aarjaneiro/planetmath huggingface dataset:
partial order with chain condition does not collapse cardinals

partial order with chain condition does not collapse cardinals

If

P

is a

partial order

which

satisfies

the

κ

chain condition

and

G

is a

generic

subset of

P

then for any

κ

<

λ

∈

𝔐

,

λ

is also a

cardinal

in

𝔐

⁢

[

G

]

, and if

cf

⁡

(

α

)

=

λ

in

𝔐

then also

cf

⁡

(

α

)

=

λ

in

𝔐

⁢

[

G

]

.

This

theorem

is the simplest way to control a notion of

forcing

, since it means that a notion of forcing does not have an effect above a certain point. Given that any

P

satisfies the

|

P

|

+

chain condition, this means that most forcings leaves all of

𝔐

above a certain point alone. (Although it is possible to get around this limit by forcing with a

proper class

.)

Title

partial order with chain condition does not collapse cardinals

Canonical name

PartialOrderWithChainConditionDoesNotCollapseCardinals

Date of creation

2013-03-22 12:53:40

Last modified on

2013-03-22 12:53:40

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 03E35

Related topic

PartialOrder

Related topic

ChainCondition

Generated on Thu Feb 8 20:19:18 2018 by

LaTeXML
proof of partial order with chain condition does not collapse cardinals

proof of partial order with chain condition does not collapse cardinals

Outline:

Given any

function

f

purporting to violate the

theorem

by being

surjective

(or

cofinal

) on

λ

, we show that there are fewer than

κ

possible values of

f

⁢

(

α

)

, and therefore only

max

⁡

(

α

,

κ

)

possible

elements

in the entire range of

f

, so

f

is not surjective (or cofinal).

Details:

Suppose

λ

>

κ

is a

cardinal

of

𝔐

that is not a cardinal in

𝔐

⁢

[

G

]

.

There is some function

f

∈

𝔐

⁢

[

G

]

and some cardinal

α

<

λ

such that

f

:

α

→

λ

is surjective. This has a name,

f

^

. For each

β

<

α

, consider

F

β

=

{

γ

<

λ

∣

p

⊩

f

^

⁢

(

β

)

=

γ

}

⁢

for some

⁢

p

∈

P

|

F

β

|

<

κ

, since any two

p

∈

P

which force different values for

f

^

⁢

(

β

)

are incompatible and

P

has no sets of incompatible elements of size

κ

.

Notice that

F

β

is

definable

in

𝔐

. Then the range of

f

must be

contained

in

F

=

⋃

i

<

α

F

i

. But

|

F

|

≤

α

⋅

κ

=

max

⁡

(

α

,

κ

)

<

λ

. So

f

cannot possibly be surjective, and therefore

λ

is not collapsed.

Now suppose that for some

α

≥

λ

>

κ

,

cf

⁡

(

α

)

=

λ

in

𝔐

and for some

η

<

λ

there is a cofinal function

f

:

η

→

α

.

We can construct

F

β

as above, and again the range of

f

is contained in

F

=

⋃

i

<

η

F

i

. But then

|

range

⁡

(

f

)

|

≤

|

F

|

≤

η

⋅

κ

<

λ

. So there is some

γ

<

α

such that

f

⁢

(

β

)

<

γ

for any

β

<

η

, and therefore

f

is not cofinal in

α

.

Title

proof of partial order with chain condition does not collapse cardinals

Canonical name

ProofOfPartialOrderWithChainConditionDoesNotCollapseCardinals

Date of creation

2013-03-22 12:53:43

Last modified on

2013-03-22 12:53:43

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

4

Author

Henry (455)

Entry type

Proof

Classification

msc 03E35

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

LaTeXML
complete partial orders do not add small subsets

complete partial orders do not add small subsets

Suppose

P

is a

κ

-complete partial order in

𝔐

. Then for any

generic

subset

G

,

𝔐

contains no bounded subsets of

κ

which are not in

𝔐

.

