aarjaneiro/planetmath huggingface dataset:
bounded recursion

bounded recursion

In this entry, the following notations are used:

ℱ

=

⋃

{

F

k

∣

k

∈

ℕ

}

, where for each

k

∈

ℕ

, and

F

k

=

{

f

∣

f

:

ℕ

k

→

ℕ

}

.

Definition

. A

function

f

∈

F

m

+

1

is said to be defined by

bounded recursion

via functions

g

∈

F

m

,

h

∈

F

m

+

2

, and

k

∈

F

m

+

1

if, for any

𝒙

∈

ℕ

m

and

y

∈

ℕ

,

1.

f

is defined by

primitive recursion

via

g

and

h

:

–

f

⁢

(

𝒙

,

0

)

=

g

⁢

(

𝒙

)

,

–

f

⁢

(

𝒙

,

y

+

1

)

=

h

⁢

(

𝒙

,

y

,

f

⁢

(

𝒙

,

y

)

)

;

2.

f

is bounded from above by

k

:

–

f

⁢

(

𝒙

,

y

)

≤

k

⁢

(

𝒙

,

y

)

.

Clearly, a function that is defined by bounded recursion is defined by primitive recursion.

Conversely

, a function is defined by primitive recursion, it is also defined by bounded recursion, since it is

bounded

by itself. However, the two

concepts

are not exactly the same. To make this precise, we first define what it means for a set of number-theoretic functions be

closed under

bounded recursion:

Definition

. A set

ℬ

⊆

ℱ

is said to be

closed under bounded recursion

if, for any

f

defined by bounded recursion via

g

,

h

,

k

∈

ℬ

, then

f

∈

ℬ

.

It is clear that

ℱ

is closed under both primitive recursion and bounded recursion, and so are

𝒫

⁢

ℛ

, the set of all

primitive recursive functions

, as well as

ℛ

∩

ℱ

, the set of all

total recursive functions

. However,

Proposition 1

.

The set

E

⁢

R

of all

elementary recursive functions

is closed under bounded recursion, but not primitive recursion.

Since the

exponential function

, the

i

-th prime function, and the function

(

x

)

y

(the

exponent

of

y

in

x

) are elementary recursive, we have the following

corollary

:

Corollary 1

.

If

f

is defined by

course-of-values recursion

via an elementary recursive function

h

, and

f

is bounded from above by an elementary recursive function

k

, then

f

is elementary recursive.

Title

bounded recursion

Canonical

name

BoundedRecursion

Date of creation

2013-03-22 19:07:01

Last modified on

2013-03-22 19:07:01

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03D20

Synonym

bounded primitive recursion

Related topic

ElementaryRecursiveFunction

Generated on Thu Feb 8 20:02:30 2018 by

LaTeXML
characterization of primitive recursive functions of one variable

characterization of primitive recursive functions of one variable

It is possible to characterize

primitive recursive functions

of one

variable

in terms of

operations

involving only

functions

of a single
variable. To describe how this goes, it is useful to first define
some notation.

Definition 1

.

Define the

constant function

K

:

N

→

N

by

K

⁢

(

n

)

=

1

for all

n

.

Definition 2

.

Define the

identity function

I

:

N

→

N

by

I

⁢

(

n

)

=

n

for all

n

.

Definition 3

.

Define the excess over square function

E

:

N

→

N

by

E

⁢

(

n

)

=

n

-

m

2

, where

m

is the largest integer such that

m

2

≤

n

.

Definition 4

.

Given a function

f

:

N

→

N

, define
the function

R

⁢

(

f

)

:

N

→

N

by the
following conditions:

•

R

⁢

(

f

)

⁢

(

0

)

=

0

•

R

⁢

(

f

)

⁢

(

n

+

1

)

=

f

⁢

(

R

⁢

(

f

)

⁢

(

n

)

)

for all integers

n

≥

0

.

Theorem 1

.

The class of primitive recursive functions of a single variable is
the smallest class

X

of functions which contains the functions

E

and

K

defined above and is

closed under

the following
three operations:

1.

If

f

∈

X

and

g

∈

X

, then

f

∘

g

∈

X

.

2.

If

f

∈

X

, then

f

+

I

∈

X

.

1

1

Here

f

+

I

has the usual meaning of

pointwise addition

—

(

f

+

I

)

⁢

(

x

)

=

f

⁢

(

x

)

+

I

⁢

(

x

)

3.

