aarjaneiro/planetmath huggingface dataset:
de Morgan’s laws

de Morgan’s laws

In

set theory

,

de Morgan’s laws

relate the three basic set

operations

to each other;
the union, the

intersection

, and the

complement

.
de Morgan’s laws are named after the
Indian-born British mathematician and logician
Augustus De Morgan (1806-1871)

[

1

]

.

If

A

and

B

are subsets of a set

X

, de Morgan’s laws state that

(

A

∪

B

)

∁

=

A

∁

∩

B

∁

,

(

A

∩

B

)

∁

=

A

∁

∪

B

∁

.

Here,

∪

denotes the union,

∩

denotes the intersection,
and

A

∁

denotes the set complement of

A

in

X

, i.e.,

A

∁

=

X

∖

A

.

Above, de Morgan’s laws are written for two sets.
In this form, they are intuitively quite clear.
For

instance

, the first claim states that an

element

that is not in

A

∪

B

is not in

A

and not in

B

. It also states that an elements not in

A

and not in

B

is not in

A

∪

B

.

For an arbitrary

collection

of subsets, de Morgan’s laws are
as follows:

Theorem

.

Let

X

be a set with subsets

A

i

⊂

X

for

i

∈

I

, where

I

is an arbitrary index-set. In other words,

I

can be finite,

countable

, or

uncountable

. Then

(

⋃

i

∈

I

A

i

)

∁

=

⋂

i

∈

I

A

i

∁

,

(

⋂

i

∈

I

A

i

)

∁

=

⋃

i

∈

I

A

i

∁

.

(proof (

http://planetmath.org/DeMorgansLawsProof

))

de Morgan’s laws in a

For

Boolean variables

x

and

y

in a

Boolean algebra

,
de Morgan’s laws state that

(

x

∧

y

)

′

=

x

′

∨

y

′

,

(

x

∨

y

)

′

=

x

′

∧

y

′

.

Not surprisingly, de Morgan’s laws form an indispensable tool
when simplifying digital circuits involving and, or, and not
gates

[

2

]

.

References

1

Wikipedia’s

http://www.wikipedia.org/wiki/Augustus_De_Morgan

entry on de Morgan, 4/2003.

2

M.M. Mano,

Computer Engineering: Hardware Design

,
Prentice Hall, 1988.

Title

de Morgan’s laws

Canonical name

DeMorgansLaws

Date of creation

2013-03-22 12:24:49

Last modified on

2013-03-22 12:24:49

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

15

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 03E30

Related topic

Set

Related topic

Complement

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

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de Morgan’s laws for sets (proof)

de Morgan’s laws for sets (proof)

Let

X

be a set with subsets

A

i

⊂

X

for

i

∈

I

, where

I

is an arbitrary index-set. In other words,

I

can be finite,

countable

, or

uncountable

. We first show that

(

∪

i

∈

I

A

i

)

′

=

∩

i

∈

I

A

i

′

,

where

A

′

denotes the

complement

of

A

.

Let us define

S

=

(

∪

i

∈

I

A

i

)

′

and

T

=

∩

i

∈

I

A

i

′

. To establish the

equality

S

=

T

, we shall
use a standard

argument

for proving equalities in

set theory

. Namely,
we show that

S

⊂

T

and

T

⊂

S

.
For the first claim, suppose

x

is an

element

in

S

.
Then

x

∉

∪

i

∈

I

A

i

, so

x

∉

A

i

for any

i

∈

I

.
Hence

x

∈

A

i

′

for all

i

∈

I

, and

x

∈

∩

i

∈

I

A

i

′

=

T

.

Conversely

, suppose

x

is an
element in

T

=

∩

i

∈

I

A

i

′

. Then

x

∈

A

i

′

for all

i

∈

I

.
Hence

x

∉

A

i

for any

i

∈

I

, so

x

∉

∪

i

∈

I

A

i

,
and

x

∈

S

.