Title

complete partial orders do not add small subsets

Canonical name

CompletePartialOrdersDoNotAddSmallSubsets

Date of creation

2013-03-22 12:53:32

Last modified on

2013-03-22 12:53:32

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

4

Author

Henry (455)

Entry type

Theorem

Classification

msc 03E40

Generated on Thu Feb 8 20:19:22 2018 by

LaTeXML
composition preserves chain condition

composition preserves chain condition

Let

κ

be a

regular cardinal

. Let

P

be a

forcing

notion satisfying the

κ

chain condition

. Let

Q

^

be a

P

-name such that

⊩

P

Q

^

is a forcing notion satisfying the

κ

chain condition

. Then

P

*

Q

satisfies

the

κ

chain condition.

Proof:

Outline

We prove that there is some

p

such that any

generic

subset of

P

including

p

also includes

κ

of the

p

i

. Then, since

Q

⁢

[

G

]

satisfies the

κ

chain condition, two of the corresponding

q

^

i

must be

compatible

. Then, since

G

is directed, there is some

p

stronger than any of these which forces this to be true, and therefore makes two elements of

S

compatible.

Let

S

=

⟨

p

i

,

q

^

i

⟩

i

<

κ

⊆

P

*

Q

.

Claim: There is some

p

∈

P

such that

p

⊩

|

{

i

∣

p

i

∈

G

^

}

|

=

κ

(Note:

G

^

=

{

⟨

p

,

p

⟩

∣

p

∈

P

}

, hence

G

^

⁢

[

G

]

=

G

)

If no

p

forces this then every

p

forces that it is not true, and therefore

⊩

P

|

{

i

∣

p

i

∈

G

}

|

≤

κ

. Since

κ

is

regular

, this means that for any generic

G

⊆

P

,

{

i

∣

p

i

∈

G

}

is bounded. For each

G

, let

f

⁢

(

G

)

be the least

α

such that

β

<

α

implies that there is some

γ

>

β

such that

p

γ

∈

G

. Define

B

=

{

α

∣

α

=

f

⁢

(

G

)

}

for some

G

.

Claim:

|

B

|

<

κ

If

α

∈

B

then there is some

p

α

∈

P

such that

p

⊩

f

⁢

(

G

^

)

=

α

, and if

α

,

β

∈

B

then

p

α

must be incompatible with

p

β

. Since

P

satisfies the

κ

chain condition, it follows that

|

B

|

<

κ

.

Since

κ

is regular,

α

=

sub

⁡

(

B

)

<

κ

. But obviously

p

α

+

1

⊩

p

α

+

1

∈

G

^

. This is a

contradiction

, so we conclude that there must be some

p

such that

p

⊩

|

{

i

∣

p

i

∈

G

^

}

|

=

κ

.

If

G

⊆

P

is any generic subset containing

p

then

A

=

{

q

^

i

⁢

[

G

]

∣

p

i

∈

G

}

must have

cardinality

κ

. Since

Q

⁢

[

G

]

satisfies the

κ

chain condition, there exist

i

,

j

<

κ

such that

p

i

,

p

j

∈

G

and there is some

q

^

⁢

[

G

]

∈

Q

⁢

[

G

]

such that

q

^

⁢

[

G

]

≤

q

^

i

⁢

[

G

]

,

q

^

j

⁢

[

G

]

. Then since

G

is directed, there is some

p

′

∈

G

such that

p

′

≤

p

i

,

p

j

,

p

and

p

′

⊩

q

^

⁢

[

G

]

≤

q

^

1

⁢

[

G

]

,

q

^

2

⁢

[

G

]

. So

⟨

p

′

,

q

^

⟩

≤

⟨

p

i

,

q

^

i

⟩

,

⟨

p

j

,

q

^

j

⟩

.

