aarjaneiro/planetmath huggingface dataset:
number

number

Number

is an abstract

concept

which is

not defined generally

in mathematics.  The numbers
can be used for counting .

In mathematics one

can define

different

kinds

of numbers; some of the most common are
.  There are also many
special kinds of e.g. :

odd numbers

,

prime numbers

,

triangular numbers

,

Fibonacci numbers

, etc.
The

algebraic integers

(

http://planetmath.org/AlgebraicNumberTheory

) are
a special kind of

complex numbers

, having

similar

(

http://planetmath.org/Number

)

divisibility

properties as the
but much richer.

Usually the numbers, which can be thought to be formed by gradually expanding the available system of numbers:

•

natural numbers

1, 2, 3, …

•

integers

(added 0 (

http://planetmath.org/Null

) and negative integers)

•

rational numbers

(added

fractional numbers

)

•

real numbers

(added

irrational numbers

)

•

complex numbers (added

imaginary numbers

)

A usual applier of the mathematics, e.g. an engineer, probably
believes that there are no other numbers than the complex
numbers.  Some school book may tell that the complex numbers
form the widest possible field (

http://planetmath.org/Field

) of numbers.  However, the mathematicians know that there exist infinitely many

extension fields

of the field

ℂ

of the complex numbers, e.g. the

rational function field

ℂ

⁢

(

X

)

or the

formal Laurent series

field

ℂ

⁢

(

(

X

)

)

.  That’s a different matter if one wants to call

numbers

the elements of the last fields.

The field

ℚ

of the rational numbers can be extended
also in another direction than the real and complex numbers:
the field of

p

-adic numbers (

http://planetmath.org/PAdicIntegers

) makes a

completion

of

ℚ

which resembles

ℝ

but which is not

contained

neither in

ℝ

in

ℂ

.

Title

number

Canonical name

Number

Date of creation

2015-11-17 12:09:30

Last modified on

2015-11-17 12:09:30

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Feature

Classification

msc 03E10

Related topic

ClassificationOfComplexNumbers

Related topic

Fraction

Related topic

CardinalNumber

Related topic

OrdinalNumber

Related topic

RuleOfProduct

Generated on Thu Feb 8 20:10:05 2018 by

LaTeXML
ordinal arithmetic

ordinal arithmetic

Ordinal arithmetic

is the

extension

of normal

arithmetic

to the transfinite ordinal numbers. The successor operation

S

⁢

x

(sometimes written

x

+

1

, although this notation risks confusion with the general

definition

of

addition

) is part of the definition of the

ordinals

, and addition is naturally defined by

recursion

over this:

•

x

+

0

=

x

•

x

+

S

⁢

y

=

S

⁢

(

x

+

y

)

•

x

+

α

=

sup

γ

<

α

⁡

(

x

+

γ

)

for

limit ordinal

α

If

x

and

y

are finite then

x

+

y

under this definition is just the usual sum, however when

x

and

y

become

infinite

, there are

differences

. In particular,

ordinal addition

is not

commutative

. For example,

ω

+

1

=

ω

+

S

⁢

0

=

S

⁢

(

ω

+

0

)

=

S

⁢

ω

but

1

+

ω

=

sup

n

<

ω

⁡

1

+

n

=

ω

Multiplication

in turn is defined by iterated addition:

•

x

⋅

0

=

0

•

x

⋅

S

⁢

y

=

x

⋅

y

+

x

•

x

⋅

α

=

sup

γ

<

α

⁡

(

x

⋅

γ

)

for limit ordinal

α

Once again this definition is

equivalent

to normal multiplication when

x

and

y

are finite, but is not commutative:

