Set Theory

The Iterative Conception

Equation form expr-017f6e18d3158b7f

′^\prime

Read as: prime

Means: prime

7 occurrences in this chapter

Equation form expr-028b89b53554c1a1

∃s∀a(a∈s↔Fa)\exists s\forall a(a \in s \liff Fa)

Read as: there exists a set s such that for every object a, a is a member of s if and only if concept capital F applies to a

Means: there exists a set s such that for every object a, a is a member of s if and only if concept capital F applies to a

1 occurrence in this chapter

Equation form expr-0a29b69cd680cf7a

0_0

Read as: subscript zero

Means: subscript zero

2 occurrences in this chapter

Equation form expr-10ce0200b450ea95

ϕ\phi

Read as: phi

Means: phi

3 occurrences in this chapter

Equation form expr-118b7990c7da2ffb

∀a(a∈ϵxF(x)↔Fa)\forall a (a \in \fregeext{x}{F(x)} \liff Fa)

Read as: for every object a, a is a member of the extension of concept capital F if and only if capital F applies to a

Means: for every object a, a is a member of the extension of concept capital F if and only if capital F applies to a

1 occurrence in this chapter

Equation form expr-1600e169f8795a45

{x:x∉x}\Setabs{x}{x \notin x}

Read as: the set of all x such that x is not a member of itself

Means: the set of all x such that x is not a member of itself

2 occurrences in this chapter

Equation form expr-2821a779042a3e3b

a∈ba \in b

Read as: a is a member of b

Means: a is a member of b

1 occurrence in this chapter

Equation form expr-333e0a1e27815d0c

GG

Read as: capital G

Means: capital G

1 occurrence in this chapter

Equation form expr-39e0eb7678e979f8

′^{\prime}

Read as: prime

Means: prime

1 occurrence in this chapter

Equation form expr-3b7498508431f01b

∀x(Fx↔x∉x)\forall x(Fx \liff x \notin x)

Read as: for every object x, concept capital F applies to x if and only if x is not a member of itself

Means: for every object x, concept capital F applies to x if and only if x is not a member of itself

1 occurrence in this chapter

Equation form expr-3de0f9d0c9350dae

∀A∀B(∀x(x∈A↔x∈B)→A=B)\lforall[A][\lforall[B][(\lforall[x][(x \in A \liff x \in B)] \lif \eq[A][B])]]

Read as: for every set capital A and every set capital B, if every object x is a member of capital A if and only if x is a member of capital B, then capital A equals capital B

Means: for every set capital A and every set capital B, if every object x is a member of capital A if and only if x is a member of capital B, then capital A equals capital B

1 occurrence in this chapter

Equation form expr-3e23e8160039594a

bb

Read as: b

Means: b

1 occurrence in this chapter

Equation form expr-3fdce894627ec6f6

R∉RR \notin R

Read as: capital R is not a member of itself

Means: capital R is not a member of itself

1 occurrence in this chapter

Equation form expr-46202d6671797c31

a∈b=df∃G(b=ϵxG(x)∧Ga)a \in b =_\text{df} \exists G(b = \fregeext{x}{G(x)} \land Ga)

Read as: object a is a member of object b is defined to mean that there exists a concept capital G such that b is the extension of G and G applies to a

Means: object a is a member of object b is defined to mean that there exists a concept capital G such that b is the extension of G and G applies to a

1 occurrence in this chapter

Equation form expr-467490627ccbada0

∃G(ϵxF(x)=ϵxG(x)∧Ga)\exists G(\fregeext{x}{F(x)} = \fregeext{x}{G(x)} \land Ga)

Read as: there exists a concept capital G such that the extension of concept capital F equals the extension of concept capital G and G applies to object a

Means: there exists a concept capital G such that the extension of concept capital F equals the extension of concept capital G and G applies to object a

1 occurrence in this chapter

Equation form expr-46c9e22099ee4bfe

1_1

Read as: subscript one

Means: subscript one

2 occurrences in this chapter

Equation form expr-4715061c5c91a712

RS={x:x∉x and x was available before S}R_S = \Setabs{x}{x \notin x \text{ and $x$ was available before $S$}}

Read as: the stage indexed Russell set capital R subscript capital S equals the set of all x such that x is not a member of itself and x was available before stage capital S