Title

composition preserves chain condition

Canonical name

CompositionPreservesChainCondition

Date of creation

2013-03-22 12:54:40

Last modified on

2013-03-22 12:54:40

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Result

Classification

msc 03E40

Classification

msc 03E35

Generated on Thu Feb 8 20:19:25 2018 by

LaTeXML
proof of complete partial orders do not add small subsets

proof of complete partial orders do not add small subsets

Take any

x

∈

𝔐

⁢

[

G

]

,

x

⊆

κ

. Let

x

^

be a name for

x

. There is some

p

∈

G

such that

p

⊩

x

^

⁢

is a subset of

⁢

κ

⁢

bounded by

⁢

λ

<

κ

Outline:

For any

q

≤

p

, we construct by

induction

a series of

elements

q

α

stronger than

p

. Each

q

α

will determine whether or not

α

∈

x

^

. Since we know the subset is bounded below

κ

, we can use the fact that

P

is

κ

complete

to find a single element stronger than

q

which fixes the exact value of

x

^

. Since the series is

definable

in

𝔐

, so is

x

^

, so we can conclude that above any element

q

≤

p

is an element which forces

x

^

∈

𝔐

. Then

p

also forces

x

^

∈

𝔐

, completing the proof.

Details:

Since

forcing

can be described within

𝔐

,

S

=

{

q

∈

P

∣

q

⊩

x

^

∈

V

}

is a set in

𝔐

. Then, given any

q

≤

p

, we can define

q

0

=

q

and for any

q

α

(

α

<

λ

),

q

α

+

1

is an element of

P

stronger than

q

α

such that either

q

α

+

1

⊩

α

+

1

∈

x

^

or

q

α

+

1

⊩

α

+

1

∉

x

^

. For limit

α

, let

q

α

′

be any upper bound of

q

β

for

α

<

β

(this exists since

P

is

κ

-complete and

α

<

κ

), and let

q

α

be stronger than

q

α

′

and

satisfy

either

q

α

+

1

⊩

α

∈

x

^

or

q

α

+

1

⊩

α

∉

x

^

. Finally let

q

*

be the upper bound of

q

α

for

α

<

λ

.

q

*

∈

P

since

P

is

κ

-complete.

Note that these elements all exist since for any

p

∈

P

and any (first-order)

sentence

ϕ

there is some

q

≤

p

such that

q

forces either

ϕ

or

¬

⁢

ϕ

.

q

*

not only forces that

x

^

is a bounded subset of

κ

, but for every

ordinal

it forces whether or not that ordinal is

contained

in

x

^

. But the set

{

α

⁢

<

λ

∣

⁢

q

*

⊩

α

∈

x

^

}

is defineable in

𝔐

, and is of course equal to

x

^

⁢

[

G

*

]

in any

generic

G

*

containing

q

*

. So

q

*

⊩

x

^

∈

𝔐

.

Since this holds for any element stronger than

p

, it follows that

p

⊩

x

^

∈

𝔐

, and therefore

x

^

⁢

[

G

]

∈

𝔐

.

Title

proof of complete partial orders do not add small subsets

Canonical name

ProofOfCompletePartialOrdersDoNotAddSmallSubsets

Date of creation

2013-03-22 12:53:35

Last modified on

2013-03-22 12:53:35

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

4

Author

Henry (455)

Entry type

Proof

Classification

msc 03E40

Generated on Thu Feb 8 20:19:27 2018 by

LaTeXML
◇ is equivalent to ♣ and continuum hypothesis

◇

is equivalent to

♣

and continuum hypothesis

If

S

is a

stationary

subset of

κ

and

λ

<

κ

implies

2

λ

≤

κ

then

◇

S

↔

♣

S

Moreover, this is best possible:

¬

⁢

◇

S

is

consistent

with

♣

S

.