ω

⋅

2

=

ω

⋅

1

+

ω

=

ω

+

ω

but

2

⋅

ω

=

sup

n

<

ω

⁡

2

⋅

n

=

ω

Both these

functions

are strongly increasing in the second

argument

and weakly increasing in the first argument. That is, if

α

<

β

then

•

γ

+

α

<

γ

+

β

•

γ

⋅

α

<

γ

⋅

β

•

α

+

γ

≤

β

+

γ

•

α

⋅

γ

≤

β

⋅

γ

Title

ordinal arithmetic

Canonical

name

OrdinalArithmetic

Date of creation

2013-03-22 13:28:52

Last modified on

2013-03-22 13:28:52

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

7

Author

Henry (455)

Entry type

Topic

Classification

msc 03E10

Related topic

AdditivelyIndecomposable

Related topic

CardinalArithmetic

Generated on Thu Feb 8 20:10:18 2018 by

LaTeXML
ordinal exponentiation

ordinal exponentiation

Let

α

,

β

be

ordinals

. We define

α

β

as follows:

α

β

:=

{

1

if

⁢

β

=

0

,

α

γ

⋅

α

if

β

is a successor ordinal and

⁢

β

=

S

⁢

γ

,

sup

⁡

{

α

γ

∣

γ

<

β

}

if

β

is a limit ordinal and

⁢

β

=

sup

⁡

{

γ

∣

γ

<

β

}

.

Some properties of

exponentiation

:

1.

0

α

=

0

if

α

>

0

2.

1

α

=

1

3.

α

1

=

α

4.

α

β

⋅

α

γ

=

α

β

+

γ

5.

(

α

β

)

γ

=

α

β

⋅

γ

6.

For any ordinals

α

,

β

with

α

>

0

and

β

>

1

, there exists a unique triple

(

γ

,

δ

,

ϵ

)

of ordinals such that

α

=

β

γ

⋅

δ

+

ϵ

where

0

<

δ

<

β

and

ϵ

<

β

δ

.

All of these properties can be proved using

transfinite induction

.

Title

ordinal exponentiation

Canonical name

OrdinalExponentiation

Date of creation

2013-03-22 17:51:09

Last modified on

2013-03-22 17:51:09

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03E10

Related topic

PropertiesOfOrdinalArithmetic

Generated on Thu Feb 8 20:10:28 2018 by

LaTeXML
ordinal number

ordinal number

An

ordinal number

is a well

ordered set

S

such that, for every

x

∈

S

,

x

=

{

z

∈

S

∣

z

<

x

}

(where

<

is the

ordering relation

on

S

).

It follows immediately from the definition that every

ordinal

is a

transitive set

.
Also note that if

a

,

b

∈

S

then we have

a

<

b

if and only if

a

∈

b

.

There is a theory of

ordinal arithmetic

which allows construction of various ordinals.
For example, all the numbers

0

,

1

,

2

, …have natural

interpretations

as ordinals,
as does the set of natural numbers (including

0

),
which in this

context

is often denoted by

ω

,
and is the first

infinite

ordinal.

Title

ordinal number

Canonical name

OrdinalNumber

Date of creation

2013-03-22 12:07:55

Last modified on

2013-03-22 12:07:55

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

8

Author

yark (2760)

Entry type

Definition

Classification

msc 03E10

Synonym

ordinal

Related topic

VonNeumannOrdinal

Generated on Thu Feb 8 20:10:30 2018 by

LaTeXML
proof of fixed points of normal functions

proof of fixed points of normal functions

Suppose

f

is a

κ

-

normal function

and consider any

α

<

κ

and define a

sequence

by

α

0

=

α

and

α

n

+

1

=

f

⁢

(

α

n

)

. Let

α

ω

=

sup

n

<

ω

⁡

α

n

. Then, since

f

is

continuous

,

f

⁢

(

α

ω

)

=

sup

n

<

ω

⁡

f

⁢

(

α

n

)

=

sup

n

<

ω

⁡

α

n

+

1

=

α

ω

So

Fix

⁡

(

f

)

is

unbounded

.