Means: the stage indexed Russell set capital R subscript capital S equals the set of all x such that x is not a member of itself and x was available before stage capital S

1 occurrence in this chapter

Equation form expr-4e07408562bedb8b

33

Read as: three

Means: three

1 occurrence in this chapter

Equation form expr-553dbacbb7e77201

FaFa

Read as: a falls under capital F

Means: a falls under capital F

1 occurrence in this chapter

Equation form expr-559aead08264d579

AA

Read as: capital A

Means: capital A

3 occurrences in this chapter

Equation form expr-5d6cde2ac5d5e460

ϵxF(x)\fregeext{x}{F(x)}

Read as: the extension of concept capital F

Means: the extension of concept capital F

2 occurrences in this chapter

Equation form expr-5feceb66ffc86f38

00

Read as: zero

Means: zero

1 occurrence in this chapter

Equation form expr-6b86b273ff34fce1

11

Read as: one

Means: one

1 occurrence in this chapter

Equation form expr-6fef795f063d3a72

∃G(∀x(Fx↔Gx)∧Ga)\exists G(\forall x(Fx \liff Gx) \land Ga)

Read as: there exists a concept capital G such that for every object x, capital F applies to x if and only if capital G applies to x, and capital G applies to object a

Means: there exists a concept capital G such that for every object x, capital F applies to x if and only if capital G applies to x, and capital G applies to object a

1 occurrence in this chapter

Equation form expr-77f8da811a896d04

∀F∀a(a∈ϵxF(x)↔Fa)\forall F \forall a(a \in \fregeext{x}{F(x)} \liff Fa)

Read as: for every concept capital F and every object a, a is a member of the extension of F if and only if F applies to a

Means: for every concept capital F and every object a, a is a member of the extension of F if and only if F applies to a

1 occurrence in this chapter

Equation form expr-80933e78f318f2a8

′′^{\prime\prime}

Read as: double prime

Means: double prime

1 occurrence in this chapter

Equation form expr-82d2ef88a99ce03d

4_4

Read as: subscript four

Means: subscript four

1 occurrence in this chapter

Equation form expr-8c2574892063f995

RR

Read as: capital R

Means: capital R

4 occurrences in this chapter

Equation form expr-8d2cacefc75ba038

∅\emptyset

Read as: the empty set

Means: the empty set

1 occurrence in this chapter

Equation form expr-8de0b3c47f112c59

SS

Read as: capital S

Means: capital S

8 occurrences in this chapter

Equation form expr-90b794d7c5784d8a

2_2

Read as: subscript two

Means: subscript two

2 occurrences in this chapter

Equation form expr-94ccaf63d8f17226

′′′^{\prime\prime\prime}

Read as: triple prime

Means: triple prime

1 occurrence in this chapter

Equation form expr-994ccdf6939990cd

R={x:x∉x}R = \Setabs{x}{x \notin x}

Read as: capital R equals the set of all x such that x is not a member of itself

Means: capital R equals the set of all x such that x is not a member of itself

2 occurrences in this chapter

Equation form expr-a523fd33164e889e

∀F∃s∀a(a∈s↔Fa)\forall F \exists s \forall a (a \in s \liff Fa)

Read as: for every concept capital F, there exists a set s such that for every object a, a is a member of s if and only if F applies to a

Means: for every concept capital F, there exists a set s such that for every object a, a is a member of s if and only if F applies to a

1 occurrence in this chapter

Equation form expr-a564c208a4edb80f

ϵxF(x)=ϵxG(x)↔∀x(Fx↔Gx)\fregeext{x}{F(x)} = \fregeext{x}{G(x)} \liff \forall x (Fx \liff Gx)

Read as: the extension of concept capital F equals the extension of concept capital G if and only if, for every object x, F applies to x if and only if G applies to x

Means: the extension of concept capital F equals the extension of concept capital G if and only if, for every object x, F applies to x if and only if G applies to x