Title

◇

is

equivalent

to

♣

and

continuum hypothesis

Canonical name

DiamondIsEquivalentToclubsuitAndContinuumHypothesis

Date of creation

2013-03-22 12:53:55

Last modified on

2013-03-22 12:53:55

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

6

Author

Henry (455)

Entry type

Theorem

Classification

msc 03E45

Synonym

diamond is equivalent to club and continuum hypothesis

Related topic

Diamond

Related topic

Clubsuit

Generated on Thu Feb 8 20:19:36 2018 by

LaTeXML
Levy collapse

Levy collapse

Given any

cardinals

κ

and

λ

in

𝔐

, we can use the

Levy collapse

to give a new model

𝔐

⁢

[

G

]

where

λ

=

κ

. Let

P

=

Levy

⁡

(

κ

,

λ

)

be the set of

partial functions

f

:

κ

→

λ

with

|

dom

⁡

(

f

)

|

<

κ

. These

functions

each give partial information about a function

F

which collapses

λ

onto

κ

.

Given any

generic

subset

G

of

P

,

𝔐

⁢

[

G

]

has a set

G

, so let

F

=

⋃

G

. Each element of

G

is a partial function, and they are all

compatible

, so

F

is a function.

dom

⁡

(

G

)

=

κ

since for each

α

<

κ

the set of

f

∈

P

such that

α

∈

dom

⁡

(

f

)

is dense (given any function without

α

, it is

trivial

to add

(

α

,

0

)

, giving a stronger function which includes

α

). Also

range

⁡

(

G

)

=

λ

since the set of

f

∈

P

such that

α

<

λ

is in the range of

f

is again dense (the domain of each

f

is bounded, so if

β

is larger than any element of

dom

⁡

(

f

)

,

f

∪

{

(

β

,

α

)

}

is stronger than

f

and includes

λ

in its domain).

So

F

is a surjective function from

κ

to

λ

, and

λ

is collapsed in

𝔐

⁢

[

G

]

. In

addition

,

|

Levy

⁡

(

κ

,

λ

)

|

=

λ

, so it

satisfies

the

λ

+

chain condition

, and therefore

λ

+

is not collapsed, and becomes

κ

+

(since for any

ordinal

between

λ

and

λ

+

there is already a surjective function to it from

λ

).

We can generalize this by

forcing

with

P

=

Levy

(

κ

,

<

λ

)

with

κ

regular

, the set of partial functions

f

:

λ

×

κ

→

λ

such that

f

⁢

(

0

,

α

)

=

0

,

|

dom

⁡

(

f

)

|

<

κ

and if

α

>

0

then

f

⁢

(

α

,

i

)

<

α

. In essence, this is the

product

of

Levy

⁡

(

κ

,

η

)

for each

η

<

λ

.

In

𝔐

⁢

[

G

]

, define

F

=

⋃

G

and

F

α

⁢

(

β

)

=

F

⁢

(

α

,

β

)

. Each

F

α

is a function from

κ

to

α

, and by the same

argument

as above

F

α

is both total and

surjective

. Moreover, it can be shown that

P

satisfies the

λ

chain condition, so

λ

does not collapse and

λ

=

κ

+

.

Title

Levy collapse

Canonical name

LevyCollapse

Date of creation

2013-04-16 22:08:32

Last modified on

2013-04-16 22:08:32

Owner

ratboy (4018)

Last modified by

e1568582 (1000182)

Numerical id

9

Author

ratboy (1000182)

Entry type

Example

Classification

msc 03E45

Generated on Thu Feb 8 20:19:38 2018 by

LaTeXML
transfinite recursion

transfinite recursion

Transfinite recursion

, roughly speaking, is a statement about the ability to define a

function

recursively using

transfinite induction

. In its most general and intuitive form, it says

Theorem 1

.

Let

G

be a (class) function on

V

, the class of all sets. Then there exists a unique (class) function

F

on

On

, the class of

ordinals

, such that

F

(

α

)

=

G

(

F

|

α

)

where

F

|

α

is the function whose domain is

seg

⁡

(

α

)

:=

{

β

∣

β

<

α

}

and whose values coincide with

F

on every

β

∈

seg

⁡

(

α

)

. In other words,

F

|

α

is the

restriction

of

F

to

α

.