If

f

∈

X

, then

R

⁢

(

f

)

∈

X

.

Title

characterization of primitive recursive functions of one variable

Canonical name

CharacterizationOfPrimitiveRecursiveFunctionsOfOneVariable

Date of creation

2013-03-22 16:45:25

Last modified on

2013-03-22 16:45:25

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

9

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 03D20

Generated on Thu Feb 8 20:02:32 2018 by

LaTeXML
definition by cases

definition by cases

Definition

A (total)

function

f

:

ℕ

k

→

ℕ

is said to be defined by cases if there are functions

f

1

,

…

,

f

m

:

ℕ

k

→

ℕ

, and

predicates

Φ

1

⁢

(

𝒙

)

,

…

,

Φ

m

⁢

(

𝒙

)

, which are pairwise exclusive

S

⁢

(

Φ

i

)

∩

S

⁢

(

Φ

j

)

=

∅

for

i

≠

j

, such that

f

⁢

(

𝒙

)

:=

{

f

1

⁢

(

𝒙

)

if

⁢

Φ

1

⁢

(

𝒙

)

,

⋯

f

m

⁢

(

𝒙

)

if

⁢

Φ

m

⁢

(

𝒙

)

.

Since

f

is a

total function

(domain is all of

ℕ

k

), we see that

S

⁢

(

Φ

1

)

∪

⋯

∪

S

⁢

(

Φ

m

)

=

ℕ

k

.

Proposition 1

.

As above, if the functions

f

1

,

…

,

f

m

:

N

k

→

N

, as well as the predicates

Φ

1

⁢

(

𝐱

)

,

…

,

Φ

m

⁢

(

𝐱

)

, are

primitive recursive

, then so is the function

f

:

N

k

→

N

defined by cases from the

f

i

and

Φ

j

.

To see this, we first need the following:

Lemma 1

.

If functions

f

1

,

…

,

f

m

:

N

k

→

N

are primitive recursive, so is

f

1

+

⋯

+

f

m

.

Proof.

By

induction

on

m

. The case when

m

=

1

is clear. Suppose the statement is true for

m

=

n

. Then

f

1

+

⋯

+

f

n

+

f

n

+

1

=

add

⁡

(

f

1

+

⋯

+

f

n

,

f

n

+

1

)

, which is primitive recursive, since

add

is, and that primitive recursiveness is preserved under

functional

composition

.
∎

Proof of Proposition 1.

f

is just

f

⁢

(

𝒙

)

:=

{

f

1

⁢

(

𝒙

)

if

⁢

𝒙

∈

S

⁢

(

Φ

1

)

,

⋯

f

m

⁢

(

𝒙

)

if

⁢

𝒙

∈

S

⁢

(

Φ

m

)

.

which can be re-written as

f

=

φ

S

⁢

(

Φ

1

)

⁢

f

1

+

⋯

+

φ

S

⁢

(

Φ

m

)

⁢

f

m

,

where

φ

S

denotes the characteristic function of set

S

. By

assumption

, both

f

i

and

φ

S

⁢

(

Φ

i

)

are primitive recursive, so is their

product

, and hence the sum of these products. As a result,

f

is primitive recursive too. ∎

Title

definition by cases

Canonical name

DefinitionByCases

Date of creation

2013-03-22 19:08:13

Last modified on

2013-03-22 19:08:13

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Example

Classification

msc 03D20

Generated on Thu Feb 8 20:02:55 2018 by

LaTeXML
recurrence relation

recurrence relation

A

recurrence relation

is an equation which gives the value of an element of a

sequence

in terms of the values of the sequence for smaller values of the position index and the position index itself. If the position

n

of a sequence

s

is denoted by

s

n

, then the next value of the sequence expressed as a recurrence relation would be of the form

s

n

+

1

=

f

⁢

(

s

1

,

s

2

,

…

,

s

n

-

1

,

s

n

,

n

)

where

f

is any

function

.

If

k

is a

positive integer

, then a sequence

s

satisfies

a

k

th order recurrence relation

if

s

n

+

1

can be written in terms of

s

n

,

…

,

s

n

-

k

+

1

whenever

n

+

1

>

k

. In other words, the recurrence relation for

s

is of the form

s

n

-

1

=

f

⁢

(

s

n

-

k

+

1

,

…

,

s

n

-

1

,

s

n

,

n

)

for some function

f

.

