aarjaneiro/planetmath huggingface dataset:
proof of Tukey’s lemma

proof of Tukey’s lemma

Let

S

be a set and

F

a set of subsets of

S

such that

F

is
of

finite character

. By Zorn’s lemma, it is enough to show that

F

is inductive. For that, it will be enough to show that if

(

F

i

)

i

∈

I

is a family of elements of

F

which is

totally ordered

by

inclusion

, then the union

U

of the

F

i

is an element of

F

as well (since

U

is an

upper bound

on the family

(

F

i

)

).
So, let

K

be a finite subset of

U

. Each element of

U

is in

F

i

for some

i

∈

I

. Since

K

is finite and
the

F

i

are totally ordered by inclusion, there is some

j

∈

I

such that all elements of

K

are in

F

j

. That is,

K

⊂

F

j

.
Since

F

is of finite character, we get

K

∈

F

, QED.

Title

proof of Tukey’s lemma

Canonical name

ProofOfTukeysLemma

Date of creation

2013-03-22 13:54:58

Last modified on

2013-03-22 13:54:58

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

4

Author

Koro (127)

Entry type

Proof

Classification

msc 03E25

Generated on Thu Feb 8 20:17:11 2018 by

LaTeXML
proof of Zermelo’s postulate

proof of Zermelo’s postulate

The following is a proof that the

axiom of choice

implies Zermelo’s

postulate

.

Proof.

Let

ℱ

be a

disjoint

family of

nonempty sets

. Let

f

:

ℱ

→

⋃

ℱ

be a

choice function

. Let

A

,

B

∈

ℱ

with

A

≠

B

. Since

ℱ

is a disjoint family of sets,

A

∩

B

=

∅

. Since

f

is a choice function,

f

⁢

(

A

)

∈

A

and

f

⁢

(

B

)

∈

B

. Thus,

f

⁢

(

A

)

∉

B

. Hence,

f

⁢

(

A

)

≠

f

⁢

(

B

)

. It follows that

f

is

injective

.

Let

C

=

{

f

⁢

(

B

)

∈

⋃

ℱ

:

B

∈

ℱ

}

. Then

C

is a set.

Let

A

∈

ℱ

. Since

f

is injective,

A

∩

C

=

{

f

⁢

(

A

)

}

.
∎

Title

proof of Zermelo’s postulate

Canonical name

ProofOfZermelosPostulate

Date of creation

2013-03-22 16:14:25

Last modified on

2013-03-22 16:14:25

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

9

Author

Wkbj79 (1863)

Entry type

Proof

Classification

msc 03E25

Generated on Thu Feb 8 20:17:14 2018 by

LaTeXML
proof of Zermelo’s well-ordering theorem

proof of Zermelo’s well-ordering theorem

Let

X

be any set and let

f

be a

choice function

on

𝒫

⁢

(

X

)

∖

{

∅

}

. Then define a

function

i

by

transfinite recursion

on the class of

ordinals

as follows:

i

⁢

(

β

)

=

f

⁢

(

X

-

⋃

γ

<

β

{

i

⁢

(

γ

)

}

)

⁢

unless

⁢

X

-

⋃

γ

<

β

{

i

⁢

(

γ

)

}

=

∅

⁢

or

⁢

i

⁢

(

γ

)

⁢

is undefined for some

⁢

γ

<

β

(the function is undefined if either of the unless clauses holds).

Thus

i

⁢

(

0

)

is just

f

⁢

(

X

)

(the least element of

X

), and

i

⁢

(

1

)

=

f

⁢

(

X

-

{

i

⁢

(

0

)

}

)

(the least element of

X

other than

i

⁢

(

0

)

).

Define by the

axiom of replacement

β

=

i

-

1

⁢

[

X

]

=

{

γ

∣

i

⁢

(

γ

)

=

x

⁢

for some

⁢

x

∈

X

}

. Since

β

is a set of ordinals, it cannot contain all the ordinals (by the

Burali-Forti paradox

).