Notice that the

theorem

above is not provable in ZF

set theory

, as

G

and

F

are both classes, not sets. In order to prove this statement, one way of getting around this difficulty is to convert both

G

and

F

into

formulas

, and modify the statement, as follows:

Let

φ

⁢

(

x

,

y

)

be a formula such that

∀

x

∃

y

∀

z

(

φ

(

x

,

z

)

↔

z

=

y

)

.

Think of

G

=

{

(

x

,

y

)

∣

φ

⁢

(

x

,

y

)

}

. Then there is a unique formula

ψ

⁢

(

α

,

z

)

(think of

F

as

{

(

α

,

z

)

∣

ψ

⁢

(

α

,

z

)

}

) such that the following two

sentences

are

derivable

using ZF axioms:

1.

∀

x

∃

y

∀

z

(

𝐎𝐧

(

x

)

∧

(

ψ

(

x

,

z

)

↔

z

=

y

)

)

,

where

𝐎𝐧

⁢

(

x

)

means “

x

is an ordinal”,

2.

∀

x

∀

y

(

𝐎𝐧

(

x

)

∧

(

ψ

(

x

,

y

)

↔

∃

f

(

A

∧

B

∧

C

∧

D

)

)

)

, where

–

A

is the formula “

f

is a function”,

–

B

is the formula “

dom

⁡

(

f

)

=

x

”,

–

C

is the formula

∀

z

(

z

∈

x

∧

φ

(

f

|

z

,

f

(

z

)

)

)

, and

–

D

is the formula

φ

⁢

(

f

,

y

)

.

A stronger form of the transfinite recursion theorem says:

Theorem 2

.

Let

φ

⁢

(

x

,

y

)

be any formula (in the

language

of set theory). Then the following is a theorem: assume that

φ

satisfies

property that, for every

x

, there is a unique

y

such that

φ

⁢

(

x

,

y

)

. If

A

be a

well-ordered set

(well-ordered by

≤

), then there is a unique function

f

defined on

A

such that

φ

(

f

|

seg

(

s

)

,

f

(

s

)

)

for every

s

∈

A

. Here,

seg

⁡

(

s

)

:=

{

t

∈

A

∣

t

<

s

}

, the

initial segment

of

s

in

A

.

The above theorem is actually a

collection

of theorems, or known as a theorem schema, where each theorem corresponds to a formula. The other

difference

between this and the previous theorem is that this theorem is provable in ZF, because the domain of the function

f

is now a set.

Title

transfinite recursion

Canonical name

TransfiniteRecursion

Date of creation

2013-03-22 17:53:51

Last modified on

2013-03-22 17:53:51

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 03E45

Classification

msc 03E10

Related topic

WellFoundedRecursion

Related topic

TransfiniteInduction

Generated on Thu Feb 8 20:19:52 2018 by

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a shorter proof: Martin’s axiom and the continuum hypothesis

a shorter proof: Martin’s axiom and the continuum hypothesis

This is another, shorter, proof for the fact that

M

⁢

A

ℵ

0

always holds.

Let

(

P

,

≤

)

be a

partially ordered set

and

𝒟

be a

collection

of subsets of

P

. We remember that a filter

G

on

(

P

,

≤

)

is

𝒟

-

generic

if

G

∩

D

≠

∅

for all

D

∈

𝒟

which are

dense in

(

P

,

≤

)

. (In this

context

“dense” means: If

D

is dense in

(

P

,

≤

)

, then for every

p

∈

P

there’s a

d

∈

D

such that

d

≤

p

.)

Let

(

P

,

≤

)

be a partially ordered set and

𝒟

a

countable

collection of

dense subsets

of

P

. Then there exists a

𝒟

-generic filter

G

on

P

. Moreover, it could be shown that for every

p

∈

P

there’s such a

𝒟

-generic filter

G

with

p

∈

G

.