Suppose

N

is a set of

fixed points

of

f

with

|

N

|

<

κ

. Then

f

⁢

(

sup

⁡

N

)

=

sup

α

∈

N

⁡

f

⁢

(

α

)

=

sup

α

∈

N

⁡

α

=

sup

⁡

N

so

sup

⁡

N

is also a fixed point of

f

, and therefore

Fix

⁡

(

f

)

is closed.

Title

proof of fixed points of normal functions

Canonical name

ProofOfFixedPointsOfNormalFunctions

Date of creation

2013-03-22 13:29:01

Last modified on

2013-03-22 13:29:01

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

4

Author

Henry (455)

Entry type

Proof

Classification

msc 03E10

Generated on Thu Feb 8 20:10:41 2018 by

LaTeXML
proof of Fodor’s lemma

proof of Fodor’s lemma

If we let

f

-

1

:

κ

→

P

⁢

(

S

)

be the

inverse

of

f

restricted to

S

then Fodor’s lemma is

equivalent

to the claim that for any

function

such that

α

∈

f

⁢

(

κ

)

→

α

>

κ

there is some

α

∈

S

such that

f

-

1

⁢

(

α

)

is

stationary

.

Then if Fodor’s lemma is false, for every

α

∈

S

there is some

club set

C

α

such that

C

α

∩

f

-

1

⁢

(

α

)

=

∅

. Let

C

=

Δ

α

<

κ

⁢

C

α

. The club sets are

closed under

diagonal intersection

, so

C

is also club and therefore there is some

α

∈

S

∩

C

. Then

α

∈

C

β

for each

β

<

α

, and so there can be no

β

<

α

such that

α

∈

f

-

1

⁢

(

β

)

, so

f

⁢

(

α

)

≥

α

, a

contradiction

.

Title

proof of Fodor’s lemma

Canonical name

ProofOfFodorsLemma

Date of creation

2013-03-22 12:53:19

Last modified on

2013-03-22 12:53:19

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

4

Author

Henry (455)

Entry type

Proof

Classification

msc 03E10

Generated on Thu Feb 8 20:10:43 2018 by

LaTeXML
proof of the existence of transcendental numbers

proof of the existence of transcendental numbers

Cantor discovered this proof.

Lemma:

Consider a

natural number

k

. Then the number of

algebraic numbers

of height

k

is finite.

Proof:

To see this, note the sum in the definition of height is

positive

. Therefore:

n

≤

k

where

n

is the degree of the

polynomial

. For a polynomial of degree

n

, there are only

n

coefficients, and the sum of their moduli is

(

k

-

n

)

, and there is only a finite number of ways of doing this (the number of ways is the number of algebraic numbers). For every polynomial with degree less than

n

, there are less ways. So the sum of all of these is also finite, and this is the number of algebraic numbers with height

k

(with some repetitions). The result follows.

Proof of the main theorem:

You can start writing a list of the algebraic numbers because you can put all the ones with height 1, then with height 2, etc, and write them in numerical order within those sets because they are

finite sets

. This implies that the set of algebraic numbers is

countable

. However, by diagonalisation, the set of real numbers is

uncountable

. So there are more

real numbers

than algebraic numbers; the result follows.

Title

proof of the

existence

of

transcendental numbers

Canonical name

ProofOfTheExistenceOfTranscendentalNumbers

Date of creation

2013-03-22 13:24:36

Last modified on

2013-03-22 13:24:36

Owner

kidburla2003 (1480)

Last modified by

kidburla2003 (1480)

Numerical id

8

Author

kidburla2003 (1480)

Entry type

Proof

Classification

msc 03E10

Related topic

AlgebraicNumbersAreCountable

Generated on Thu Feb 8 20:10:45 2018 by

LaTeXML
proof of theorems in additively indecomposable

proof of theorems in additively indecomposable

•

ℍ

is closed.

Let

{

α

i

∣

i

<

κ

}

be some

increasing sequence

of elements of

ℍ

and let

α

=

sup

⁡

{

α

i

∣

i

<

κ

}

. Then for any

x

,

y

<

α

, it must be that

x

<

α

i

and

y

<

α

j

for some

i

,

j

<

κ

. But then

x

+

y

<

α

max

⁡

{

i

,

j

}

<

α

.