Since the ordinals are well ordered, there is a least ordinal

α

not in

β

, and therefore

i

⁢

(

α

)

is undefined. It cannot be that the second unless clause holds (since

α

is the least such ordinal) so it must be that

X

-

⋃

γ

<

α

{

i

⁢

(

γ

)

}

=

∅

, and therefore for every

x

∈

X

there is some

γ

<

α

such that

i

⁢

(

γ

)

=

x

. Since we already know that

i

is

injective

, it is a

bijection

between

α

and

X

, and therefore establishes a

well-ordering

of

X

by

x

<

X

y

↔

i

-

1

⁢

(

x

)

<

i

-

1

⁢

(

y

)

.

The reverse is simple. If

C

is a set of

nonempty sets

, select any well

ordering

of

⋃

C

. Then a choice function is just

f

⁢

(

a

)

=

the least member of

a

under that well ordering

.

Title

proof of Zermelo’s well-ordering theorem

Canonical name

ProofOfZermelosWellorderingTheorem

Date of creation

2013-03-22 12:59:07

Last modified on

2013-03-22 12:59:07

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

9

Author

Henry (455)

Entry type

Proof

Classification

msc 03E25

Generated on Thu Feb 8 20:17:16 2018 by

LaTeXML
proof that a relation is union of functions if and only if AC

proof that a relation is union of functions if and only if AC

Theorem

.

A

relation

R

is the union of a set of

functions

each of which
has the same domain as

R

if and only if for each set

A

of

nonempty sets

, there is a

choice function

on

A

.

Proof.

Suppose that

R

is a relation with

dom

⁡

(

R

)

=

A

⁢

and

⁢

rng

⁡

(

R

)

⊆

B

. Let

g

:

A

→

𝒫

⁢

(

B

)

be given
by

a

↦

R

⁢

[

{

a

}

]

. There is be a choice function

c

on

g

⁢

[

A

]

. Let

f

=

c

∘

g

, and for each pair

⟨

a

,

b

⟩

∈

R

, let

f

a

⁢

b

send

a

to

b

and agree
with

f

elsewhere. Let

F

=

{

f

a

⁢

b

∣

⟨

a

,

b

⟩

∈

R

}

. Clearly

⋃

F

⊆

A

×

B

, so suppose

⟨

u

,

v

⟩

∈

⋃

F

; then there is a pair

⟨

a

,

b

⟩

∈

R

such that

⟨

u

,

v

⟩

∈

f

a

⁢

b

∈

F

. Either

⟨

u

,

v

⟩

=

⟨

a

,

b

⟩

, or

v

=

f

⁢

(

u

)

=

c

∘

g

⁢

(

u

)

=

c

⁢

(

R

⁢

[

{

u

}

]

)

∈

R

⁢

[

{

u

}

]

. In each case,

⟨

u

,

v

⟩

∈

R

.
Thus,

⋃

F

⊆

R

.

For each pair

⟨

a

,

b

⟩

∈

R

,

⟨

a

,

b

⟩

∈

f

a

⁢

b

∈

F

, so

R

⊆

⋃

F

. Therefore,

R

=

⋃

F

.

Suppose that

A

is set of nonempty sets. Let

R

=

⋃

{

{

a

}

×

a

∣

a

∈

A

}

. A set

x

is an

element of

dom

⁡

(

R

)

if and only if

x

∈

{

a

}

for some

a

∈

A

. Thus,

dom

⁡

(

R

)

=

A

. There is a set

F

of functions, each of which has
domain

A

, such that

R

=

⋃

F

. Let

f

∈

F

; then

dom

⁡

(

f

)

=

A

, and for each pair

⟨

a

,

f

⁢

(

a

)

⟩

∈

f

,

⟨

a

,

f

⁢

(

a

)