Proof.

Let

D

1

,

…

,

D

n

,

…

be the dense subsets in

𝒟

. Furthermore let

p

0

=

p

. Now we can choose for every

1

≤

n

<

ω

an

element

p

n

∈

P

such that

p

n

≤

p

n

-

1

and

p

n

∈

D

n

. If we now consider the set

G

:=

{

q

∈

P

∣

∃

n

<

ω

⁢

s.t.

⁢

p

n

≤

q

}

, then it is easy to check that

G

is a

𝒟

-generic filter on

P

and

p

∈

G

obviously. This

completes

the proof.
∎

Title

a shorter proof: Martin’s axiom and the

continuum hypothesis

Canonical name

AShorterProofMartinsAxiomAndTheContinuumHypothesis

Date of creation

2013-03-22 13:53:58

Last modified on

2013-03-22 13:53:58

Owner

x_bas (2940)

Last modified by

x_bas (2940)

Numerical id

11

Author

x_bas (2940)

Entry type

Proof

Classification

msc 03E50

Defines

𝒟

-generic

Defines

generic

Defines

dense

Generated on Thu Feb 8 20:19:54 2018 by

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continuum hypothesis

continuum hypothesis

The

continuum hypothesis

that there is no

cardinal number

κ

such that

ℵ

0

<

κ

<

2

ℵ

0

.

An

equivalent

statement is that

ℵ

1

=

2

ℵ

0

.

It is known to be of the axioms of ZFC.

The continuum hypothesis can also be stated as: there is no subset of the

real numbers

which has

cardinality

strictly between

that of the reals and that of the integers. It is from this that the name comes, since the set of real numbers is also known as the

continuum

.

Title

continuum hypothesis

Canonical name

ContinuumHypothesis

Date of creation

2013-03-22 12:05:29

Last modified on

2013-03-22 12:05:29

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

14

Author

rspuzio (6075)

Entry type

Axiom

Classification

msc 03E50

Synonym

CH

Related topic

AxiomOfChoice

Related topic

ZermeloFraenkelAxioms

Related topic

GeneralizedContinuumHypothesis

Defines

continuum

Generated on Thu Feb 8 20:19:56 2018 by

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generalized continuum hypothesis

generalized continuum hypothesis

The

generalized continuum hypothesis

states that for any

infinite

cardinal

λ

there is no cardinal

κ

such that

λ

<

κ

<

2

λ

.

An

equivalent

condition is that

ℵ

α

+

1

=

2

ℵ

α

for every

ordinal

α

.
Another equivalent condition is that

ℵ

α

=

ℶ

α

for every ordinal

α

.

Like the

continuum hypothesis

, the generalized continuum hypothesis is known to be

independent

of the axioms of ZFC.

Title

generalized continuum hypothesis

Canonical name

GeneralizedContinuumHypothesis

Date of creation

2013-03-22 12:05:31

Last modified on

2013-03-22 12:05:31

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

15

Author

yark (2760)

Entry type

Axiom

Classification

msc 03E50

Synonym

generalised continuum hypothesis

Synonym

GCH

Related topic

AlephNumbers

Related topic

BethNumbers

Related topic

ContinuumHypothesis

Related topic

Cardinality

Related topic

CardinalExponentiationUnderGCH

Related topic

ZermeloFraenkelAxioms

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

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Martin’s axiom

Martin’s axiom

For any cardinal

κ

,

Martin’s Axiom for

κ

(

M

⁢

A

κ

) states that if

P

is a

partial order

satisfying ccc then given any set of

κ

dense (

http://planetmath.org/DenseInAPoset

) subsets of

P

, there is a directed (

http://planetmath.org/DirectedSet

) subset intersecting each such subset.

Martin’s Axiom

states that

M

⁢

A

κ

holds for every

κ

<

2

ℵ

0

.