The second claim,

(

∩

i

∈

I

A

i

)

′

=

∪

i

∈

I

A

i

′

,

follows by applying the first claim to the sets

A

i

′

.

Title

de Morgan’s laws for sets (proof)

Canonical name

DeMorgansLawsForSetsproof

Date of creation

2013-03-22 13:32:16

Last modified on

2013-03-22 13:32:16

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Proof

Classification

msc 03E30

Generated on Thu Feb 8 20:18:12 2018 by

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maximality principle

maximality principle

Let

S

be a

collection

of sets. If, for each chain

C

⊆

S

, there exists an

X

∈

S

such that every element of

C

is a subset of

X

, then

S

contains a

maximal element

. This is known as the

maximality principle

.

The maximality principle is

equivalent

to the

axiom of choice

.

Title

maximality principle

Canonical name

MaximalityPrinciple

Date of creation

2013-03-22 12:26:18

Last modified on

2013-03-22 12:26:18

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

9

Author

akrowne (2)

Entry type

Theorem

Classification

msc 03E30

Classification

msc 03E25

Synonym

maximal principle

Related topic

ZornsLemma

Related topic

AxiomOfChoice

Related topic

WellOrderingPrinciple

Related topic

TukeysLemma

Related topic

ZermelosPostulate

Related topic

HaudorffsMaximumPrinciple

Generated on Thu Feb 8 20:18:24 2018 by

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Tarski’s axiom

Tarski’s axiom

Tarski proposed the following axiom for

set theory

:

For every set

S

, there exists a set

U

which enjoys the following

properties

:

•

S

is an element of

U

•

For every

element

X

∈

U

, every subset of

X

is also an element of

U

.

•

For every element

X

∈

U

, the

power set

of

X

is also an element of

U

.

•

Every subset of

U

whose

cardinality

is less than the cardinality of

U

is an element of

U

.

This axiom implies the

axiom of choice

. It also implies the

existence

of

inaccessible cardinal

numbers.

Title

Tarski’s axiom

Canonical

name

TarskisAxiom

Date of creation

2013-03-22 15:37:25

Last modified on

2013-03-22 15:37:25

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 03E30

Generated on Thu Feb 8 20:18:29 2018 by

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union

union

The

union

of two sets

A

and

B

is the set which contains all

x

∈

A

and all

x

∈

B

, denoted

A

∪

B

. In the

Venn diagram

below,

A

∪

B

is the entire area shaded in blue.

We can extend this to any (finite or

infinite

) family

(

A

i

)

i

∈

I

, writing

⋃

i

∈

I

A

i

for the union of this family. Formally, for a family

(

A

i

)

i

∈

I

of sets:

x

∈

⋃

i

∈

I

A

i

⇔

⋁

i

∈

I

(

x

∈

A

i

)

Alternatively, and equivalently,

x

∈

⋃

i

∈

I

A

i

⇔

∃

i

∈

I

⁢

such that

⁢

x

∈

A

i

This

characterization

makes it much clearer that if

I

is itself the

empty set

(that is, if we are taking the union of an empty family), then the union is empty; that is,

⋃

i

∈

∅

A

i

=

∅

Often

elements of

sets are taken from some

universe

U

of

elements

under consideration (for example, the real numbers

ℝ

, or living things on the planet, or words in a particular book). When this is the case, it is meaningful to discuss the

complement

of a set: if

A

is a set of elements from some universe

U

, then the complement of

A

is the set

A

C

=

U

\

A

=

{

x

∈

U

∣

x

∉

A

}

From an axiomatic point of view, the

existence

of the union is guaranteed by the

axiom of union

.