1 occurrence in this chapter

Equation form expr-af6b0b2279da0a6e

{x:ϕ(x)}\Setabs{x}{\phi(x)}

Read as: the set of all x such that phi holds of x

Means: the set of all x such that phi holds of x

3 occurrences in this chapter

Equation form expr-b3ca23795ebd1e38

{∅}\{\emptyset\}

Read as: the singleton set containing the empty set

Means: the singleton set containing the empty set

1 occurrence in this chapter

Equation form expr-b4668eacf7fe6b79

A≈B\cardeq{A}{B}

Read as: capital A and capital B are equinumerous

Means: capital A and capital B are equinumerous

1 occurrence in this chapter

Equation form expr-c1e0d6344f9474d8

RSR_S

Read as: capital R subscript capital S

Means: capital R subscript capital S

1 occurrence in this chapter

Equation form expr-c76dfd53588b64aa

∃G\exists G

Read as: there exists a concept capital G

Means: there exists a concept capital G

1 occurrence in this chapter

Equation form expr-ca978112ca1bbdca

aa

Read as: a

Means: a

2 occurrences in this chapter

Equation form expr-cbcf8adbc34332ab

R={x:x∉x}R = \Setabs{x}{x\notin x}

Read as: capital R equals the set of all x such that x is not a member of itself

Means: capital R equals the set of all x such that x is not a member of itself

1 occurrence in this chapter

Equation form expr-d21b2c2986892de7

{∅,{∅}}\{\emptyset, \{\emptyset\}\}

Read as: the set containing the empty set and the singleton containing the empty set

Means: the set containing the empty set and the singleton containing the empty set

1 occurrence in this chapter

Equation form expr-d4735e3a265e16ee

22

Read as: two

Means: two

1 occurrence in this chapter

Equation form expr-d870c8f345891827

{{∅}}\{\{\emptyset\}\}

Read as: the singleton set containing the singleton set containing the empty set

Means: the singleton set containing the singleton set containing the empty set

1 occurrence in this chapter

Equation form expr-df7e70e5021544f4

BB

Read as: capital B

Means: capital B

3 occurrences in this chapter

Equation form expr-ebe091a297bc654f

R∈RR \in R

Read as: capital R is a member of itself

Means: capital R is a member of itself

1 occurrence in this chapter

Equation form expr-f677ab7c3c7a26f2

a∈ϵxF(x)a \in \fregeext{x}{F(x)}

Read as: object a is a member of the extension of concept capital F

Means: object a is a member of the extension of concept capital F

1 occurrence in this chapter

Equation form expr-f67ab10ad4e4c531

FF

Read as: capital F

Means: capital F

7 occurrences in this chapter

Equation form expr-f9550b866cb1a375

¬∃x∀z(Rzx↔¬Rzz).\lnot \exists x \forall z(Rzx \liff \lnot R zz).

Read as: it is not the case that there exists an object x such that, for every object z, capital R relates z to x if and only if capital R does not relate z to itself

Means: it is not the case that there exists an object x such that, for every object z, capital R relates z to x if and only if capital R does not relate z to itself

1 occurrence in this chapter

Equation form expr-fce0a176cac9bac4

′′′′^{\prime\prime\prime\prime}

Read as: quadruple prime

Means: quadruple prime

1 occurrence in this chapter

Equation form expr-fffdff4b07a9d973

3_3

Read as: subscript three

Means: subscript three

1 occurrence in this chapter

Axiom of Extensionality

Sets with exactly the same elements are the same set. Equivalently, for any sets capital A and capital B, agreement about membership of every object x implies that A equals B.

Source

Naive Comprehension principle

The source principle says that, for any formula phi, the set of all x for which phi holds exists. The following theorem explains why this unrestricted principle is inconsistent.

Source

Russell's Paradox

There is no Russell set whose members are exactly the sets that are not members of themselves. The source proof assumes such a set and derives that it is a member of itself exactly when it is not.

Source

Predicative Comprehension principle

For every formula phi that quantifies only over sets, the source proposes a new set prime of all x for which phi holds. The surrounding discussion keeps set primes outside the earlier set domain, then explains why further prime levels and a simple theory of types would be required.

Source

Cumulative iterative stage hierarchy

A pointed hierarchy begins at stage zero and widens through stages one to six. Horizontal stage boundaries mark the successive levels, and dotted extensions on both sides show that the hierarchy continues upward. The complete geometry and every printed stage label are linearized in the diagram.

Source

Stage hierarchy with urelements

The hierarchy has a flat base at stage zero rather than a point, representing the source's option of beginning with basic urelements. It widens through stages one to six, with dotted extensions showing continuation. The complete geometry and every printed stage label are linearized in the diagram.

Source

Membership in a Frege extension

For every concept capital F and every object a, object a is a member of the extension of F if and only if F applies to a. The proof uses Frege's definition of membership and Basic Law Five.

