aarjaneiro/planetmath huggingface dataset:
tree (set theoretic)

tree (set theoretic)

In a set , a

tree

is defined to be a set

T

and a

relation

<

T

⊆

T

×

T

such that:

•

<

T

is a

partial ordering

of

T

•

For any

t

∈

T

,

{

s

∈

T

∣

s

<

T

t

}

is

well-ordered

The nodes (elements of the tree) immediately greater than a node are termed its

children

, the node immediately less is its

parent

(if it exists), any node less is an

ancestor

and any node greater is a

descendant

. A node with no ancestors is a

root

.

The partial ordering

represents

from the root, and the well-ordering requirement prohibits any loops or splits below a node (that is, each node has at most one parent, and therefore at most one grand-parent, and so on).

Conversely

, if

a

<

T

d

then there is exactly one

b

≤

T

d

such that

a

<

T

b

and there is nothing between

a

and

b

. Note that there is no requirement that a node have a parent–there could be an infinitely long branch

a

1

<

T

a

2

<

T

a

2

<

T

⋯

with each

a

i

<

T

b

.

Since there is generally no requirement that the tree be

connected

, the null ordering makes any set into a tree, although the tree is a

trivial

one, since each element of the set forms a single node with no children.

Since the set of ancestors of any node is well-ordered, we can it with an

ordinal

. We call this the

height

, and write:

ht

⁡

(

t

)

=

o

.

t

.

⁡

(

{

s

∈

T

∣

s

<

T

t

}

)

. This all accords with usage: a root has height

0

, something immediately above the root has height

1

, and so on. We can then assign a height to the tree itself, which we define to be the least number greater than the height of any element of the tree. For finite trees this is just one greater than the height of its tallest

element

, but

infinite

trees may not have a tallest element, so we define

ht

⁡

(

T

)

=

sup

⁡

{

ht

⁡

(

t

)

+

1

∣

t

∈

T

}

.

For every

α

<

T

ht

⁡

(

T

)

we define the

α

-th

level

to be the set

T

α

=

{

t

∈

T

∣

ht

⁡

(

t

)

=

α

}

. So of course

T

0

is all roots of the tree. If

α

<

T

ht

⁡

(

T

)

then

T

⁢

(

α

)

is the subtree of elements with height less than

α

:

t

∈

T

⁢

(

α

)

↔

x

∈

T

∧

ht

⁡

(

t

)

<

α

.

We call a tree a

κ

-tree for any

cardinal

κ

if

|

T

|

=

κ

and

ht

⁡

(

T

)

=

κ

. If

κ

is finite, the only way to do this is to have a single branch of length

κ

.

Title

tree (set theoretic)

Canonical

name

TreesetTheoretic

Date of creation

2013-03-22 12:52:19

Last modified on

2013-03-22 12:52:19

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

14

Author

Henry (455)

Entry type

Definition

Classification

msc 03E05

Synonym

tree

Related topic

Antichain

Related topic

Aronszajn

Related topic

SuslinTree

Defines

level

Defines

root

Defines

height

Defines

tree

Defines

child

Defines

children

Defines

parent

Defines

ancestor

Defines

descendant

Defines

α

-th level

Defines

κ

-tree

Generated on Thu Feb 8 20:07:34 2018 by

LaTeXML
aleph numbers

aleph numbers

The

aleph numbers

are

infinite

cardinal numbers

defined by

transfinite recursion

, as described below.
They are written

ℵ

α

, where

ℵ

is aleph,
the first letter of the Hebrew

alphabet

,
and

α

is an

ordinal number

.
Sometimes we write

ω

α

instead of

ℵ

α

,
usually to emphasise that it is an

ordinal

.

To start the transfinite recursion,
we define

ℵ

0

to be the first infinite ordinal.
This is the

cardinality

of

countably infinite

sets, such as

ℕ

and

ℚ

.
For each ordinal

α

,
the cardinal number

ℵ

α

+

1

is defined to be
the least ordinal of cardinality greater than

ℵ

α

.
For each

limit ordinal

δ

,
we define

ℵ

δ

=

⋃

α

∈

δ

ℵ

α

.