Note that the sets

A

i

may be, but need not be,

disjoint

. Unions

satisfy

some basic

properties

that are

obvious

from the

definitions

:

•

Idempotency

:

A

∪

A

=

A

•

A

∪

A

C

=

U

where

U

is the

universe

of

A

•

Commutativity

:

A

∪

B

=

B

∪

A

•

Associativity

:

(

A

∪

B

)

∪

C

=

A

∪

(

B

∪

C

)

Here are some examples of set unions:

{

1

,

2

}

∪

{

3

,

4

}

=

{

1

,

2

,

3

,

4

}

{

b

l

u

e

,

g

r

e

e

n

}

∪

∅

=

{

b

l

u

e

,

g

r

e

e

n

}

{

x

∈

ℤ

∣

x

⩾

1

}

∪

{

x

∈

ℤ

∣

x

⩽

-

1

}

=

{

x

∈

ℤ

∣

x

≠

0

}

{

1

,

2

}

∪

{

1

,

4

}

=

{

1

,

2

,

4

}

{

x

∈

ℝ

∣

x

⩾

2

}

∪

{

x

∈

ℝ

∣

x

⩽

2

}

=

ℝ

⋃

n

∈

ℤ

n

>

0

{

p

n

∣

p

∈

ℤ

}

=

ℚ

The first three of these are the union of

disjoint sets

, while the latter three are not - in those cases, the sets overlap each other.

Title

union

Canonical

name

Union

Date of creation

2013-03-22 12:14:19

Last modified on

2013-03-22 12:14:19

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

13

Author

rm50 (10146)

Entry type

Definition

Classification

msc 03E30

Related topic

Intersection

Generated on Thu Feb 8 20:18:32 2018 by

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universe

universe

A

universe

𝐔

is a

nonempty set

satisfying the following axioms:

1.

If

x

∈

𝐔

and

y

∈

x

, then

y

∈

𝐔

.

2.

If

x

,

y

∈

𝐔

, then

{

x

,

y

}

∈

𝐔

.

3.

If

x

∈

𝐔

, then the

power set

𝒫

⁢

(

x

)

∈

𝐔

.

4.

If

{

x

i

|

i

∈

I

∈

𝐔

}

is a family of elements of

𝐔

, then

∪

i

∈

I

x

i

∈

𝐔

.

From these axioms, one can deduce the following

properties

:

1.

If

x

∈

𝐔

, then

{

x

}

∈

𝐔

.

2.

If

x

is a subset of

y

∈

𝐔

, then

x

∈

𝐔

.

3.

If

x

,

y

∈

𝐔

, then the

ordered pair

(

x

,

y

)

=

{

{

x

,

y

}

,

x

}

is in

𝐔

.

4.

If

x

,

y

∈

𝐔

, then

x

∪

y

and

x

×

y

are in

𝐔

.

5.

If

{

x

i

|

i

∈

I

∈

𝐔

}

is a family of elements of

𝐔

, then the

product

∏

i

∈

I

x

i

is in

𝐔

.

6.

If

x

∈

𝐔

, then the

cardinality

of

x

is

strictly

less than
the cardinality of

𝐔

. In particular,

𝐔

∉

𝐔

.

In order for

uncountable

universes to exist, it is necessary to adopt an extra axiom for

set theory

. This is usually phrased as:

Axiom 1

.

For every

cardinal

α

, there exists a

strongly inaccessible

cardinal

β

>

α

.

This axiom cannot be proven using the axioms ZFC. But it seems (according to Bourbaki) that it probably cannot be proven not to lead to a

contradiction

.

One usually also assumes

Axiom 2

.

For every set

X

, there is no infinite descending chain

⋯

∈

x

2

∈

x

1

∈

X

; this is called being

artinian

.

This axiom does not affect the

consistency

of ZFC, that is, ZFC is

consistent

if and only if ZFC with this axiom added is consistent. This is also known as the

axiom of foundation

, and it is often included with ZFC. If it is not accepted, then one can for all practical purposes restrict oneself to working within the class of artinian sets.

Finally, one must be careful when using

relations

within universes; the details are too technical for Bourbaki to work out (!), but see the appendix to Exposé 1 of

[

SGA4

]

for more detail.

The standard reference for universes is

[

SGA4

]

.

References

SGA4

Grothendieck et al.