An example of a recurrence relation is

s

n

+

1

=

s

n

+

(

n

+

1

)

,

which is the recurrence relation for the sum of the

integers

from

1

to

n

+

1

. This could also be expressed as

s

n

=

s

n

-

1

+

n

keeping in mind that, as long as we set the proper initial values of the sequence, the recurrence relation indices can have any

constant

amount added or subtracted. Note that this is a first order recurrence relation.

As another example of a recurrence relation, the

Fibonacci sequence

satisfies the recurrence relation

s

n

+

1

=

s

n

+

s

n

-

1

.

Note that this is a

second order

recurrence relation.

Title

recurrence relation

Canonical name

RecurrenceRelation

Date of creation

2013-03-22 11:56:04

Last modified on

2013-03-22 11:56:04

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

13

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 03D20

Classification

msc 11B37

Synonym

difference equation

Related topic

BerlekampMasseyAlgorithm

Related topic

Equation

Related topic

FiniteDifference

Defines

first order

Defines

second order

Defines

kth order

Generated on Thu Feb 8 20:04:43 2018 by

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computable number

computable number

A

real number

r

is called

computable

if there exists some

terminating

algorithm

(or

Turing machine

) that can approximate it to arbitrary precision. Specifically, the algorithm takes a

natural number

n

as input and produces an

integer

m

as output such that

m

-

1

n

<

r

<

m

+

1

n

Alternatively, and equivalently, one may say that

r

is computable if there exists a

recursive function

F

such that the above

inequality

holds when

m

=

F

⁢

(

n

)

.

A

complex number

is called computable if its real and

imaginary parts

are computable.

The computable complex numbers form an algebraically closed field, and arguably this field contains all the numbers we ever need in practice. It contains all

algebraic numbers

as well as many known transcendental

constants

, such as

π

and

e

for example. There are however many real numbers which are not computable: the set of all

computable numbers

is

countable

(because the set of algorithms is) while the set of real numbers is

uncountable

(as shown by Cantor’s diagonal argument).

Every computable number is

definable

, but not vice versa. An example of a definable, non-computable real is Chaitin’s constant,

Ω

.

Computable numbers were introduced by Alan Turing in 1936. Turing’s original definition differed from the one given above. Turing called a real number

r

computable if there is a Turing machine which on input

n

produces the

n

-th decimal digit of

r

. By distinguishing the cases of

rational and irrational

r

, one can show that Turing’s definition is

equivalent

to the definition given above. However, there does not exist any general algorithm which transforms approximating Turing machines (in the sense of our definition) into digit-enumerating Turing machines (in the sense of Turing’s definition).

Title

computable number

Canonical name

ComputableNumber

Date of creation

2013-03-22 13:33:51

Last modified on

2013-03-22 13:33:51

Owner

AxelBoldt (56)

Last modified by

AxelBoldt (56)

Numerical id

15

Author

AxelBoldt (56)

Entry type

Topic

Classification

msc 03D25

Classification

msc 68Q05

Synonym

recursive number

Defines

computable

Defines

computable real number

Generated on Thu Feb 8 20:05:15 2018 by

LaTeXML
index set

index set

In computability theory, a set

A

⊆

ω

is called an

index set

if for all

x

,

y

,

x

∈

A

,

φ

x

=

φ

y

⟹

y

∈

A

.

φ

x

stands for the

partial function

with Gödel number (or index)

x

.

Thus, if

A

is an index set and

φ

x

=

φ

y

, then either

x

,

y

∈

A

or

x

,

y

∉

A

. Intuitively, if

A

contains the Gödel index

x

of a partial function

φ

, then

A

contains all indices for the partial function. (Recall that there are

ℵ

0

Gödel numbers for each partial function.)

It is instructive to compare the notion of an index set in computability theory with that of an

indexing

set.

Title

index set

Canonical name

IndexSet

Date of creation

2013-03-22 18:09:48

Last modified on

2013-03-22 18:09:48

Owner

yesitis (13730)

Last modified by

yesitis (13730)

Numerical id

5

Author

yesitis (13730)

Entry type

Definition

Classification

msc 03D25

Generated on Thu Feb 8 20:05:17 2018 by

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index set theorem

index set theorem

Index Set Theorem

:

If

A

is an

index set

and

A

≠

∅

,

ω

, then either

K

≤

1

A

or

K

≤

1

A

∁

.