As a

consequence

of the

Well-Ordering Principle

(

http://planetmath.org/ZermelosWellOrderingTheorem

),
every

infinite set

is equinumerous with an aleph number.
Every infinite cardinal is therefore an aleph.
More precisely, for every infinite cardinal

κ

there is exactly one ordinal

α

such that

κ

=

ℵ

α

.

Title

aleph numbers

Canonical

name

AlephNumbers

Date of creation

2013-03-22 14:15:39

Last modified on

2013-03-22 14:15:39

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

6

Author

yark (2760)

Entry type

Definition

Classification

msc 03E10

Synonym

alephs

Related topic

GeneralizedContinuumHypothesis

Related topic

BethNumbers

Generated on Thu Feb 8 20:07:36 2018 by

LaTeXML
another proof of cardinality of the rationals

another proof of cardinality of the rationals

If we have a

rational number

p

/

q

with

p

and

q

having no common factor,
and each expressed in base 10 then we can view

p

/

q

as a base 11

integer

,
where the digits are

0

,

1

,

2

,

…

,

9

and

/

. That is, slash (

/

) is a symbol for a
digit. For example, the rational 3/2 corresponds to the integer

3

⋅

11

2

+

10

⋅

11

+

2

.
The rational

-

3

/

2

corresponds to the integer

-

(

3

⋅

11

2

+

10

⋅

11

+

2

)

.

This gives a

one-to-one

map into the
integers so the

cardinality

of the rationals is at most the cardinality of
the integers. So the rationals are

countable

.

Title

another proof of cardinality of the rationals

Canonical name

AnotherProofOfCardinalityOfTheRationals

Date of creation

2013-03-22 16:01:49

Last modified on

2013-03-22 16:01:49

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

10

Author

Mathprof (13753)

Entry type

Proof

Classification

msc 03E10

Generated on Thu Feb 8 20:07:59 2018 by

LaTeXML
beth numbers

beth numbers

The

beth numbers

are

infinite

cardinal numbers

defined in a

similar

manner to the

aleph numbers

, as described below.
They are written

ℶ

α

, where

ℶ

is beth,
the second letter of the Hebrew

alphabet

,
and

α

is an

ordinal number

.

We define

ℶ

0

to be the first infinite cardinal (that is,

ℵ

0

).
For each

ordinal

α

,
we define

ℶ

α

+

1

=

2

ℶ

α

.
For each

limit ordinal

δ

,
we define

ℶ

δ

=

⋃

α

∈

δ

ℶ

α

.

Note that

ℶ

1

is the

cardinality

of the

continuum

.

For any ordinal

α

the

inequality

ℵ

α

⩽

ℶ

α

holds.
The

Generalized Continuum Hypothesis

is

equivalent

to the assertion that

ℵ

α

=

ℶ

α

for every ordinal

α

.

For every limit ordinal

δ

,
the cardinal

ℶ

δ

is a

strong limit cardinal

.
Every

uncountable

strong limit cardinal arises in this way.

Title

beth numbers

Canonical name

BethNumbers

Date of creation

2013-03-22 14:17:09

Last modified on

2013-03-22 14:17:09

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Definition

Classification

msc 03E10

Related topic

AlephNumbers

Related topic

GeneralizedContinuumHypothesis

Generated on Thu Feb 8 20:08:01 2018 by

LaTeXML
bijection between unit interval and unit square

bijection between unit interval and unit square

The real numbers in the open unit interval

I

=

(

0

,

1

)

can be uniquely represented by their

decimal expansions

, when these must not end in an

infinite

string of 9’s.  Correspondingly, the

elements of

the open unit square

I

×

I

are represented by the pairs of such decimal expansions.

Let

P

:=

(

0

.

x

1

x

2

x

3

…

,

0

.

y

1

y

2

y

3

…

)

be such a pair representing an arbitrary point in

I

×

I

and let

p

:=

0

.

x

1

⁢

y

1

⁢

x

2

⁢

y

2

⁢

x

3

⁢

y

3

⁢

…

Then it’s apparent that

P

↦

p

(1)

is an

injective

mapping

from

I

×

I

to

I

.  Thus

|

I

×

I

|

≤

|

I

|

.