⟩

∈

{

a

}

×

a

; i.e.,

f

⁢

(

a

)

∈

a

.
Each such

f

is, thus, a choice function on

A

.
∎

Title

proof that a relation is union of functions if and only if AC

Canonical name

ProofThatARelationIsUnionOfFunctionsIfAndOnlyIfAC

Date of creation

2013-03-22 16:34:23

Last modified on

2013-03-22 16:34:23

Owner

ratboy (4018)

Last modified by

ratboy (4018)

Numerical id

4

Author

ratboy (4018)

Entry type

Proof

Classification

msc 03E25

Generated on Thu Feb 8 20:17:20 2018 by

LaTeXML
relation as union of functions

relation as union of functions

Every

relation

R

is a union of

functions

with the same domain as

R

.

This

proposition

of Doran is

equivalent

(

http://planetmath.org/Equivalent3

) to the

Axiom of Choice

(

http://planetmath.org/AxiomOfChoice

).

References

1

R. S. Doran:

“A

Theorem

which is Equivalent to the Axiom of Choice”.  –

Elemente der Mathematik

27

(1972).

Title

relation as union of functions

Canonical name

RelationAsUnionOfFunctions

Date of creation

2013-03-22 15:24:08

Last modified on

2013-03-22 15:24:08

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 03E25

Related topic

AxiomOfChoice

Generated on Thu Feb 8 20:17:22 2018 by

LaTeXML
selector

selector

Let

S

be a

collection

of sets. A

selector

of

S

is by definition a set

{

f

⁢

(

R

)

|

R

∈

S

}

where

f

is a

choice function

on

S

(that is

f

⁢

(

R

)

∈

R

).

There is a different selector of

S

for every choice function

f

.

Title

selector

Canonical name

Selector

Date of creation

2013-03-22 16:35:46

Last modified on

2013-03-22 16:35:46

Owner

porton (9363)

Last modified by

porton (9363)

Numerical id

7

Author

porton (9363)

Entry type

Definition

Classification

msc 03E25

Related topic

ChoiceFunction

Related topic

Collection

Generated on Thu Feb 8 20:17:24 2018 by

LaTeXML
Tukey’s lemma

Tukey’s lemma

Each nonempty family of

finite character

has a

maximal element

.

Here, by a maximal element we a maximal element with respect to the

inclusion

ordering

:

A

≤

B

iff

A

⊆

B

. This lemma is

equivalent

to the

axiom of choice

.

Title

Tukey’s lemma

Canonical name

TukeysLemma

Date of creation

2013-03-22 13:13:34

Last modified on

2013-03-22 13:13:34

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

6

Author

Koro (127)

Entry type

Theorem

Classification

msc 03E25

Related topic

AxiomOfChoice

Related topic

MaximalityPrinciple

Related topic

ZornsLemma

Related topic

ZermelosPostulate

Related topic

KuratowskisLemma

Generated on Thu Feb 8 20:17:39 2018 by

LaTeXML
well ordered set

well ordered set

A

well-ordered

set is a

totally ordered set

in which every nonempty subset has a least member.

An example of well-ordered set is the set of positive integers with the standard order relation

(

ℤ

+

,

<

)

, because any nonempty subset of it has least member. However,

ℝ

+

(the

positive

reals) is not a well-ordered set with the usual order, because

(

0

,

1

)

=

{

x

:

0

<

x

<

1

}

is a nonempty subset but it doesn’t contain a least number.

A

well-ordering

of a set

X

is the result of defining a

binary relation

≤

on

X

to itself in such a way that

X

becomes well-ordered with respect to

≤

.