In the statement of the

theorem

,

K

is the halting set

{

x

:

φ

x

⁢

(

x

)

⁢

c

⁢

o

⁢

n

⁢

v

⁢

e

⁢

r

⁢

g

⁢

e

⁢

s

}

,

≤

1

is the one-one reducibility (or 1-reducibility)

relation symbol

, and

A

∁

stands for the

complement

of the set

A

(relative to

ω

).

Title

index set theorem

Canonical name

IndexSetTheorem

Date of creation

2013-03-22 18:09:51

Last modified on

2013-03-22 18:09:51

Owner

yesitis (13730)

Last modified by

yesitis (13730)

Numerical id

5

Author

yesitis (13730)

Entry type

Theorem

Classification

msc 03D25

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

LaTeXML
primitive recursive number

primitive recursive number

A special of

computable numbers

is so-called the

primitive recursive numbers

. Informally, these are numbers that can be measured by

primitive recursive functions

to an arbitrary degree of precision.

Definition

. A non-negative

real number

r

is said to be

primitive recursive

if there is a primitive recursive function

f

:

ℕ

→

ℕ

such that

f

⁢

(

n

)

=

{

[

r

]

⁢

⁢

(

the integer part of

⁢

r

)

,

if

⁢

n

=

0

,

n

th

⁢

digit of

⁢

r

⁢

when

⁢

r

⁢

is expressed in its decimal representation,

if

⁢

n

≠

0

.

A real number

r

is

primitive recursive

if

|

r

|

is, and a

complex number

x

+

y

⁢

i

is

primitive recursive

if both

x

and

y

are.

Clearly, any

integer

is primitive recursive. It is easy to see that all

rational numbers

are primitive recursive too, as the decimal representation of a rational number is

periodic

, so if

r

=

[

r

]

.

a

1

⁢

⋯

⁢

a

k

¯

,

we can define

f

so that

f

⁢

(

n

)

=

{

[

r

]

,

if

⁢

n

=

0

,

a

i

if

⁢

n

≠

0

⁢

and

⁢

n

≡

i

(

mod

k

)

.

Here, we assume that

r

is non-negative.

In

addition

, we can show that

n

is primitive recursive for any non-negative integer

n

.

Proof.

Suppose

r

=

n

. Write

r

in its decimal representation

r

=

n

0

.

n

1

⁢

n

2

⁢

⋯

⁢

n

k

⁢

⋯

Then

n

0

=

[

n

]

. Multiply

r

by

10

to get its decimal representation

10

⁢

r

=

n

0

⁢

n

1

.

n

2

⁢

⋯

⁢

n

k

⁢

⋯

Then

10

⁢

n

0

+

n

1

=

[

10

⁢

r

]

=

[

100

⁢

n

]

, so that

n

1

=

[

100

⁢

n

]

-

10

⁢

n

0

By

induction

, we see that

n

k

+

1

=

[

100

k

+

1

⁢

n

]

-

10

⁢

(

10

k

⁢

n

0

+

10

k

-

1

⁢

n

1

+

⋯

+

n

k

)

.

Define

f

:

ℕ

2

→

ℕ

by

f

⁢

(

n

,

m

)

=

n

m

. Then

f

⁢

(

n

,

0

)

is primitive recursive. Next,

f

⁢

(

n

,

m

)

=

[

100

m

⁢

n

]

-

10

⁢

∑

i

=

0

m

-

1

10

m

-

1

-

i

⁢

f

⁢

(

n

,

i

)

=

h

⁢

(

n

,

m

,

f

¯

⁢

(

n

,

m

)

)

,

where

h

⁢

(

x

,

y

,

z

)

=

[

100

x

⁢

y

]

⁢

-

˙

⁢

10

⁢

∑

i

=

0

y

⁢

-

˙

⁢

1

10

y

⁢

-

˙

⁢

s

⁢

(

i

)

⁢

(

z

)

i

which is primitive recursive (all of the

operations

, including the

bounded sum

are primitive recursive). Since

f

is defined by

course-of-values recursion

via

h

,

f

is primitive recursive also.
∎

Remark

. It can be shown that

π

is primitive recursive. A proof of this can be found in the link below.

References

1

S. G. Simpson,

http://www.math.psu.edu/simpson/courses/math558/fom.pdf

Foundations of Mathematics

. (2009).

