, is, there exists y, phi of x and y. End display. 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: two absoluteness conditions. First, for all x and z in capital S, phi of x and z holds if and only if its relativization to capital S holds. Second, for every x in capital S, there exists a y satisfying phi of x and y if and only if there exists such a y in capital S satisfying the relativized formula. End conditions. 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: for every x in capital A, there exists a y such that phi holds of x and y if and only if there exists such a y in capital S. The complete surrounding source prose remains in the continuous listener stream.

Source

Theorem three in this chapter

This source theorem contains, in source order: set theory Z F; then the theory of capital T; then the theory of capital T set theory Z F is derivable; then the theory of capital T; then the membership relation; then the theory of capital T set theory Z F is derivable; then the theory of capital T phi is derivable; then phi belongs to set theory Z F. The complete surrounding source prose remains in the continuous listener stream.

Source

Display math nine in this chapter

This source display math contains, in source order: the theory of capital T there exists capital M, psi of capital M applied to tag applied to star is derivable. The complete surrounding source prose remains in the continuous listener stream.

Source

Display math ten in this chapter

This source display math contains, in source order: source-ordered derivation in theory capital T. First, capital T derives that every capital X satisfying psi also satisfies, inside capital X, that some capital N satisfies psi. Using this and the referenced equation, capital T derives a capital M satisfying psi in which some capital N satisfies psi. In particular, capital T derives such a capital M containing a capital N that is internally transitive and satisfies theta through the displayed relativizations. Finally, by transitivity, capital T derives a capital M satisfying psi that contains a transitive capital N satisfying theta. End derivation. The complete surrounding source prose remains in the continuous listener stream.

Source

Proposition one in this chapter

This source proposition contains, in source order: the theory of capital T; then set theory Z; then the theory of capital T set theory Z F is derivable; then the theory of capital T. The complete surrounding source prose remains in the continuous listener stream.

Source

Exercise two in this chapter

This source exercise contains, in source order: sigma; then set theory Z derives for every capital X, open scope, capital X, is transitive implies sigma superscript capital X, close scope. The complete surrounding source prose remains in the continuous listener stream. The exercise is preserved as stated and no solution is supplied.

Source

Exercise three in this chapter

This source exercise contains, in source order: phi belongs to set theory Z; then set theory Z F derives phi superscript capital V subscript omega plus omega. The complete surrounding source prose remains in the continuous listener stream. The exercise is preserved as stated and no solution is supplied.

Source

Exercise four 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

Cross-reference reference-001349

chapter “Ordinals”

Source occurrence

Cross-reference reference-001350

chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001351

section “ℵ-Fixed Points” in chapter “Cardinal Arithmetic”

Source occurrence

Cross-reference reference-001352

Penelope Maddy

Source occurrence

Cross-reference reference-001353

1988

Source occurrence

Cross-reference reference-001354

1988

Source occurrence

Cross-reference reference-001355

corollary two in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001356

section “Replacement” in chapter “Ordinals”

Source occurrence

Cross-reference reference-001357

theorem five in chapter “Ordinals”

Source occurrence

Cross-reference reference-001358

section “The General Idea of an Ordinal” in chapter “Ordinals”

Source occurrence

Cross-reference reference-001359

theorem five in chapter “Ordinals”

Source occurrence

Cross-reference reference-001360

theorem five in chapter “Ordinals”

Source occurrence

Cross-reference reference-001361

section “ℵ-Fixed Points” in chapter “Cardinal Arithmetic”

Source occurrence

Cross-reference reference-001362

(George Boolos, 1971, 229)

Source occurrence

Cross-reference reference-001363

Richard Montague (1965)

Source occurrence

Cross-reference reference-001364

Dana Scott (1974)

Source occurrence

Cross-reference reference-001365

Michael Potter (2004)

Source occurrence

Cross-reference reference-001366

Tim Button (2021)