But since

I

×

I

contains more than one horizontal open segment equally long as

I

(and accordingly there is a natural

injection

from

I

to

I

×

I

), we must have also

|

I

×

I

|

≥

|

I

|

.

The

conclusion

is that

|

I

×

I

|

=

|

I

|

,

i.e. that the sets

I

×

I

and

I

have equal

cardinalities

,
and the Schröder

-

Bernstein theorem even garantees a

bijection

between the sets.

Remark 1.

Georg Cantor utilised

continued fractions

for constructing such a bijection between the unit interval and the unit square; cf. e.g.

http://www.maa.org/pubs/AMM-March11_Cantor.pdf

this MAA article.

Remark 2.

Since the mapping

g

:

I

→

ℝ

defined by

g

⁢

(

x

)

=

tan

⁡

(

π

⁢

x

-

π

2

)

is

bijective

, we can conclude that the sets

ℝ

and

ℝ

×

ℝ

, i.e. the set of the points of a line and the set of the points of a plane, have the same cardinalities.  This common cardinality is

2

ℵ

0

.

Title

bijection between unit interval and unit square

Canonical name

BijectionBetweenUnitIntervalAndUnitSquare

Date of creation

2015-02-03 21:45:39

Last modified on

2015-02-03 21:45:39

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

18

Author

pahio (2872)

Entry type

Result

Classification

msc 03E10

Related topic

JuliusKonig

Related topic

BijectionBetweenClosedAndOpenInterval

Generated on Thu Feb 8 20:08:03 2018 by

LaTeXML
Cantor normal form

Cantor normal form

Ordinal Normal Form

(Cantor).

For

ordinal numbers

α

≥

2

and

γ

≥

1

there is a unique

n

such that there exist unique

β

0

>

⋯

>

β

n

and

0

<

δ

0

<

α

,

…

,

0

<

δ

n

<

α

such that

γ

=

α

β

0

⋅

δ

0

+

⋯

+

α

β

n

⋅

δ

n

.

This

theorem

is often referred to as the

Cantor

Normal Form

of

γ

in the base of

α

.

Title

Cantor normal form

Canonical name

CantorNormalForm

Date of creation

2013-03-22 15:33:01

Last modified on

2013-03-22 15:33:01

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

9

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 03E10

Generated on Thu Feb 8 20:08:18 2018 by

LaTeXML
Cantor’s diagonal argument

Cantor’s diagonal argument

One of the starting points in
Cantor’s development of

set theory

was his discovery that there are
different degrees of

infinity

. The

rational numbers

, for example, are

countably infinite

; it is

possible

to

enumerate

all the rational
numbers by means of an

infinite

list. By contrast, the real numbers
are

uncountable

. it is

impossible

to enumerate them by means of an
infinite list. These discoveries underlie the idea of

cardinality

,
which is expressed by saying that two sets have the same cardinality
if there exists a

bijective

correspondence between them.

In essence, Cantor discovered two

theorems

: first, that the set of
real numbers has the same cardinality as the

power set

of the
naturals; and second, that a set and its power set have a different
cardinality (see Cantor’s theorem). The proof of the second result
is based on the celebrated diagonalization

argument

.

Cantor showed that for every given infinite

sequence

of real numbers

x

1

,

x

2

,

x

3

,

…

it is possible to construct a real number

x

that is not on that list. Consequently, it is impossible to enumerate
the real numbers; they are uncountable. No generality is lost if we
suppose that all the numbers on the list are between

0

and

1

.
Certainly, if this subset of the real numbers in uncountable, then the
full set is uncountable as well.

Let us write our sequence as a table of

decimal expansions

:

0

.

d

11

d

12

d

13

d

14

…

0

.

d

21

d

22

d

23

d

24

…

0

.

d

31

d

32

d

33

d

34

…

0

.

d

41

d

42

d

43

d

44

…

⋮

⋮

⋮

⋮

⋮

⋱

where

x

n

=

0

.

d

n

⁢

1

⁢

d

n

⁢

2

⁢

d

n

⁢

3

⁢

d

n

⁢

4

⁢

…

,

and the expansion avoids an infinite trailing string of
the digit

9

.

