Martin’s axiom is consistent

Martin’s axiom is consistent

If

κ

is an

uncountable

strong limit cardinal

such that for any

λ

<

κ

,

κ

λ

=

κ

then it is

consistent

that

2

ℵ

0

=

κ

and MA. This is shown by using

finite support iterated forcing

to construct a model of ZFC in which this is true. Historically, this proof was the motivation for developing iterated forcing.

Outline

The proof uses the convenient fact that

M

⁢

A

κ

holds as long as it holds for all partial orders smaller than

κ

. Given the conditions on

κ

, there are at most

κ

names for these partial orders. At each step in the

forcing

, we force with one of these names. The result is that the actual

generic

subset we add

intersects

every dense subset of every partial order.

Construction of

P

κ

Q

^

α

will be constructed by

induction

with three conditions:

|

P

α

|

≤

κ

for all

α

≤

κ

,

⊩

P

α

Q

^

α

⊆

ℳ

, and

P

α

satisfies

the ccc. Note that a partial ordering on a

cardinal

λ

<

κ

is a

function

from

λ

×

λ

to

{

0

,

1

}

, so there are at most

2

λ

<

κ

of them. Since a canonical name for a partial ordering of a cardinal is just a function from

P

α

to that cardinal, there are at most

κ

2

λ

≤

κ

of them.

At each of the

κ

steps, we want to deal with one of these possible partial orderings, so we need to

partition

the

κ

steps in to

κ

steps for each of the

κ

cardinals less than

κ

. In

addition

, we need to include every

P

α

name for any level. Therefore, we partion

κ

into

⟨

S

γ

,

δ

⟩

γ

,

δ

<

κ

for each cardinal

δ

, with each

S

γ

,

δ

having

cardinality

κ

and the added condition that

η

∈

S

γ

,

δ

implies

η

≥

γ

. Then each

P

γ

name for a partial ordering of

δ

is assigned some index

η

∈

S

γ

,

δ

, and that partial order will be dealt with at stage

Q

η

.

Formally, given

Q

^

β

for

β

<

α

,

P

α

can be constructed and the

P

α

names for partial orderings of each cardinal

δ

enumerated by the

elements of

S

α

,

δ

.

α

∈

S

γ

,

δ

for some

γ

α

and

δ

α

, and

α

≥

γ

α

so some canonical

P

γ

α

name for a partial order

≤

^

α

of

δ

α

has already been assigned to

α

.

Since

≤

^

α

is a

P

γ

α

name, it is also a

P

α

name, so

Q

^

α

can be defined as

⟨

δ

α

,

≤

^

α

⟩

if

⊩

P

α

⟨

δ

α

,

≤

^

α

⟩

satisfies the ccc

and by the

trivial

partial order

⟨

1

,

{

⟨

1

,

1

⟩

}

⟩

otherwise. Obviously this satisfies the ccc, and so

P

α

+

1

does as well. Since

Q

^

α

is either trivial or a cardinal together with a canonical name,

⊩

P

α

Q

^

α

⊆

ℳ

. Finally,

|

P

α

+

q

|

≤

∑

n

|

α

|

n

⋅

(

sup

i

⁡

|

Q

^

i

|

)

n

≤

κ

.

Proof that

M

⁢

A

λ

holds for

λ

<

κ

Lemma: It suffices to show that

M

⁢

A

λ

holds for partial order with size

≤

λ

Proof.

Suppose

P

is a partial order with

|

P

|

>

κ

and let

⟨

D

α

⟩

α

<

λ

be dense subsets of

P

. Define functions

f

i

:

P

→

D

α

for

α

⁢

κ

with

f

α

⁢

(

p

)

≥

p

(obviously such

elements

exist since

D

α

is dense). Let

g

:

P

×

P

→

P

be a function such that

g

⁢

(

p

,

q

)

≥

p

,

q

whenever

p

and

q

are

compatible

. Then pick some element

q

∈

P

and let

Q

be the

closure

of

{

q

}

under

f

α

and

g

with the same

ordering

as

P

(restricted to

Q

).

Since there are only

κ

functions being used, it must be that

|

Q

|

≤

κ

. If

p

∈

Q

then

f

α

⁢

(

p

)

≥

p

and clearly

f

α

⁢

(

p

)

∈

Q

∩

D

α

, so each

D

α

∩

Q

is dense in

Q

. In addition,

Q

is ccc: if

A

is an

antichain

in

Q

and

p

1

,

p

2

∈

A

then

p

1

,

p

2

are incompatible in

Q

. But if they were compatible in

P

then

g

⁢

(

p

1

,

p

2

)

≥

p

1

,

p

2

would be an element of

Q

, so they must be incompatible in

P

. Therefore

A

is an antichain in

P

, and therefore must have

countable

cardinality, since

P

satisfies the ccc.

By

assumption

, there is a directed

G

⊆

Q

such that

G

∩

(

D

α

∩

Q

)

≠

∅

for each

α

<

κ

, and therefore

M

⁢

A

λ

holds in full.
∎

Now we must prove that, if

G

is a generic subset of

P

κ

,

R

some partial order with

|

R

|

≤

λ

and

⟨

D

α

⟩

α

<

λ

are dense subsets of

R

then there is some directed subset of

R

intersecting each

D

α

.