Title

primitive recursive number

Canonical name

PrimitiveRecursiveNumber

Date of creation

2013-03-22 19:06:06

Last modified on

2013-03-22 19:06:06

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03D25

Classification

msc 68Q05

Related topic

BombellisMethodOfComputingSquareRoots

Generated on Thu Feb 8 20:05:29 2018 by

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recursively enumerable

recursively enumerable

For a

language

L

, TFAE:

•

There exists a

Turing machine

f

such that

∀

x

.

(

x

∈

L

)

⇔

the computation

f

⁢

(

x

)

terminates

.

•

There exists a

total recursive function

f

:

ℕ

→

L

which is

onto

.

•

There exists a total recursive function

f

:

ℕ

→

L

which is

one-to-one

and onto

.

A language

L

fulfilling

any

(and therefore

all

) of the above conditions is called

recursively enumerable

.

Examples

1.

Any

recursive language

.

2.

The set of

encodings

of Turing machines which halt when given no input.

3.

The set of encodings of

theorems

of

Peano arithmetic

.

4.

The set of integers

n

for which the hailstone sequence starting at

n

reaches 1.

(We don’t know if this set is recursive, or even if it is

N

; but a

trivial

program

shows it is recursively enumerable.)

Title

recursively enumerable

Canonical name

RecursivelyEnumerable

Date of creation

2013-03-22 12:44:32

Last modified on

2013-03-22 12:44:32

Owner

ariels (338)

Last modified by

ariels (338)

Numerical id

6

Author

ariels (338)

Entry type

Definition

Classification

msc 03D25

Synonym

semi-recursive

Related topic

HaltingProblem

Related topic

TuringComputable

Defines

semi-recursive

Defines

recursively enumerable function

Generated on Thu Feb 8 20:05:31 2018 by

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Ackermann function

Ackermann function

Ackermann’s

function

A

⁢

(

x

,

y

)

is defined by the

recurrence relations

A

⁢

(

0

,

y

)

=

y

+

1

A

⁢

(

x

+

1

,

0

)

=

A

⁢

(

x

,

1

)

A

⁢

(

x

+

1

,

y

+

1

)

=

A

⁢

(

x

,

A

⁢

(

x

+

1

,

y

)

)

Ackermann’s function is an example of a

recursive function

that is not

primitive recursive

, but is instead

μ

-

recursive

(that is,

Turing-computable

).

Ackermann’s function grows extremely fast. In fact, we find that

A

⁢

(

0

,

y

)

=

y

+

1

A

⁢

(

1

,

y

)

=

2

+

(

y

+

3

)

-

3

A

⁢

(

2

,

y

)

=

2

⋅

(

y

+

3

)

-

3

A

⁢

(

3

,

y

)

=

2

y

+

3

-

3

A

⁢

(

4

,

y

)

=

2

2

.

⋅

2

-

3

(

y

+

3

⁢

exponentiations

)

⋯

… and at this point conventional notation breaks down, and we need to employ something like

Conway notation

or

Knuth notation

for large numbers.

Ackermann’s function wasn’t actually written in this form by its namesake, Wilhelm Ackermann. Instead, Ackermann found that the

z

-fold

exponentiation

of

x

with

y

was an example of a recursive function which was not primitive recursive. Later this was simplified by Rosza Peter to a function of two variables,

similar

to the one given above.

Title

Ackermann function

Canonical

name

AckermannFunction

Date of creation

2013-03-22 12:33:13

Last modified on

2013-03-22 12:33:13

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

9

Author

akrowne (2)

Entry type

Definition

Classification

msc 03D75

Generated on Thu Feb 8 20:05:33 2018 by

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halting problem

halting problem

The

halting problem

is to determine, given a particular input to a particular

computer

program

, whether the program will terminate after a finite number of steps.

The

consequences

of a solution to the halting problem are far-reaching. Consider some

predicate

P

⁢

(

x

)

regarding

natural numbers

; suppose we

conjecture

that

P

⁢

(

x

)

holds for all

x

∈

ℕ

. (Goldbach’s conjecture, for example, takes this form.) We can write a program that will count up through the natural numbers and terminate upon finding some

n

such that

P

⁢

(

n

)

is false; if the conjecture holds in general, then our program will never terminate. Then,

without

running

the program

, we could pass it along to a halting program to prove or disprove the conjecture.