Seminaires en Geometrie Algebrique 4

, Tome 1, Exposé 1 (or the appendix to Exposé 1, by N. Bourbaki for more detail and a large number of results there described as “ne pouvant servir à rien”). SGA4 is

http://www.math.mcgill.ca/ archibal/SGA/SGA.html

available on the Web. (It is in French.)

Title

universe

Canonical

name

Universe

Date of creation

2013-03-22 13:31:13

Last modified on

2013-03-22 13:31:13

Owner

archibal (4430)

Last modified by

archibal (4430)

Numerical id

8

Author

archibal (4430)

Entry type

Definition

Classification

msc 03E30

Classification

msc 18A15

Related topic

Small

Generated on Thu Feb 8 20:18:34 2018 by

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well-foundedness and axiom of foundation

well-foundedness and axiom of foundation

Recall that a

relation

R

on a class

C

is

well-founded

if

1.

For any

x

∈

C

, the

collection

{

y

∈

C

∣

y

⁢

R

⁢

x

}

is a set, and

2.

for any non-empty

B

⊆

C

, there is an

element

z

∈

B

such that if

y

⁢

R

⁢

z

, then

y

∉

B

.

z

is called an

R

-

minimal element

of

B

. It is clear that the membership relation

∈

in the class of all sets

satisfies

the first condition above.

Theorem 1

.

Given ZF,

∈

is a well-founded relation iff the

Axiom of Foundation

(AF) is true.

We will prove this using one of the

equivalent

versions of AF: for every non-empty set

A

, there is an

x

∈

A

such that

x

∩

A

=

∅

.

Proof.

Suppose

∈

is well-founded and

A

a non-empty set. We want to find

x

∈

A

such that

x

∩

A

=

∅

. Since

∈

is well-founded, there is a

∈

-

minimal set

x

such that

x

∈

A

. Since no set

y

such that

y

∈

x

and

y

∈

A

(otherwise

x

would not be

∈

-

minimal

), we have that

x

∩

A

=

∅

.

Conversely

, suppose that AF is true. Let

A

be any non-empty set. We want to find a

∈

-minimal element in

A

. Let

x

∈

A

such that

x

∩

A

=

∅

. Then

x

is

∈

-minimal in

A

, for otherwise there is

y

∈

A

such that

y

∈

x

, which implies

y

∈

x

∩

A

=

∅

, a

contradiction

.
∎

Title

well-foundedness and axiom of foundation

Canonical name

WellfoundednessAndAxiomOfFoundation

Date of creation

2013-03-22 17:25:34

Last modified on

2013-03-22 17:25:34

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 03E30

Generated on Thu Feb 8 20:18:57 2018 by

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chain condition

chain condition

A

partial order

P

satisfies

the

κ

-

chain condition

if for any

S

⊆

P

with

|

S

|

=

κ

then there exist distinct

x

,

y

∈

S

and some

p

such that

p

≤

x

and

p

≤

y

.

If

κ

=

ℵ

1

then

P

is said to satisfy the

countable chain condition

(c.c.c.)

Title

chain condition

Canonical name

ChainCondition

Date of creation

2013-03-22 12:53:38

Last modified on

2013-03-22 12:53:38

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

7

Author

Henry (455)

Entry type

Definition

Classification

msc 03E35

Related topic

PartialOrder

Related topic

PartialOrderWithChainConditionDoesNotCollapseCardinals

Defines

chain condition

Defines

countable chain condition

Defines

κ

-chain condition

Defines

c.c.c.

Defines

ccc

Generated on Thu Feb 8 20:18:58 2018 by

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composition of forcing notions

composition of forcing notions

Suppose

P

is a

forcing

notion in

𝔐

and

Q

^

is some

P

-name such that

⊩

P

Q

^

is a forcing notion

.