Title

well

ordered set

Canonical

name

WellOrderedSet

Date of creation

2013-03-22 11:47:22

Last modified on

2013-03-22 11:47:22

Owner

drini (3)

Last modified by

drini (3)

Numerical id

16

Author

drini (3)

Entry type

Definition

Classification

msc 03E25

Classification

msc 06A05

Classification

msc 81T17

Classification

msc 81T13

Classification

msc 81T75

Classification

msc 81T45

Classification

msc 81T10

Classification

msc 81T05

Classification

msc 42-02

Classification

msc 55R15

Classification

msc 47D03

Classification

msc 55U35

Classification

msc 55U40

Classification

msc 47D08

Classification

msc 55-02

Classification

msc 18-00

Synonym

well-ordered

Synonym

well-ordered set

Related topic

WellOrderingPrinciple

Related topic

NaturalNumbersAreWellOrdered

Defines

well-ordering

Generated on Thu Feb 8 20:17:41 2018 by

LaTeXML
well-ordering principle for natural numbers proven from the principle of finite induction

well-ordering principle for natural numbers proven from the principle of finite induction

Let

S

be a

nonempty set

of

natural numbers

. We show that there is an

a

∈

S

such that for all

b

∈

S

,

a

≤

b

. Suppose not, then

(

*

)

∀

a

∈

S

,

∃

b

∈

S

b

<

a

.

We will use the

principle of finite induction

(the strong form) to show that

S

is empty, a contradition.

Fix any natural number

n

, and suppose that for all natural numbers

m

<

n

,

m

∈

ℕ

∖

S

. If

n

∈

S

, then (*) implies that there is an

element

b

∈

S

such that

b

<

n

. This would be incompatible with the

assumption

that for all natural numbers

m

<

n

,

m

∈

ℕ

∖

S

.
Hence, we conclude that

n

is not in

S

.

Therefore, by

induction

, no natural number is a member of

S

. The set is empty.

Title

well-ordering principle for natural numbers

proven from the principle of finite induction

Canonical

name

WellorderingPrincipleForNaturalNumbersProvenFromThePrincipleOfFiniteInduction

Date of creation

2013-03-22 16:38:02

Last modified on

2013-03-22 16:38:02

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Proof

Classification

msc 03E25

Related topic

NaturalNumbersAreWellOrdered

Generated on Thu Feb 8 20:17:43 2018 by

LaTeXML
well-ordering principle implies axiom of choice

well-ordering principle implies axiom of choice

Theorem

.

The

well-ordering principle

implies the

axiom of choice

.

Proof.

Let

C

be a

collection

of

nonempty sets

. Then

⋃

S

∈

C

S

is a set. By the well-ordering principle,

⋃

S

∈

C

S

is

well-ordered

under some

relation

<

. Since each

S

is a nonempty subset of

⋃

S

∈

C

S

, each

S

has a least member

m

S

with respect to the relation

<

.

Define

f

:

C

→

⋃

S

∈

C

S

by

f

⁢

(

S

)

=

m

S

. Then

f

is a

choice function

. Hence, the axiom of choice holds.
∎

Title

well-ordering principle implies axiom of choice

Canonical name

WellorderingPrincipleImpliesAxiomOfChoice

Date of creation

2013-03-22 16:07:46

Last modified on

2013-03-22 16:07:46

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

7

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 03E25

Related topic

AxiomOfChoice

Related topic

ZermelosWellOrderingTheorem

Generated on Thu Feb 8 20:17:46 2018 by

LaTeXML
Zermelo’s postulate

Zermelo’s postulate

If

ℱ

is a

disjoint

family of

nonempty sets

, then there is a set

C

which has exactly one element of each

A

∈

ℱ

(i.e such that

A

∩

C

is a

singleton

for each

A

∈

ℱ

.)

This is one of the many

propositions

that are

equivalent

to the

axiom of choice

.