•

ℍ

is

unbounded

.

Consider any

α

, and define a

sequence

by

α

0

=

S

⁢

α

and

α

n

+

1

=

α

n

+

α

n

. Let

α

ω

=

sup

n

<

ω

⁡

α

n

be the limit of this sequence. If

x

,

y

<

α

ω

then it must be that

x

<

α

i

and

y

<

α

j

for some

i

,

j

<

ω

, and therefore

x

+

y

<

α

max

⁡

{

i

,

j

}

+

1

. Note that

α

ω

is, in fact, the next element of

ℍ

, since every

element

in the sequence is clearly additively

decomposable

.

•

f

ℍ

⁢

(

α

)

=

ω

α

.

Since

0

is not in

ℍ

, we have

f

ℍ

⁢

(

0

)

=

1

.

For any

α

+

1

, we have

f

ℍ

⁢

(

α

+

1

)

is the least

additively indecomposable

number greater than

f

ℍ

⁢

(

α

)

. Let

α

0

=

S

⁢

f

ℍ

⁢

(

α

)

and

α

n

+

1

=

α

n

+

α

n

=

α

n

⋅

2

. Then

f

ℍ

⁢

(

α

+

1

)

=

sup

n

<

ω

⁡

α

n

=

sup

n

<

ω

⁡

S

⁢

α

⋅

2

⁢

⋯

⁢

2

=

f

ℍ

⁢

(

α

)

⋅

ω

. The limit case is

trivial

since

ℍ

is closed and unbounded, so

f

ℍ

is continuous.

Title

proof of theorems in additively indecomposable

Canonical name

ProofOfTheoremsInAdditivelyIndecomposable

Date of creation

2013-03-22 13:29:07

Last modified on

2013-03-22 13:29:07

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Proof

Classification

msc 03E10

Classification

msc 03F15

Generated on Thu Feb 8 20:10:47 2018 by

LaTeXML
proof that countable unions are countable

proof that countable unions are countable

Let

C

be a

countable set

of countable sets. We will show that

∪

C

is

countable

.

Let

P

be the set of

positive

primes (

http://planetmath.org/Prime

).

P

is

countably infinite

, so there is a

bijection

between

P

and

ℕ

. Since there is a bijection between

C

and a subset of

ℕ

, there must in turn be a one-to-one function

f

:

C

→

P

.

Each

S

∈

C

is countable, so there exists a bijection between

S

and some subset of

ℕ

. Call this

function

g

, and define a new function

h

S

:

S

→

ℕ

such that for all

x

∈

S

,

h

S

⁢

(

x

)

=

f

⁢

(

S

)

g

⁢

(

x

)

Note that

h

S

is

one-to-one

. Also note that for any distinct pair

S

,

T

∈

C

, the range of

h

S

and the range of

h

T

are

disjoint

due to the

fundamental theorem of arithmetic

.

We may now define a one-to-one function

h

:

∪

C

→

ℕ

, where, for each

x

∈

∪

C

,

h

⁢

(

x

)

=

h

S

⁢

(

x

)

for some

S

∈

C

where

x

∈

S

(the choice of

S

is irrelevant, so long as it contains

x

). Since the range of

h

is a subset of

ℕ

,

h

is a bijection into that set and hence

∪

C

is countable.

Title

proof that countable unions are countable

Canonical name

ProofThatCountableUnionsAreCountable

Date of creation

2013-03-22 12:00:01

Last modified on

2013-03-22 12:00:01

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

9

Author

Koro (127)

Entry type

Proof

Classification

msc 03E10

Generated on Thu Feb 8 20:10:49 2018 by

LaTeXML
proof that the rationals are countable

proof that the rationals are countable

Suppose we have a

rational number

α

=

p

/

q

in

lowest terms

with

q

>

0

. Define the “height” of this number as

h

⁢

(

α

)

=

|

p

|

+

q

. For example,

h

⁢

(

0

)

=

h

⁢

(

0

1

)

=

1

,

h

⁢

(

-

1

)

=

h

⁢

(

1

)

=

2

, and

h

⁢

(

-

2

)

=

h

⁢

(

-

1

2

)

=

h

⁢

(

1

2

)

=

h

⁢

(

2

)

=

3

.