For each

n

=

1

,

2

,

…

we choose a digit

c

n

that is different from

d

n

⁢

n

and not equal to

9

, and consider the real number

x

with
decimal expansion

0

.

c

1

⁢

c

2

⁢

c

3

⁢

…

By construction, this number

x

is different from every member of the
given sequence. After all, for every

n

, the number

x

differs from
the number

x

n

in the

n

th

decimal
digit.
The claim is proven.

Title

Cantor’s diagonal argument

Canonical name

CantorsDiagonalArgument

Date of creation

2013-03-22 12:22:03

Last modified on

2013-03-22 12:22:03

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

9

Author

rmilson (146)

Entry type

Topic

Classification

msc 03E10

Synonym

Cantor’s diagonalization

Related topic

CantorsTheorem

Generated on Thu Feb 8 20:08:20 2018 by

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cardinal exponentiation under GCH

cardinal exponentiation under GCH

Many results about

cardinal exponentiation

can neither be proved nor disproved in ZFC. If, however, we allow ourselves to use GCH in addition to ZFC, then we have the following

theorem

, which gives an essentially

complete

description of the way cardinal exponentiation involving

infinite

cardinals

works.

Theorem

.

Assume the

Generalized Continuum Hypothesis

holds.
Let

κ

and

λ

be cardinals,
at least one of which is infinite,
and such that

κ

>

0

and

λ

>

1

.
Then

λ

κ

=

{

κ

+

,

𝑖𝑓

λ

≤

κ

+

;

λ

+

,

𝑖𝑓

cf

⁡

(

λ

)

≤

κ

≤

λ

;

λ

,

𝑖𝑓

κ

<

cf

⁡

(

λ

)

.

Here,

cf

⁡

(

λ

)

is the

cofinality

of

λ

, and

λ

+

is the

cardinal successor

of

λ

.

Title

cardinal exponentiation under GCH

Canonical name

CardinalExponentiationUnderGCH

Date of creation

2013-03-22 14:54:04

Last modified on

2013-03-22 14:54:04

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Theorem

Classification

msc 03E10

Related topic

CardinalArithmetic

Related topic

GeneralizedContinuumHypothesis

Generated on Thu Feb 8 20:08:26 2018 by

LaTeXML
cardinal number

cardinal number

A

cardinal number

is an

ordinal number

S

with the

property

that

S

≤

X

for every ordinal number

X

which has the same

cardinality

as

S

. Cardinal numbers have the property that for every set

A

, there exists a unique cardinal number having the same number of

elements

as

A

.

Title

cardinal number

Canonical

name

CardinalNumber

Date of creation

2013-03-22 12:07:58

Last modified on

2013-03-22 12:07:58

Owner

djao (24)

Last modified by

djao (24)

Numerical id

8

Author

djao (24)

Entry type

Definition

Classification

msc 03E10

Synonym

cardinal

Related topic

Cardinality

Related topic

CardinalArithmetic

Generated on Thu Feb 8 20:08:28 2018 by

LaTeXML
cardinal successor

cardinal successor

The

cardinal successor

of a

cardinal

κ

is the least cardinal greater than

κ

. It is denoted by

κ

+

.

Title

cardinal successor

Canonical name

CardinalSuccessor

Date of creation

2013-03-22 14:04:43

Last modified on

2013-03-22 14:04:43

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

6

Author

yark (2760)

Entry type

Definition

Classification

msc 03E10

Related topic

SuccessorCardinal

Generated on Thu Feb 8 20:08:29 2018 by

LaTeXML
cardinality of a countable union

cardinality of a countable union

Let

C

be a

countable

collection

of

countable sets

. Then

∪

C

is countable.