If

|

R

|

<

λ

then

λ

additional elements can be added greater than any other element of

R

to make

|

R

|

=

λ

, and then since there is an order isomorphism into some partial order of

λ

, assume

R

is a partial ordering of

λ

. Then let

D

=

{

⟨

α

,

β

⟩

∣

α

∈

D

β

}

.

Take canonical names so that

R

=

R

^

⁢

[

G

]

,

D

=

D

^

⁢

[

G

]

and

D

i

=

D

^

i

⁢

[

G

]

for each

i

<

λ

and:

⊩

P

κ

R

^

⁢

is a partial ordering satisfying ccc and

D

^

⊆

λ

×

λ

⁢

and

D

α

^

⁢

is dense in

⁢

R

^

For any

α

,

β

there is a maximal antichain

D

α

,

β

⊆

P

κ

such that if

p

∈

D

α

,

β

then either

p

⊩

P

κ

α

≤

R

^

β

or

p

⊩

P

κ

α

≰

R

^

β

and another maximal antichain

E

α

,

β

⊆

P

κ

such that if

p

∈

E

α

,

β

then either

p

⊩

P

κ

⟨

α

,

β

⟩

∈

D

^

or

p

⊩

P

κ

⟨

α

,

β

⟩

∉

D

^

. These antichains determine the value of those two

formulas

.

Then, since

κ

cf

⁡

κ

>

κ

and

κ

μ

=

κ

for

μ

<

κ

, it must be that

cf

⁡

κ

=

κ

, so

κ

is

regular

. Then

γ

=

sup

(

{

α

+

1

∣

α

∈

dom

(

p

)

,

p

∈

⋃

α

,

β

<

λ

D

α

,

β

∪

E

α

,

β

)

<

κ

, so

D

α

,

β

,

E

α

,

β

⊆

P

γ

, and therefore the

P

κ

names

R

^

and

D

^

are also

P

γ

names.

Lemma: For any

γ

,

G

γ

=

{

p

↾

γ

∣

p

∈

G

}

is a generic subset of

P

γ

Proof.

First, it is directed, since if

p

1

↾

γ

,

p

2

↾

γ

∈

G

γ

then there is some

p

∈

G

such that

p

≤

p

1

,

p

2

, and therefore

p

↾

γ

∈

G

γ

and

p

≤

p

1

↾

γ

,

p

2

↾

γ

.

Also, it is generic. If

D

is a dense subset of

P

γ

then

D

κ

=

{

p

∈

P

κ

∣

p

≤

q

∈

D

}

is dense in

P

κ

, since if

p

∈

P

κ

then there is some

d

≤

p

↾

, but then

d

is compatible with

p

, so

d

∪

p

∈

D

κ

. Therefore there is some

p

∈

D

κ

∩

G

κ

, and so

p

↾

∈

D

∩

G

γ

.
∎

Since

R

^

and

D

^

are

P

γ

names,

R

^

⁢

[

G

]

=

R

^

⁢

[

G

γ

]

=

R

and

D

^

⁢

[

G

]

=

D

^

⁢

[

G

γ

]

=

D

, so

V

⁢

[

G

γ

]

⊨

R

^

⁢

is a partial ordering of

⁢

λ

⁢

satisfying the ccc and

D

α

^

⁢

is dense in

⁢

R

^

Then there must be some

p

∈

G

γ

such that

p

⊩

P

γ

R

^

⁢

is a partial ordering of

⁢

λ

⁢

satisfying the ccc

Let

A

p

be a maximal antichain of

P

γ

such that

p

∈

A

p

, and define

≤

^

*

as a

P

γ

name with

(

p

,

m

)

∈

≤

^

*

for each

m

∈

R

^

and

(

a

,

n

)

∈

≤

^

*

if

n

=

(

α

,

β

)

where

α

<

β

<

λ

and

p

≠

a

∈

A

p

. That is,

≤

^

*

⁢

[

G

]

=

R

when

p

∈

G

and

≤

^

*

[

G

]

=

∈

↾

λ

otherwise. Then this is the name for a partial ordering of

λ

, and therefore there is some

η

∈

S

γ

,

λ

such that

≤

^

*

=

≤

^

η

, and

η

≥

γ

. Since

p

∈

G

γ

⊆

G

η

,

Q

^

η

⁢

[

G

η

]

=

≤

^

η

⁢

[

G

η

]

=

R

.

Since

P

η

+

1

=

P

η

*

Q

η

, we know that

G

Q

η

⊆

Q

η

is generic since

http://planetmath.org/node/3258

forcing with the

composition

is

equivalent

to successive forcing. Since

D

i

∈

V

⁢

[

G

γ

]

⊆

V

⁢

[

G

η

]

and is dense, it follows that

D

i

∩

G

Q

η

≠

∅

and since

G

Q

η

is a subset of

R

in

P

κ

,

M

⁢

A

λ

holds.

Proof that

2

ℵ

0

=

κ

The relationship between Martin’s axiom and the

continuum hypothesis

tells us that

2

ℵ

0

≥

κ

. Since

2

ℵ

0

was less than

κ

in

V

, and since

|

P

κ

|

=

κ

adds at most

κ

elements, it must be that

2

ℵ

0

=

κ

.

Title

Martin’s axiom is consistent

Canonical name

MartinsAxiomIsConsistent

Date of creation

2013-03-22 13:21:59

Last modified on

2013-03-22 13:21:59

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Result

Classification

msc 03E50

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