Title

Zermelo’s

postulate

Canonical

name

ZermelosPostulate

Date of creation

2013-03-22 13:13:36

Last modified on

2013-03-22 13:13:36

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

6

Author

Koro (127)

Entry type

Theorem

Classification

msc 03E25

Related topic

AxiomOfChoice

Related topic

MaximalityPrinciple

Related topic

TukeysLemma

Related topic

ZornsLemma

Related topic

KuratowskisLemma

Generated on Thu Feb 8 20:17:48 2018 by

LaTeXML
Zermelo’s well-ordering theorem

Zermelo’s well-ordering theorem

If

X

is any set whatsoever, then there exists a

well-ordering

of

X

. The well-ordering theorem is

equivalent

to the

Axiom of Choice

.

Title

Zermelo’s well-ordering theorem

Canonical name

ZermelosWellorderingTheorem

Date of creation

2013-03-22 12:58:55

Last modified on

2013-03-22 12:58:55

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

5

Author

yark (2760)

Entry type

Theorem

Classification

msc 03E25

Synonym

well-ordering principle

Related topic

EquivalenceOfTheAxiomOfChoiceAndTheWellOrderingTheorem

Related topic

EquivalenceOfTheAxiomOfChoiceAndTheWellOrderingTheorem2

Related topic

HaudorffsMaximumPrinciple

Related topic

ZornsLemmeAndTheWellOrderingTheoremEquivalenceOfHaudorffsMaximumPrinciple

Related topic

ZornsLemmaAndTheWellOrderingTheoremEquiv

Generated on Thu Feb 8 20:17:50 2018 by

LaTeXML
axiom of extensionality

axiom of extensionality

If

X

and

Y

have the same

elements

, then

X

=

Y

.

The

Axiom of Extensionality

is one of the axioms of

Zermelo-Fraenkel set theory

.
In symbols, it reads:

∀

u

(

u

∈

X

↔

u

∈

Y

)

→

X

=

Y

.

Note that the

converse

,

X

=

Y

→

∀

u

(

u

∈

X

↔

u

∈

Y

)

is an axiom of the

predicate calculus

. Hence we have,

X

=

Y

↔

∀

u

(

u

∈

X

↔

u

∈

Y

)

.

Therefore the Axiom of Extensionality expresses the most fundamental notion of a set: a set is determined by its elements.

Title

axiom of extensionality

Canonical name

AxiomOfExtensionality

Date of creation

2013-03-22 13:42:40

Last modified on

2013-03-22 13:42:40

Owner

Sabean (2546)

Last modified by

Sabean (2546)

Numerical id

5

Author

Sabean (2546)

Entry type

Axiom

Classification

msc 03E30

Synonym

extensionality

Generated on Thu Feb 8 20:17:55 2018 by

LaTeXML
axiom of infinity

axiom of infinity

There exists an

infinite set

.

The

Axiom of Infinity

is an axiom of

Zermelo-Fraenkel set theory

.
At first glance, this axiom seems to be ill-defined. How are we to know what
constitutes an infinite set when we have not yet defined the notion of a

finite set

? However, once we have a theory of

ordinal numbers

in hand, the axiom makes sense.

Meanwhile, we can give a

definition

of finiteness that does not rely upon
the

concept

of number. We do this by introducing the notion of an

inductive
set

. A set

S

is said to be inductive if

∅

∈

S

and for every

x

∈

S

,

x

∪

{

x

}

∈

S

. We may then state the
Axiom of Infinity as follows:

There exists an inductive set.

In symbols:

∃

S

[

∅

∈

S

∧

(

∀

x

∈

S

)

[

x

∪

{

x

}

∈

S

]

]

We shall then be able to prove that the following conditions are

equivalent

:

1.

There exists an inductive set.

2.

There exists an infinite set.

3.

The least nonzero

limit ordinal

,

ω

, is a set.

Title

axiom of infinity

Canonical

name

AxiomOfInfinity

Date of creation

2013-03-22 13:43:52

Last modified on

2013-03-22 13:43:52

Owner

Sabean (2546)

Last modified by

Sabean (2546)

Numerical id

6

Author

Sabean (2546)

Entry type

Axiom

Classification

msc 03E30

Synonym

infinity

Generated on Thu Feb 8 20:17:57 2018 by

LaTeXML
axiom of pairing

axiom of pairing

For any

a

and

b

there exists a set

{

a

,

b

}

that contains exactly

a

and

b

.