Note that the set of numbers with a given height is finite. The rationals can now be partitioned into classes by height, and the numbers in each class can be ordered by way of increasing numerators. Thus it is possible to assign a

natural number

to each of the rationals by starting with

0

,

-

1

,

1

,

-

2

,

-

1

2

,

1

2

,

2

,

-

3

,

…

and progressing through classes of increasing heights. This assignment constitutes a

bijection

between

ℕ

and

ℚ

and proves that

ℚ

is

countable

.

A corollary is that the

irrational numbers

are

uncountable

, since the union of the irrationals and the rationals is

ℝ

, which is uncountable.

Title

proof that the rationals are countable

Canonical name

ProofThatTheRationalsAreCountable

Date of creation

2013-03-22 11:59:53

Last modified on

2013-03-22 11:59:53

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

9

Author

alozano (2414)

Entry type

Proof

Classification

msc 03E10

Related topic

RationalNumber

Related topic

IrrationalNumber

Related topic

Countable

Related topic

Irrational

Generated on Thu Feb 8 20:10:51 2018 by

LaTeXML
properties of ordinal arithmetic

properties of ordinal arithmetic

Let

On

be the class of

ordinals

, and

α

,

β

,

γ

,

δ

∈

𝐎𝐧

. Then the following

properties

are satisfied:

1.

(

additive identity

):

α

+

0

=

0

+

α

=

α

(proof (

http://planetmath.org/ExampleOfTransfiniteInduction

))

2.

(

associativity

of

addition

):

α

+

(

β

+

γ

)

=

(

α

+

β

)

+

γ

3.

(

multiplicative identity

):

α

⋅

1

=

1

⋅

α

=

α

4.

(multiplicative zero):

α

⋅

0

=

0

⋅

α

=

0

5.

(

associativity of multiplication

):

α

⋅

(

β

⋅

γ

)

=

(

α

⋅

β

)

⋅

γ

6.

(

left distributivity

):

α

⋅

(

β

+

γ

)

=

α

⋅

β

+

α

⋅

γ

7.

(

existence

and uniqueness of

subtraction

): if

α

≤

β

, then there is a unique

γ

such that

α

+

γ

=

β

8.

(existence and uniqueness of

division

): for any

α

,

β

with

β

≠

0

, there exists a unique pair of ordinals

γ

,

δ

such that

α

=

β

⋅

δ

+

γ

and

γ

<

β

.

Conspicuously absent from the above list of properties are the

commutativity

laws, as well as right distributivity of

multiplication

over addition. Below are some

counterexamples

:

•

ω

+

1

≠

1

+

ω

=

ω

, for the former has a top

element

and the latter does not.

•

ω

⋅

2

≠

2

⋅

ω

, for the former is

ω

+

ω

, which consists an element

α

such that

β

<

α

for all

β

<

ω

, and the latter is

2

⋅

sup

⁡

{

n

∣

n

<

ω

}

=

sup

⁡

{

2

⋅

n

∣

n

<

ω

}

=

sup

⁡

{

n

∣

n

<

ω

}

, which is just

ω

, and which does not consist such an element

α

•

(

1

+

1

)

⋅

ω

≠

1

⋅

ω

+

1

⋅

ω

, for the former is

2

⋅

ω

and the latter is

ω

⋅

2

, and the rest of the follows from the previous counterexample.

All of the properties above can be proved using

transfinite induction

. For a proof of the first property, please see this link (

http://planetmath.org/ExampleOfTransfiniteInduction

).