Then take a set of

P

-names

Q

such that given a

P

name

Q

~

of

Q

,

⊩

P

Q

~

=

Q

^

(that is, no matter which

generic

subset

G

of

P

we force with, the names in

Q

correspond precisely to the elements of

Q

^

⁢

[

G

]

). We can define

P

*

Q

=

{

⟨

p

,

q

^

⟩

∣

p

∈

P

,

q

^

∈

Q

}

We can define a

partial order

on

P

*

Q

such that

⟨

p

1

,

q

^

1

⟩

≤

⟨

p

2

,

q

^

2

⟩

iff

p

1

≤

P

p

2

and

p

1

⊩

q

^

1

≤

Q

^

q

^

2

. (A note on

interpretation

:

q

1

and

q

2

are

P

names; this requires only that

q

^

1

≤

q

^

2

in generic subsets contain

p

1

, so in other generic subsets that fact could fail.)

Then

P

*

Q

^

is itself a forcing notion, and it can be shown that forcing by

P

*

Q

^

is

equivalent

to forcing first by

P

and then by

Q

^

⁢

[

G

]

.

Title

composition of forcing notions

Canonical name

CompositionOfForcingNotions

Date of creation

2013-03-22 12:54:20

Last modified on

2013-03-22 12:54:20

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

4

Author

Henry (455)

Entry type

Definition

Classification

msc 03E35

Classification

msc 03E40

Related topic

Forcing

Generated on Thu Feb 8 20:19:01 2018 by

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equivalence of forcing notions

equivalence of forcing notions

Let

P

and

Q

be two

forcing

notions such that given any

generic

subset

G

of

P

there is a generic subset

H

of

Q

with

𝔐

⁢

[

G

]

=

𝔐

⁢

[

H

]

and vice-versa. Then

P

and

Q

are

equivalent

.

Since if

G

∈

𝔐

⁢

[

H

]

,

τ

⁢

[

G

]

∈

𝔐

for any

P

-name

τ

, it follows that if

G

∈

𝔐

⁢

[

H

]

and

H

∈

𝔐

⁢

[

G

]

then

𝔐

⁢

[

G

]

=

𝔐

⁢

[

H

]

.

Title

equivalence of forcing notions

Canonical name

EquivalenceOfForcingNotions

Date of creation

2013-03-22 12:54:24

Last modified on

2013-03-22 12:54:24

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Definition

Classification

msc 03E35

Classification

msc 03E40

Synonym

equivalent

Related topic

Forcing

Related topic

ProofThatForcingNotionsAreEquivalentToTheirComposition

Generated on Thu Feb 8 20:19:03 2018 by

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FS iterated forcing preserves chain condition

FS iterated forcing preserves chain condition

Let

κ

be a

regular cardinal

and let

⟨

Q

^

β

⟩

β

<

α

be a

finite support iterated forcing

where for every

β

<

α

,

⊩

P

β

Q

^

β

has the

κ

chain condition

.

By

induction

:

P

0

is the

empty set

.

If

P

α

satisfies

the

κ

chain condition then so does

P

α

+

1

, since

P

α

+

1

is

equivalent

to

P

α

*

Q

α

and

composition

preserves the

κ

chain condition for

regular

κ

.

Suppose

α

is a

limit ordinal

and

P

β

satisfies the

κ

chain condition for all

β

<

α

. Let

S

=

⟨

p

i

⟩

i

<

κ

be a subset of

P

α

of size

κ

. The domains of the elements of

p

i

form

κ

finite subsets of

α

, so if

cf

⁡

(

α

)

>

κ

then these are

bounded

, and by the

inductive hypothesis

, two of them are

compatible

.

Otherwise, if

cf

⁡

(

α

)

<

κ

, let

⟨

α

j

⟩

j

<

cf

⁡

(

α

)

be an increasing sequence of

ordinals

cofinal

in

α

. Then for any

i

<

κ

there is some

n

⁢

(

i

)

<

cf

⁡

(

α

)

such that

dom

⁡

(

p

i

)

⊆

α

n

⁢

(

i

)

. Since

κ

is regular and this is a

partition

of

κ

into fewer than

κ

pieces, one piece must have size

κ

, that is, there is some

j

such that

j

=

n

⁢

(

i

)

for

κ

values of

i

, and so

{

p

i

∣

n

⁢

(

i

)

=

j

}

is a set of conditions of size

κ

contained

in

P

α

j

, and therefore contains compatible members by the induction hypothesis.