Title

cardinality

of a countable union

Canonical name

CardinalityOfACountableUnion

Date of creation

2013-03-22 11:59:59

Last modified on

2013-03-22 11:59:59

Owner

vampyr (22)

Last modified by

vampyr (22)

Numerical id

5

Author

vampyr (22)

Entry type

Theorem

Classification

msc 03E10

Related topic

A_nAreCountableSoIsA_1XXA_nIfA_1

Generated on Thu Feb 8 20:08:33 2018 by

LaTeXML
cardinality of the rationals

cardinality of the rationals

The set of rational numbers

ℚ

is

countable

, and therefore its

cardinality

is

ℵ

0

.

Title

cardinality of the

rationals

Canonical name

CardinalityOfTheRationals

Date of creation

2013-03-22 11:59:47

Last modified on

2013-03-22 11:59:47

Owner

quadrate (89)

Last modified by

quadrate (89)

Numerical id

6

Author

quadrate (89)

Entry type

Theorem

Classification

msc 03E10

Synonym

power of the rationals

Related topic

Countable

Related topic

RationalNumber

Related topic

Irrational

Related topic

AlgebraicNumbersAreCountable

Related topic

A_nAreCountableSoIsA_1XXA_nIfA_1

Generated on Thu Feb 8 20:08:39 2018 by

LaTeXML
club

club

If

κ

is a

cardinal

then a set

C

⊆

κ

is

closed

iff for any

S

⊆

C

and

α

<

κ

,

sup

⁡

(

S

∩

α

)

=

α

then

α

∈

C

. (That is, if the limit of some

sequence

in

C

is less than

κ

then the limit is also in

C

.)

If

κ

is a cardinal and

C

⊆

κ

then

C

is

unbounded

if, for any

α

<

κ

, there is some

β

∈

C

such that

α

<

β

.

If a set is both closed and unbounded then it is a

club

set.

Title

club

Canonical name

Club

Date of creation

2013-03-22 12:53:01

Last modified on

2013-03-22 12:53:01

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Definition

Classification

msc 03E10

Defines

club

Defines

closed

Defines

unbounded

Defines

closed unbounded

Defines

closed set

Defines

unbounded set

Defines

closed unbounded set

Defines

club set

Generated on Thu Feb 8 20:08:41 2018 by

LaTeXML
club filter

club filter

If

κ

is a

regular

uncountable

cardinal

then

club

⁡

(

κ

)

, the filter of all sets containing a club subset of

κ

, is a

κ

-

complete

filter

closed under

diagonal intersection

called the

club filter

.

To see that this is a filter, note that

κ

∈

club

⁡

(

κ

)

since it is obviously both closed and

unbounded

. If

x

∈

club

⁡

(

κ

)

then any subset of

κ

containing

x

is also in

club

⁡

(

κ

)

, since

x

, and therefore anything containing it, contains a club set.

It is a

κ

complete filter because the

intersection

of fewer than

κ

club sets is a club set. To see this, suppose

⟨

C

i

⟩

i

<

α

is a

sequence

of club sets where

α

<

κ

. Obviously

C

=

⋂

C

i

is closed, since any sequence which appears in

C

appears in every

C

i

, and therefore its limit is also in every

C

i

. To show that it is unbounded, take some

β

<

κ

. Let

⟨

β

1

,

i

⟩

be an

increasing sequence

with

β

1

,

1

>

β

and

β

1

,

i

∈

C

i

for every

i

<

α

. Such a sequence can be constructed, since every

C

i

is unbounded. Since

α

<

κ

and

κ

is regular, the limit of this sequence is less than

κ

. We call it

β

2

, and define a new sequence

⟨

β

2

,

i

⟩

similar

to the previous sequence. We can repeat this process, getting a sequence of sequences

⟨

β

j

,

i

⟩

where each element of a sequence is greater than every member of the previous sequences. Then for each

i

<

α

,

⟨

β

j

,

i

⟩

is an increasing sequence

contained

in

C

i

, and all these sequences have the same limit (the limit of

⟨

β

j

,

i

⟩

). This limit is then contained in every

C

i

, and therefore

C

, and is greater than

β

.

To see that

club

⁡

(

κ

)

is closed under diagonal intersection, let

⟨

C

i

⟩

,

i

<

κ

be a sequence, and let

C

=

Δ

i

<

κ

⁢

C

i

. Since the diagonal intersection contains the intersection, obviously

C

is unbounded. Then suppose

S

⊆

C

and

sup

⁡

(

S

∩

α

)

=

α

. Then

S

⊆

C

β

for every

β

≥

α

, and since each

C

β

is closed,

α

∈

C

β

, so

α

∈

C

.