For properties of the

arithmetic

regarding

exponentiation

of ordinals, please refer to this link (

http://planetmath.org/OrdinalExponentiation

).

Title

properties of ordinal arithmetic

Canonical

name

PropertiesOfOrdinalArithmetic

Date of creation

2013-03-22 17:51:05

Last modified on

2013-03-22 17:51:05

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Result

Classification

msc 03E10

Related topic

OrdinalExponentiation

Generated on Thu Feb 8 20:11:01 2018 by

LaTeXML
Schröder-Bernstein theorem

Schröder-Bernstein theorem

Let

S

and

T

be sets.
If there are

injections

S

→

T

and

T

→

S

,
then there is a

bijection

S

→

T

.

The Schröder-Bernstein

theorem

is useful
for proving many results about

cardinality

,
since it replaces one hard problem (finding a bijection between

S

and

T

)
with two generally easier problems (finding two injections).

Title

Schröder-Bernstein theorem

Canonical name

SchroderBernsteinTheorem

Date of creation

2013-03-22 12:21:46

Last modified on

2013-03-22 12:21:46

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Theorem

Classification

msc 03E10

Synonym

Schroeder-Bernstein theorem

Synonym

Cantor-Schroeder-Bernstein theorem

Synonym

Cantor-Schröder-Bernstein theorem

Synonym

Cantor-Bernstein theorem

Related topic

AnInjectionBetweenTwoFiniteSetsOfTheSameCardinalityIsBijective

Related topic

ProofOfSchroederBernsteinTheoremUsingTarskiKnasterTheorem

Generated on Thu Feb 8 20:11:15 2018 by

LaTeXML
Schroeder-Bernstein theorem, proof of

Schroeder-Bernstein theorem, proof of

We first prove as a lemma that for any

B

⊂

A

, if there is an

injection

f

:

A

→

B

, then there is also a

bijection

h

:

A

→

B

.

Inductively define a

sequence

(

C

n

)

of subsets of

A

by

C

0

=

A

∖

B

and

C

n

+

1

=

f

⁢

(

C

n

)

.
Now let

C

=

⋃

k

=

0

∞

C

k

, and define

h

:

A

→

B

by

h

⁢

(

z

)

=

{

f

⁢

(

z

)

,

z

∈

C

z

,

z

∉

C

.

If

z

∈

C

, then

h

⁢

(

z

)

=

f

⁢

(

z

)

∈

B

. But if

z

∉

C

, then

z

∈

B

, and so

h

⁢

(

z

)

∈

B

. Hence

h

is

well-defined

;

h

is
injective by construction. Let

b

∈

B

. If

b

∉

C

, then

h

⁢

(

b

)

=

b

. Otherwise,

b

∈

C

k

=

f

⁢

(

C

k

-

1

)

for some

k

>

0

, and
so there is some

a

∈

C

k

-

1

such that

h

⁢

(

a

)

=

f

⁢

(

a

)

=

b

. Thus

h

is

bijective

; in particular, if

B

=

A

, then

h

is simply the

identity
map

on

A

.

To prove the

theorem

, suppose

f

:

S

→

T

and

g

:

T

→

S

are injective. Then the

composition

g

⁢

f

:

S

→

g

⁢

(

T

)

is also
injective. By the lemma, there is a bijection

h

′

:

S

→

g

⁢

(

T

)

.
The injectivity of

g

implies that

g

-

1

:

g

⁢

(

T

)

→

T

exists
and is bijective. Define

h

:

S

→

T

by

h

⁢

(

z

)

=

g

-

1

⁢

h

′

⁢

(

z

)

; this
map is a bijection, and so

S

and

T

have the same

cardinality

.