The

Axiom of Pairing

is one of the axioms of

Zermelo-Fraenkel set theory

. In symbols, it reads:

∀

a

∀

b

∃

c

∀

x

(

x

∈

c

↔

x

=

a

∨

x

=

b

)

.

Using the

Axiom of Extensionality

, we see that the set

c

is unique, so it makes sense to define the pair

{

a

,

b

}

=

the unique

c

such that

∀

x

(

x

∈

c

↔

x

=

a

∨

x

=

b

)

.

Using the Axiom of Pairing, we may define, for any set

a

, the

singleton

{

a

}

=

{

a

,

a

}

.

We may also define, for any set

a

and

b

, the

ordered pair

(

a

,

b

)

=

{

{

a

}

,

{

a

,

b

}

}

.

Note that this

definition

satisfies

the condition

(

a

,

b

)

=

(

c

,

d

)

⁢

iff

⁢

a

=

c

⁢

and

⁢

b

=

d

.

We may define the ordered

n

-tuple recursively

(

a

1

,

…

,

a

n

)

=

(

(

a

1

,

…

,

a

n

-

1

)

,

a

n

)

.

Title

axiom of pairing

Canonical

name

AxiomOfPairing

Date of creation

2013-03-22 13:42:43

Last modified on

2013-03-22 13:42:43

Owner

Sabean (2546)

Last modified by

Sabean (2546)

Numerical id

7

Author

Sabean (2546)

Entry type

Axiom

Classification

msc 03E30

Synonym

pairing

Generated on Thu Feb 8 20:17:59 2018 by

LaTeXML
axiom of power set

axiom of power set

The

axiom of power set

is an axiom of

Zermelo-Fraenkel set theory

which

postulates

that for any set

X

there exists a set

𝒫

⁢

(

X

)

, called the

power set

of

X

, consisting of all subsets of

X

. In symbols, it reads:

∀

X

∃

𝒫

(

X

)

∀

u

(

u

∈

𝒫

(

X

)

↔

u

⊆

X

)

.

In the above,

u

⊆

X

is defined as

∀

z

(

z

∈

u

→

z

∈

X

)

. By the

extensionality

axiom, the set

𝒫

⁢

(

X

)

is unique.

The Power Set Axiom allows us to define the

Cartesian product

of two sets

X

and

Y

:

X

×

Y

=

{

(

x

,

y

)

:

x

∈

X

∧

y

∈

Y

}

.

The Cartesian product is a set since

X

×

Y

⊆

𝒫

⁢

(

𝒫

⁢

(

X

∪

Y

)

)

.

We may define the Cartesian product of any finite

collection

of sets recursively:

X

1

×

⋯

×

X

n

=

(

X

1

×

⋯

×

X

n

-

1

)

×

X

n

.

Title

axiom of power set

Canonical name

AxiomOfPowerSet

Date of creation

2013-03-22 13:43:03

Last modified on

2013-03-22 13:43:03

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

11

Author

mathcam (2727)

Entry type

Axiom

Classification

msc 03E30

Synonym

power set axiom

Synonym

powerset axiom

Synonym

axiom of powerset

Generated on Thu Feb 8 20:18:02 2018 by

LaTeXML
axiom of union

axiom of union

For any

X

there exists a set

Y

=

⋃

X

.

The

Axiom of Union

is an axiom of

Zermelo-Fraenkel set theory

. In symbols, it reads

∀

X

∃

Y

∀

u

(

u

∈

Y

↔

∃

z

(

z

∈

X

∧

u

∈

z

)

)

.

Notice that this means that

Y

is the set of elements of all elements of

X

. More succinctly, the union of any set of sets is a set. By

Extensionality

, the set

Y

is unique.