Title

club filter

Canonical name

ClubFilter

Date of creation

2013-03-22 12:53:11

Last modified on

2013-03-22 12:53:11

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Definition

Classification

msc 03E10

Defines

club filter

Generated on Thu Feb 8 20:08:43 2018 by

LaTeXML
countable

countable

A set

S

is

countable

if there exists a

bijection

between

S

and some subset of

ℕ

.

All

finite sets

are countable.

Title

countable

Canonical

name

Countable

Date of creation

2013-03-22 11:59:00

Last modified on

2013-03-22 11:59:00

Owner

vampyr (22)

Last modified by

vampyr (22)

Numerical id

6

Author

vampyr (22)

Entry type

Definition

Classification

msc 03E10

Related topic

ProofThatTheRationalsAreCountable

Related topic

CantorSet

Related topic

AverageValueOfFunction

Related topic

NumerableSet

Related topic

SierpinskiSetOfEuclideanPlane

Generated on Thu Feb 8 20:08:45 2018 by

LaTeXML
countably infinite

countably infinite

A set

S

is

countably infinite

if there is a

bijection

between

S

and

ℕ

.

As the name implies, any countably infinite set is both

countable

and

infinite

.

Countably infinite sets are also sometimes called

denumerable

.

Title

countably infinite

Canonical name

CountablyInfinite

Date of creation

2013-03-22 11:59:05

Last modified on

2013-03-22 11:59:05

Owner

vampyr (22)

Last modified by

vampyr (22)

Numerical id

7

Author

vampyr (22)

Entry type

Definition

Classification

msc 03E10

Synonym

denumerable

Related topic

NumerableSet

Generated on Thu Feb 8 20:08:46 2018 by

LaTeXML
equivalence class of equinumerous sets is not a set

equivalence class of equinumerous sets is not a set

Recall that two sets are equinumerous iff there is a

bijection

between them.

Proposition 1

.

Let

A

be a non-empty set, and

E

⁢

(

A

)

the class of all sets equinumerous to

A

. Then

E

⁢

(

A

)

is a

proper class

.

Proof.

E

⁢

(

A

)

≠

∅

since

A

is in

E

⁢

(

A

)

. Since

A

≠

∅

, pick an

element

a

∈

A

, and let

B

=

A

-

{

a

}

. Then

C

:=

{

y

∣

y

⁢

is a set, and

⁢

y

∉

B

}

is a proper class, for otherwise

C

∪

B

would be the “set” of all sets, which is impossible. For each

y

in

C

, the set

F

⁢

(

y

)

:=

B

∪

{

y

}

is in one-to-one correspondence with

A

, with the bijection

f

:

F

⁢

(

y

)

→

A

given by

f

⁢

(

x

)

=

x

if

x

∈

B

, and

f

⁢

(

y

)

=

a

. Therefore

E

⁢

(

A

)

contains

F

⁢

(

y

)

for every

y

in the proper class

C

. Furthermore, since

F

⁢

(

y

1

)

≠

F

⁢

(

y

2

)

whenever

y

1

≠

y

2

, we have that

E

⁢

(

A

)

is a proper class as a result.
∎

Remark

. In the proof above, one can think of

F

as a

class function

from

C

to

E

⁢

(

A

)

, taking every

y

∈

C

into

F

⁢

(

y

)

. This

function

is

one-to-one

, so

C

embeds in

E

⁢

(

A

)

, and hence

E

⁢

(

A

)

is a proper class.

Title

equivalence class

of equinumerous sets is not a set

Canonical name

EquivalenceClassOfEquinumerousSetsIsNotASet

Date of creation

2013-03-22 18:50:34

Last modified on

2013-03-22 18:50:34

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

4

Author

CWoo (3771)

Entry type

Result

Classification

msc 03E10

Generated on Thu Feb 8 20:08:56 2018 by

LaTeXML