Title

Martin’s axiom

Canonical

name

MartinsAxiom

Date of creation

2013-03-22 12:54:55

Last modified on

2013-03-22 12:54:55

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

6

Author

Henry (455)

Entry type

Definition

Classification

msc 03E50

Defines

MA

Generated on Thu Feb 8 20:20:06 2018 by

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Sierpiński set of Euclidean plane

Sierpiński set of Euclidean plane

A subset

S

of

ℝ

2

is called a

Sierpiński set

of the plane, if every line

parallel

to the

x

-axis

intersects

S

only in countably many points and every line parallel to the

y

-axis avoids

S

in only countably many points:

{

x

∈

ℝ

⋮

(

x

,

y

)

∈

S

}

is countable for all

y

∈

ℝ

{

y

∈

ℝ

⋮

(

x

,

y

)

∉

S

}

is countable for all

x

∈

ℝ

The

existence

of Sierpiński sets is

equivalent

(

http://planetmath.org/Equivalent3

) with the

continuum hypothesis

, as is proved in [1].

References

1

Gerald Kuba

:  “Wie plausibel ist die Kontinuumshypothese?”.  –

Elemente der Mathematik

61

(2006).

Title

Sierpiński set of

Euclidean plane

Canonical name

SierpinskiSetOfEuclideanPlane

Date of creation

2013-05-18 23:13:46

Last modified on

2013-05-18 23:13:46

Owner

pahio (2872)

Last modified by

unlord (1)

Numerical id

11

Author

pahio (1)

Entry type

Definition

Classification

msc 03E50

Related topic

Countable

Defines

Sierpinski set

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

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inaccessible cardinal

inaccessible cardinal

A

limit cardinal

κ

is a

strong limit cardinal

if for any

λ

<

κ

,

2

λ

<

κ

.

A

regular

(

http://planetmath.org/cofinality

) limit cardinal

κ

is called

weakly inaccessible

, and a regular strong limit cardinal is called

inaccessible

.

Title

inaccessible cardinal

Canonical name

InaccessibleCardinal

Date of creation

2013-03-22 12:52:46

Last modified on

2013-03-22 12:52:46

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

7

Author

Henry (455)

Entry type

Definition

Classification

msc 03E55

Related topic

CardinalNumber

Defines

inaccessible

Defines

weakly inaccessible

Defines

strongly inaccessible

Defines

strong limit

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

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measurable and real-valued measurable cardinals

measurable and real-valued measurable cardinals

Let

κ

be an

uncountable

cardinal

. Then

1.

κ

is

measurable

if there exists a nonprincipal

κ

-

complete

ultrafilter

U

on

κ

;

2.

κ

is

real-valued measurable

if there exists a nontrivial

κ

-

additive

measure

μ

on

κ

.

If

κ

is measurable, then it is real-valued measurable. This is so because the ultrafilter

U

and its dual ideal

I

induce a two-valued measure

μ

on

κ

where every member of

U

is mapped to 1 and every member of

I

is mapped to 0. Since

U

is

κ

-complete,

I

is also

κ

-complete. It can then be proved that if

I

μ

–the ideal of those sets whose measures are 0–is

κ

-complete, then

I

μ

is

κ

-additive.

On the

converse

side, if

κ

is not real-valued measurable, then

κ

≤

2

ℵ

0

. It can be shown that if

κ

is real-valued measurable, then it is

regular

; a further result is that

κ

is

weakly inaccessible

.

Inaccessible cardinals

are in some sense ”large.”

Title

measurable and real-valued measurable cardinals

Canonical name

MeasurableAndRealvaluedMeasurableCardinals

Date of creation

2013-03-22 18:54:53

Last modified on

2013-03-22 18:54:53

Owner

yesitis (13730)

Last modified by

yesitis (13730)

Numerical id

4

Author

yesitis (13730)

Entry type

Definition

Classification

msc 03E55

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