Martin’s axiom and the continuum hypothesis

Martin’s axiom and the continuum hypothesis

M

⁢

A

ℵ

0

always holds

Given a

countable

collection

of dense subsets of a

partial order

, we can selected a set

⟨

p

n

⟩

n

<

ω

such that

p

n

is in the

n

-th dense subset, and

p

n

+

1

≤

p

n

for each

n

. Therefore

C

⁢

H

implies

M

⁢

A

.

If

M

⁢

A

κ

then

2

ℵ

0

>

κ

, and in fact

2

κ

=

2

ℵ

0

κ

≥

ℵ

0

, so

2

κ

≥

2

ℵ

0

, hence it will suffice to find an surjective function from

P

⁡

(

ℵ

0

)

to

P

⁡

(

κ

)

.

Let

A

=

⟨

A

α

⟩

α

<

κ

, a

sequence

of

infinite subsets

of

ω

such that for any

α

≠

β

,

A

α

∩

A

β

is finite.

Given any subset

S

⊆

κ

we will

construct

a

function

f

:

ω

→

{

0

,

1

}

such that a unique

S

can be recovered from each

f

.

f

will have the property that if

i

∈

S

then

f

⁢

(

a

)

=

0

for finitely many

elements

a

∈

A

i

, and if

i

∉

S

then

f

⁢

(

a

)

=

0

for infinitely many elements of

A

i

.

Let

P

be the partial order (under

inclusion

) such that each element

p

∈

P

satisfies

:

•

p

is a

partial function

from

ω

to

{

0

,

1

}

•

There exist

i

1

,

…

,

i

n

∈

S

such that for each

j

<

n

,

A

i

j

⊆

dom

⁡

(

p

)

•

There is a finite subset of

ω

,

w

p

, such that

w

p

=

dom

⁡

(

p

)

-

⋃

j

<

n

A

i

j

•

For each

j

<

n

,

p

⁢

(

a

)

=

0

for finitely many elements of

A

i

j

This satisfies ccc. To see this, consider any

uncountable

sequence

S

=

⟨

p

α

⟩

α

<

ω

1

of elements of

P

. There are only countably many finite subsets of

ω

, so there is some

w

⊆

ω

such that

w

=

w

p

for uncountably many

p

∈

S

and

p

↾

w

is the same for each such element. Since each of these function’s domain covers only a finite number of the

A

α

, and is

1

on all but a finite number of elements in each, there are only a countable number of different

combinations

available, and therefore two of them are

compatible

.

Consider the following groups of dense subsets:

•

D

n

=

{

p

∈

P

∣

n

∈

dom

⁡

(

p

)

}

for

n

<

ω

. This is obviously dense since any

p

not already in

D

n

can be extended to one which is by adding

⟨

n

,

1

⟩

•

D

α

=

{

p

∈

P

∣

dom

⁡

(

p

)

⊇

A

α

}

for

α

∈

S

. This is dense since if

p

∉

D

α

then

p

∪

{

⟨

a

,

1

⟩

∣

a

∈

A

α

∖

dom

⁡

(

p

)

}

is.

•

For each

α

∉

S

,

n

<

ω

,

D

n

,

α

=

{

p

∈

P

∣

m

≥

n

∧

p

⁢

(

m

)

=

0

}

for some

m

<

ω

. This is dense since if

p

∉

D

n

,

α

then

dom

⁡

(

p

)

∩

A

α

=

A

α

∩

(

w

p

∪

⋃

j

A

i

j

)

. But

w

p

is finite, and the

intersection

of

A

α

with any other

A

i

is finite, so this intersection is finite, and hence

bounded

by some

m

.

A

α

is

infinite

, so there is some

m

≤

x

∈

A

α

. So

p

∪

{

⟨

x

,

0

⟩

}

∈

D

n

,

α

.

By

M

⁢

A

κ

, given any set of

κ

dense subsets of

P

, there is a

generic

G

which intersects all of them. There are a total of

ℵ

0

+

|

S

|

+

(

κ

-

|

S

|

)

⋅

ℵ

0

=

κ

dense subsets in these three groups, and hence some generic

G

intersecting all of them. Since

G

is directed,

g

=

⋃

G

is a partial function from

ω

to

{

0

,

1

}

. Since for each

n

<

ω

,

G

∩

D

n

is non-empty,

n

∈

dom

⁡

(

g

)

, so

g

is a

total function

. Since

G

∩

D

α

for

α

∈

S

is non-empty, there is some element of

G

whose domain contains all of

A

α

and is

0

on a finite number of them, hence

g

⁢

(

a

)

=

0

for a finite number of

a

∈

A

α

. Finally, since

G

∩

D

n

,

α

for each

n

<

ω

,

α

∉

S

, the set of

n

∈

A

α

such that

g

⁢

(

n

)

=

0

is

unbounded

, and hence infinite. So

g

is as promised, and

2

κ

=

2

ℵ

0

.

Title

Martin’s axiom and the

continuum hypothesis

Canonical name

MartinsAxiomAndTheContinuumHypothesis

Date of creation

2013-03-22 12:55:05

Last modified on

2013-03-22 12:55:05

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

4

Author

Henry (455)

Entry type

Result

Classification

msc 03E50

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