Title

Schroeder-Bernstein theorem

, proof of

Canonical

name

SchroederBernsteinTheoremProofOf

Date of creation

2013-03-22 12:49:56

Last modified on

2013-03-22 12:49:56

Owner

mps (409)

Last modified by

mps (409)

Numerical id

19

Author

mps (409)

Entry type

Proof

Classification

msc 03E10

Synonym

proof of Cantor-Bernstein theorem

Synonym

proof of Cantor-Schroeder-Bernstein theorem

Related topic

AnInjectionBetweenTwoFiniteSetsOfTheSameCardinalityIsBijective

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

LaTeXML
sequence

sequence

Sequences

Given any set

X

, a

sequence

in

X

is a

function

f

:

ℕ

→

X

from the set of natural numbers to

X

. Sequences are usually written with

subscript

notation:

x

0

,

x

1

,

x

2

⁢

…

, instead of

f

⁢

(

0

)

,

f

⁢

(

1

)

,

f

⁢

(

2

)

⁢

…

.

Generalized sequences

One can generalize the above definition to any arbitrary

ordinal

. For any set

X

, a

generalized sequence

or

transfinite sequence

in

X

is a function

f

:

ω

→

X

where

ω

is any

ordinal number

. If

ω

is a finite ordinal, then we say the sequence is a

finite sequence

.

Title

sequence

Canonical name

Sequence

Date of creation

2013-03-22 11:50:33

Last modified on

2013-03-22 11:50:33

Owner

djao (24)

Last modified by

djao (24)

Numerical id

11

Author

djao (24)

Entry type

Definition

Classification

msc 03E10

Classification

msc 40-00

Related topic

ConvergentSequence

Defines

generalized sequence

Defines

transfinite sequence

Defines

finite sequence

Generated on Thu Feb 8 20:11:19 2018 by

LaTeXML
stationary set

stationary set

If

κ

is a

cardinal

,

C

⊆

κ

, and

C

intersects

every club in

κ

then

C

is

stationary

. If

C

is not stationary then it is

thin

.

Title

stationary set

Canonical name

StationarySet

Date of creation

2013-03-22 12:53:03

Last modified on

2013-03-22 12:53:03

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

4

Author

Henry (455)

Entry type

Definition

Classification

msc 03E10

Related topic

FodorsLemma

Defines

stationary

Defines

stationary set

Defines

thin

Defines

thin set

Generated on Thu Feb 8 20:11:21 2018 by

LaTeXML
subsets of countable sets are countable

subsets of countable sets are countable

The definition of

countable sets

would not serve us well if it did not conform with our intuition about countable sets. So let us prove that countability is in a sense hereditary.

Theorem 1

.

Every subset of a countable set is itself

countable

.

Proof.

Let

B

⊆

A

and

A

countable with

f

:

A

→

K

,

K

⊆

ℕ

a

bijective function

as in the definition of countable sets.

Let us consider

f

|

B

, the

function

f

restricted to

B

, i.e.

f

|

B

:

B

→

f

⁢

(

B

)

. Then

f

|

B

is trivially onto, but also

one-to-one

(

f

was one-to-one!). So we have a bijective function from

B

onto

f

⁢

(

B

)

⊆

K

⊆

ℕ

, which the proof.
∎

Title

subsets of countable sets are countable

Canonical name

SubsetsOfCountableSetsAreCountable

Date of creation

2013-03-22 15:45:56

Last modified on

2013-03-22 15:45:56

Owner

beke (12826)

Last modified by

beke (12826)

Numerical id

7

Author

beke (12826)

Entry type

Corollary

Classification

msc 03E10

Generated on Thu Feb 8 20:11:23 2018 by

LaTeXML
successor

successor

Given a set

S

, the

successor

of

S

is the set

S

∪

{

S

}

. One often denotes the successor of

S

by

S

′

.

Title

successor

Canonical name

Successor

Date of creation

2013-03-22 11:49:58

Last modified on

2013-03-22 11:49:58

Owner

djao (24)

Last modified by

djao (24)

Numerical id

7

Author

djao (24)

Entry type

Definition

Classification

msc 03E10

Classification

msc 70K75

Related topic

NaturalNumber

Generated on Thu Feb 8 20:11:24 2018 by

LaTeXML