forcing

forcing

Forcing

is the method used by Paul Cohen to prove the independence of
the

continuum hypothesis

(CH). In fact, the method was used by Cohen to
prove that CH could be violated.

Adding a set to a model of

set theory

via forcing is

similar

to adjoining a new

element

to a field. Suppose we have a field

k

, and we want to add to this field an
element

α

such

α

2

=

-

1

. We see
that we cannot simply drop a new

α

in

k

, since then we are
not guaranteed that we still have a field. Neither can we simply
assume that

k

already has such an element. The standard way of
doing this is to start by adjoining a

generic

indeterminate

X

, and
impose a constraint on

X

, saying that

X

2

+

1

=

0

. What we do is take
the

quotient

k

⁢

[

X

]

/

(

X

2

+

1

)

, and make a field out of it by taking the
quotient field. We then obtain

k

⁢

(

α

)

, where

α

is the

equivalence class

of

X

in the quotient.
The general case of this is the

theorem

of algebra saying that every

polynomial

p

over a field

k

has a root in some

extension field

.

We can rephrase this and say that “it is

consistent

with standard
field theory that

-

1

have a

square root

”.

When the theory we consider is ZFC, we run in exactly the same
problem : we can’t just add a “new” set and pretend it has the
required properties, because then we may violate something else, like

foundation

. Let

𝔐

be a

transitive

model of set theory, which we
call the

ground model

. We want to “add a new set”

S

to

𝔐

in
such a way that the

extension

𝔐

′

has

𝔐

as a

subclass

, and the
properties of

𝔐

are preserved, and

S

∈

𝔐

′

.

The first step is to “approximate” the new set using

elements of

𝔐

. This is the analogue of finding the

irreducible polynomial

in
the

algebraic

example. The set

P

of such “approximations” can be
ordered by how much information the approximations give : let

p

,

q

∈

P

, then

p

≤

q

if and only if

p

“is

stronger

than”

q

. We
call this set a set of

forcing conditions

. Furthermore, it is required that the set

P

itself and the order relation be elements of

𝔐

.

Since

P

is a

partial order

, some of its subsets have interesting
properties. Consider

P

as a

topological space

with the

order
topology

. A subset

D

⊆

P

is

dense

in

P

if and only
if for every

p

∈

P

, there is

d

∈

D

such that

d

≤

p

. A
filter in

P

is said to be

𝔐

-generic

if and only if it

intersects

every one of the

dense subsets

of

P

which are in

𝔐

. An

𝔐

-generic
filter in

P

is also referred to as a

generic set of conditions

in the literature. In general, even though

P

is a set in

𝔐

, generic filters are not elements of

𝔐

.

If

P

is a set of forcing conditions, and

G

is a generic set of
conditions in

P

, all in the ground model

𝔐

, then we define

𝔐

⁢

[

G

]

to be the least model of ZFC that contains

G

. The
big theorem is this :

Theorem.

𝔐

⁢

[

G

]

is a model of ZFC, and has the same

ordinals

as

𝔐

, and

𝔐

⊆

𝔐

⁢

[

G

]

.

The way to prove that we can violate CH using a generic extension is
to add many new “subsets of

ω

” in the following way : let

𝔐

be a transitive model of ZFC, and let

(

P

,

≤

)

be the set (in

𝔐

) of all

functions

f

whose domain is a finite subset of

ℵ

2

×

ℵ

0

, and whose range is the set

{

0

,

1

}

. The

ordering

here is

p

≤

q

if and only if

p

⊃

q

. Let

G

be a generic set of conditions in

P

. Then

⋃

G

is a

total function

whose domain is

ℵ

2

×

ℵ

0

, and range is

{

0

,

1

}

. We can see this

f

as coding

ℵ

2

new functions

f

α

:

ℵ

0

→

{

0

,

1

}

,

α

<

ℵ

2

,
which are subsets of omega. These
functions are all distinct.

(

P

,

≤

)

http://planetmath.org/node/3242

doesn’t collapse

cardinals

since it

satisfies

the

countable chain condition

. Thus

ℵ

2

𝔐

⁢

[

G

]

=

ℵ

2

𝔐

and CH is false in

𝔐

⁢

[

G

]

.

All this relies on a proper definition of the satisfaction relation in

𝔐

⁢

[

G

]

, and the

forcing relation

. Details can be found in Thomas Jech’s book

Set Theory

.

Title

forcing

Canonical name

Forcing

Date of creation

2013-03-22 12:44:17

Last modified on

2013-03-22 12:44:17

Owner

ratboy (4018)

Last modified by

ratboy (4018)

Numerical id

12

Author

ratboy (4018)

Entry type

Definition

Classification

msc 03E50

Classification

msc 03E35

Classification

msc 03E40

Related topic

ForcingRelation

Related topic

CompositionOfForcingNotions

Related topic

EquivalenceOfForcingNotions

Related topic

FieldAdjunction

Defines

forcing

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