Source

Naive Comprehension from Basic Law Five

For every concept capital F there exists a set s whose members are exactly the objects to which F applies. The proof existentially generalizes from the preceding lemma about Frege extensions.

Source

Cross-reference reference-001174

the part on sets, functions, and relations

Source occurrence

Cross-reference reference-001175

the part on set theory

Source occurrence

Cross-reference reference-001176

the part on sets, functions, and relations

Source occurrence

Cross-reference reference-001177

the section Russell's Paradox

Source occurrence

Cross-reference reference-001178

the section on Frege's Basic Law Five

Source occurrence

Cross-reference reference-001179

Jean van Heijenoort (1967), pp. 124–8

Source occurrence

Cross-reference reference-001180

(Alfred North Whitehead and Bertrand Russell, 1910, p. 37)

Source occurrence

Cross-reference reference-001181

the section Russell's Paradox again

Source occurrence

Cross-reference reference-001182

Frank Plumpton Ramsey

Source occurrence

Cross-reference reference-001183

(Frank Plumpton Ramsey, 1925)

Source occurrence

Cross-reference reference-001184

Øystein Linnebo (2010)

Source occurrence

Cross-reference reference-001185

(Joseph R. Shoenfield, 1977, p. 323)

Source occurrence

Cross-reference reference-001186

the part on set theory

Source occurrence

Cross-reference reference-001187

the section The Story in More Detail

Source occurrence

Cross-reference reference-001188

the section The Cumulative Iterative Approach

Source occurrence

Cross-reference reference-001189

the chapter The Size of Sets

Source occurrence

Cross-reference reference-001190

Michael Potter 2004, pp. vi, 24, 50–1

Source occurrence

Cross-reference reference-001191

(Kenneth Kunen, 1980, p. 8)

Source occurrence

Cross-reference reference-001192

the section Russell's Paradox again

Source occurrence

Cross-reference reference-001193

Richard Kimberly Heck (2012), pp. 8–9

Source occurrence

Cross-reference reference-001194

the lemma on membership in Frege extensions

Source occurrence

Source disclosures

Source-generated case expression tr065-source-macro-0001

00

Read as: zero

Read in context source

Source-generated case expression tr065-source-macro-0002

11

Read as: one

Read in context source

Source-generated case expression tr065-source-macro-0003

22

Read as: two

Read in context source

Source-generated case expression tr065-source-macro-0004

33

Read as: three

Read in context source

Source-generated case expression tr065-source-macro-0005

44

Read as: four

Read in context source

Source-generated case expression tr065-source-macro-0006

55

Read as: five

Read in context source

Source-generated case expression tr065-source-macro-0007

66

Read as: six

Read in context source

Source-generated case expression tr065-source-macro-0008

00

Read as: zero

Read in context source

Source-generated case expression tr065-source-macro-0009

11

Read as: one

Read in context source

Source-generated case expression tr065-source-macro-0010

22

Read as: two

Read in context source

Source-generated case expression tr065-source-macro-0011

33

Read as: three

Read in context source

Source-generated case expression tr065-source-macro-0012

44

Read as: four

Read in context source

Source-generated case expression tr065-source-macro-0013

55

Read as: five

Read in context source

Source-generated case expression tr065-source-macro-0014

66

Read as: six

Read in context source

Ordered structures

Cumulative iterative stage hierarchy

Structure: diagram tikz.

Cumulative iterative hierarchy diagram. The solid outer boundary has a single pointed bottom at stage zero and widens upward. Six horizontal solid boundaries mark stages one through six. The printed labels from bottom to top are zero, then one, then two, then three, then four, then five, and finally six. Dotted extensions continue both sloping sides beyond stage six. No individual set, urelement, membership arrow, or quantity of objects is printed. End diagram.

Read the source-bound structure in context

Stage hierarchy with urelements

Structure: diagram tikz.

Hierarchy with urelements diagram. The solid outer boundary has a flat base at stage zero and widens upward. Six horizontal solid boundaries mark stages one through six. The printed labels from bottom to top are zero, then one, then two, then three, then four, then five, and finally six. Dotted extensions continue both sloping sides beyond stage six. No individual set, urelement, membership arrow, or quantity of objects is printed. End diagram.

Read the source-bound structure in context