Finally, if

cf

⁡

(

α

)

=

κ

, let

C

=

⟨

α

j

⟩

j

<

κ

be a

strictly increasing

,

continuous

sequence

cofinal in

α

. Then for every

i

<

κ

there is some

n

⁢

(

i

)

<

κ

such that

dom

⁡

(

p

i

)

⊆

α

n

⁢

(

i

)

. When

n

⁢

(

i

)

is a limit ordinal, since

C

is continuous, there is also (since

dom

⁡

(

p

i

)

is finite) some

f

⁢

(

i

)

<

i

such that

dom

⁡

(

p

i

)

∩

[

α

f

⁢

(

i

)

,

α

i

)

=

∅

. Consider the set

E

of

elements

i

such that

i

is a limit ordinal and for any

j

<

i

,

n

⁢

(

j

)

<

i

. This is a club, so by Fodor’s lemma there is some

j

such that

{

i

∣

f

⁢

(

i

)

=

j

}

is

stationary

.

For each

p

i

such that

f

⁢

(

i

)

=

j

, consider

p

i

′

=

p

i

↾

j

. There are

κ

of these, all members of

P

j

, so two of them must be compatible, and hence those two are also compatible in

P

.

Title

FS iterated forcing preserves chain condition

Canonical name

FSIteratedForcingPreservesChainCondition

Date of creation

2013-03-22 12:57:14

Last modified on

2013-03-22 12:57:14

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

4

Author

Henry (455)

Entry type

Result

Classification

msc 03E35

Classification

msc 03E40

Generated on Thu Feb 8 20:19:05 2018 by

LaTeXML
forcing relation

forcing relation

If

𝔐

is a

transitive

model of

set theory

and

P

is a

partial order

then we can define a

forcing relation

:

p

⊩

P

ϕ

⁢

(

τ

1

,

…

,

τ

n

)

(

p

forces

ϕ

⁢

(

τ

1

,

…

,

τ

n

)

)

for any

p

∈

P

, where

τ

1

,

…

,

τ

n

are

P

- names.

Specifically, the

relation

holds if for every generic filter

G

over

P

which contains

p

,

𝔐

⁢

[

G

]

⊨

ϕ

⁢

(

τ

1

⁢

[

G

]

,

…

,

τ

n

⁢

[

G

]

)

That is,

p

forces

ϕ

if every of

𝔐

by a generic filter over

P

containing

p

makes

ϕ

true.

If

p

⊩

P

ϕ

holds for every

p

∈

P

then we can write

⊩

P

ϕ

to mean that for any

generic

G

⊆

P

,

𝔐

⁢

[

G

]

⊨

ϕ

.

Title

forcing relation

Canonical name

ForcingRelation

Date of creation

2013-03-22 12:53:28

Last modified on

2013-03-22 12:53:28

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Definition

Classification

msc 03E35

Classification

msc 03E40

Related topic

Forcing

Defines

forcing relation

Defines

forces

Generated on Thu Feb 8 20:19:07 2018 by

LaTeXML
iterated forcing

iterated forcing

We can define an

iterated forcing

of length

α

by

induction

as follows:

Let

P

0

=

∅

.

Let

Q

^

0

be a

forcing

notion.

For

β

≤

α

,

P

β

is the set of all

functions

f

such that

dom

⁡

(

f

)

⊆

β

and for any

i

∈

dom

⁡

(

f

)

,

f

⁢

(

i

)

is a

P

i

-name for a member of

Q

^

i

. Order

P

β

by the rule

f

≤

g

iff

dom

⁡

(

g

)

⊆

dom

⁡

(

f

)

and for any

i

∈

dom

⁡

(

f

)

,

g

↾

i

⊩

f

⁢

(

i

)

≤

Q

^

i

g

⁢

(

i

)

. (Translated, this means that any

generic

subset including

g

restricted to

i

forces that

f

⁢

(

i

)

, an element of

Q

^

i

, be less than

g

⁢

(

i

)

.)