Source occurrence

Cross-reference reference-001367

Michael Potter 2004

Source occurrence

Cross-reference reference-001368

Tim Button 2021

Source occurrence

Cross-reference reference-001369

Michael Potter 2004, §13.2

Source occurrence

Cross-reference reference-001370

chapter “The Iterative Conception”

Source occurrence

Cross-reference reference-001371

chapter “Steps towards Z”

Source occurrence

Cross-reference reference-001372

(1989, p. 19)

Source occurrence

Cross-reference reference-001373

section “Russell's Paradox (again)” in chapter “The Iterative Conception”

Source occurrence

Cross-reference reference-001374

section “The General Idea of an Ordinal” in chapter “Ordinals”

Source occurrence

Cross-reference reference-001375

theorem five in chapter “Ordinals”

Source occurrence

Cross-reference reference-001376

theorem five in chapter “Ordinals”

Source occurrence

Cross-reference reference-001377

theorem five in chapter “Ordinals”

Source occurrence

Cross-reference reference-001378

Michael Potter (2004), §13.2

Source occurrence

Cross-reference reference-001379

section “Appendix: Finite axiomatizability” in chapter “Replacement”

Source occurrence

Cross-reference reference-001380

Joseph R. Shoenfield (1977)

Source occurrence

Cross-reference reference-001381

Michael Potter (2004), p. 223

Source occurrence

Cross-reference reference-001382

Joseph R. Shoenfield

Source occurrence

Cross-reference reference-001383

Luca Incurvati (2020), 90–5

Source occurrence

Cross-reference reference-001384

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

Source occurrence

Cross-reference reference-001385

definition one in chapter “Replacement”

Source occurrence

Cross-reference reference-001386

Richard Montague (1961)

Source occurrence

Cross-reference reference-001387

Azriel Lévy (1960)

Source occurrence

Cross-reference reference-001388

section “Appendix: Results surrounding Replacement” in chapter “Replacement”

Source occurrence

Cross-reference reference-001389

Luca Incurvati (2020), 95–100

Source occurrence

Cross-reference reference-001390

theorem “Reflection Schema” in chapter “Replacement”

Source occurrence

Cross-reference reference-001391

lemma one in chapter “Replacement”

Source occurrence

Cross-reference reference-001392

equation (*) in chapter “Replacement”

Source occurrence

Cross-reference reference-001393

Richard Montague (1961)

Source occurrence

Cross-reference reference-001394

Azriel Lévy (1960)

Source occurrence

Cross-reference reference-001395

section “Z and ZF: A Milestone” in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001396

Azriel Lévy (1960), first part of Theorem 2

Source occurrence

Cross-reference reference-001397

Azriel Lévy (1960), Theorem 6

Source occurrence

Cross-reference reference-001398

the lemma on reflect

Source occurrence

Cross-reference reference-001399

Richard Montague (1961)

Source occurrence

Cross-reference reference-001400

theorem “Reflection Schema” in chapter “Replacement”

Source occurrence

Cross-reference reference-001401

equation (*) in chapter “Replacement”

Source occurrence

Cross-reference reference-001402

equation (*) in chapter “Replacement”

Source occurrence

Cross-reference reference-001403

Michael Potter (2004), 223

Source occurrence

Cross-reference reference-001404

theorem three in chapter “Replacement”

Source occurrence

Cross-reference reference-001405

theorem three in chapter “Replacement”

Source occurrence

Cross-reference reference-001406

theorem three in chapter “Replacement”

Source occurrence

Cross-reference reference-001407

theorem three in chapter “Replacement”

Source occurrence

Cross-reference reference-001408

theorem three in chapter “Replacement”

Source occurrence

Cross-reference reference-001409

proposition one in chapter “Replacement”

Source occurrence

Cross-reference reference-001410

theorem three in chapter “Replacement”

Source occurrence

Cross-reference reference-001411

proposition one in chapter “Replacement”

Source occurrence

Cross-reference reference-001412

section “Replacement and “Absolute Infinity”” in chapter “Replacement”

Source occurrence

Cross-reference reference-001413

theorem five in chapter “Ordinals”

Source occurrence

Source disclosures