Y

is called the

union

of

X

.

In particular, the Axiom of Union, along with the

Axiom of Pairing

allows us to define

X

∪

Y

=

⋃

{

X

,

Y

}

,

as well as the triple

{

a

,

b

,

c

}

=

{

a

,

b

}

∪

{

c

}

and therefore the

n

-tuple

{

a

1

,

…

,

a

n

}

=

{

a

1

}

∪

⋯

∪

{

a

n

}

Title

axiom of union

Canonical name

AxiomOfUnion

Date of creation

2013-03-22 13:42:49

Last modified on

2013-03-22 13:42:49

Owner

Sabean (2546)

Last modified by

Sabean (2546)

Numerical id

8

Author

Sabean (2546)

Entry type

Axiom

Classification

msc 03E30

Synonym

union

Generated on Thu Feb 8 20:18:04 2018 by

LaTeXML
axiom schema of separation

axiom schema of separation

Let

ϕ

⁢

(

u

,

p

)

be a

formula

. For any

X

and

p

, there exists a set

Y

=

{

u

∈

X

:

ϕ

⁢

(

u

,

p

)

}

.

The

Axiom Schema of Separation

is an

axiom schema

of

Zermelo-Fraenkel set theory

. Note that it

represents

infinitely many

individual

axioms, one for each formula

ϕ

. In symbols, it reads:

∀

X

∀

p

∃

Y

∀

u

(

u

∈

Y

↔

u

∈

X

∧

ϕ

(

u

,

p

)

)

.

By

Extensionality

, the set

Y

is unique.

The Axiom Schema of Separation implies that

ϕ

may depend on more than one parameter

p

.

We may show by

induction

that if

ϕ

⁢

(

u

,

p

1

,

…

,

p

n

)

is a formula, then

∀

X

∀

p

1

⋯

∀

p

n

∃

Y

∀

u

(

u

∈

Y

↔

u

∈

X

∧

ϕ

(

u

,

p

1

,

…

,

p

n

)

)

holds, using the Axiom Schema of Separation and the

Axiom of Pairing

.

Another

consequence

of the Axiom Schema of Separation is that a

subclass

of any set is a set. To see this, let

𝐂

be the class

𝐂

=

{

u

:

ϕ

⁢

(

u

,

p

1

,

…

,

p

n

)

}

. Then

∀

X

∃

Y

(

𝐂

∩

X

=

Y

)

holds, which means that the

intersection

of

𝐂

with any set is a set. Therefore, in particular, the intersection of two sets

X

∩

Y

=

{

x

∈

X

:

x

∈

Y

}

is a set. Furthermore the difference of two sets

X

-

Y

=

{

x

∈

X

:

x

∉

Y

}

is a set and, provided there exists at least one set, which is guaranteed by the

Axiom of Infinity

, the

empty set

is a set. For if

X

is a set, then

∅

=

{

x

∈

X

:

x

≠

x

}

is a set.

Moreover, if

𝐂

is a nonempty class, then

⋂

𝐂

is a set, by

Separation

.

⋂

𝐂

is a subset of every

X

∈

𝐂

.

Lastly, we may use Separation to show that the class of all sets,

V

, is not a set, i.e.,

V

is a

proper class

. For example, suppose

V

is a set. Then by Separation

V

′

=

{

x

∈

V

:

x

∉

x

}

is a set and we have reached a

Russell paradox

.

Title

axiom schema of separation

Canonical name

AxiomSchemaOfSeparation

Date of creation

2013-03-22 13:42:46

Last modified on

2013-03-22 13:42:46

Owner

Sabean (2546)

Last modified by

Sabean (2546)

Numerical id

18

Author

Sabean (2546)

Entry type

Axiom

Classification

msc 03E30

Synonym

separation schema

Synonym

separation

Generated on Thu Feb 8 20:18:07 2018 by

LaTeXML