For

β

<

α

,

Q

^

β

is a forcing notion in

P

β

(so

⊩

P

β

Q

^

β

is a forcing notion

).

Then the

sequence

⟨

Q

^

β

⟩

β

<

α

is an iterated forcing.

If

P

β

is restricted to finite functions that it is called a

finite support iterated forcing

(FS), if

P

β

is restricted to

countable

functions, it is called a

countable support iterated function

(CS), and in general if each function in each

P

β

has size less than

κ

then it is a

<

κ

-support iterated forcing

.

Typically we construct the sequence of

Q

^

β

’s by induction, using a function

F

such that

F

⁢

(

⟨

Q

^

β

⟩

β

<

γ

)

=

Q

^

γ

.

Title

iterated forcing

Canonical name

IteratedForcing

Date of creation

2013-03-22 12:54:47

Last modified on

2013-03-22 12:54:47

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Definition

Classification

msc 03E35

Classification

msc 03E40

Defines

FS

Defines

CS

Defines

finite support

Defines

finite support iterated forcing

Defines

countable support

Defines

countable support iterated forcing

Defines

support iterated forcing

Generated on Thu Feb 8 20:19:14 2018 by

LaTeXML
iterated forcing and composition

iterated forcing and composition

There is a

function

satisfying

forcings

are

equivalent

if one is

dense in

the other

f

:

P

α

*

Q

α

→

P

α

+

1

.

Proof

Let

f

⁢

(

⟨

g

,

q

^

⟩

)

=

g

∪

{

⟨

α

,

q

^

⟩

}

. This is obviously a member of

P

α

+

1

, since it is a

partial function

from

α

+

1

(and if the domain of

g

is less than

α

then so is the domain of

f

⁢

(

⟨

g

,

q

^

⟩

)

), if

i

<

α

then obviously

f

⁢

(

⟨

g

,

q

^

⟩

)

applied to

i

satisfies

the definition of

iterated forcing

(since

g

does), and if

i

=

α

then the definition is satisfied since

q

^

is a name in

P

i

for a member of

Q

i

.

f

is order preserving, since if

⟨

g

1

,

q

^

1

⟩

≤

⟨

g

2

,

q

^

2

⟩

, all the appropriate

characteristics

of a function carry over to the image, and

g

1

↾

α

⊩

P

i

q

^

1

≤

q

^

2

(by the definition of

≤

in

*

).

If

⟨

g

1

,

q

^

1

⟩

and

⟨

g

2

,

q

^

2

⟩

are

incomparable

then either

g

1

and

g

2

are incomparable, in which case whatever prevents them from being compared applies to their images as well, or

q

^

1

and

q

^

2

aren’t compared appropriately, in which case again this prevents the images from being compared.

Finally, let

g

be any element of

P

α

+

1

. Then

g

↾

α

∈

P

α

. If

α

∉

dom

⁡

(

g

)

then this is just

g

, and

f

⁢

(

⟨

g

,

q

^

⟩

)

≤

g

for any

q

^

. If

α

∈

dom

⁡

(

g

)

then

f

(

⟨

g

↾

α

,

g

(

α

)

⟩

)

=

g

. Hence

f

⁢

[

P

α

*

Q

α

]

is dense in

P

α

+

1

, and so these are equivalent.

Title

iterated forcing and composition

Canonical name

IteratedForcingAndComposition

Date of creation

2013-03-22 12:54:51

Last modified on

2013-03-22 12:54:51

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

6

Author

Henry (455)

Entry type

Result

Classification

msc 03E35

Classification

msc 03E40

Generated on Thu Feb 8 20:19:16 2018 by

LaTeXML