In 1936, Alan Turing proved that the halting problem is undecideable; the

argument

is presented here informally. Consider a hypothetical program that decides the halting the problem:

Algorithm

Halt

(P, I)

Input

: A computer program

P

and some input

I

for

P

Output

: True if

P

halts on

I

and false otherwise

The implementation of the algorithm, as it turns out, is irrelevant. Now consider another program:

Algorithm

Break

(x)

Input

: Code

x

Output

:

begin

if

IsValidCode(x) and Halt(x,x)

then

while

true

do

nothing

else

return true

end

If our halting solution determines that Break(Break) halts, then it will immediately enter an infinite loop; otherwise, Break will return immediately. We must conclude that the Halt program does not decide the halting problem.
So for any attempted solution to the halting problem, we can find some input which breaks that solution.

Title

halting problem

Canonical

name

HaltingProblem

Date of creation

2013-03-22 12:14:28

Last modified on

2013-03-22 12:14:28

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

12

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 03D75

Related topic

RecursivelyEnumerable

Generated on Thu Feb 8 20:06:13 2018 by

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weak AI thesis

weak AI thesis

Weak AI (

artificial intelligence

) thesis: A digital computer is a powerful tool for
studying intelligence and developing useful technology, and it enables us to formulate and test hypotheses in a more

rigorous

and precise fashion. However a AI

program

is not itself a cognitive process, at most it is a

simulation

of such process.

References

1. Mooney RJ (1999)

http://www.cs.utexas.edu/ mooney/cs343/slide-handouts/philosophy.4.pdf

Philosophical

Arguments

Against AI.

2. Searle JR. (1980)

http://members.aol.com/NeoNoetics/MindsBrainsPrograms.html

Minds, brains, and programs. Behavioral and Brain Sciences 3(3): 417-457.

Title

weak AI thesis

Canonical name

WeakAIThesis

Date of creation

2013-03-22 17:07:15

Last modified on

2013-03-22 17:07:15

Owner

dankomed (17058)

Last modified by

dankomed (17058)

Numerical id

9

Author

dankomed (17058)

Entry type

Topic

Classification

msc 03D80

Synonym

weak artificial intelligence thesis

Related topic

StrongAIThesis

Generated on Thu Feb 8 20:06:32 2018 by

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complete ultrafilter and partitions

complete ultrafilter and partitions

If

U

is an

ultrafilter

on a set

S

, then

U

is

κ

-

complete

⇔

there is no

partition

of

S

into

κ

-many pieces for which each piece

X

α

of the partition is not in

U

.

We prove the case of

σ

-completeness; the case of arbitrary

infinite

cardinality

follows closely. For the

⇒

direction, let

P

be a partition of

S

into

ω

many pieces, all of which do not belong to

U

, and write

S

=

⋃

n

=

1

ω

X

n

to illustrate this partition. Now,

∅

=

S

∁

=

⋂

n

=

1

ω

X

n

∁

. Since, by our

assumption

, each of the

X

n

do not belong to

U

, we have

X

n

∁

∈

U

for each

n

<

ω

as

U

is an ultrafilter. Thus,

(

⋂

n

=

1

ω

X

n

∁

)

∈

U

by

σ

-completeness. This, however, means

∅

∈

U

, contradicting the definition of a filter.

Note that the

converse

states that every partition

P

of

S

into

ω

-many pieces has a (unique) piece

X

1

∈

U

. To prove this, let

Y

n

be a

collection

of

ω

many members of

U

and let

Y

=

⋂

n

=

1

ω

Y

n

. Now consider the partition

{

P

ι

:

ι

≤

ω

}

of

S

∖

Y

:

for each

s

∈

S

∖

Y

, put

s

∈

P

ι

if

ι

is the least index for which

s

∉

Y

ι

.

It is easy to verify that each

s

∈

S

∖

Y

belongs to a

unique

P

ι

, the collection of

P

ι

’s is indeed a partition of

S

∖

Y

.

Along with

Y

,

{

P

ι

:

ι

≤

ω

}

partitions

S

into

ℵ

0

=

ω

many pieces. A (unique) piece of this partition belongs in

U

:

P

ι

⁣

*

∈

U

or

Y

∈

U

. But,

P

ι

∩

Y

ι

=

∅

∉

U

by the definition of

P

ι

. This excludes the possibility for the former to belong in

U

(cf.

alternative characterization of filter

) and so

Y

∈

U

.

Thus, starting from an arbitrary collection

{

Y

n

}

of

ω

-many members of

U

, we have identified a partition of

S

for which the unique piece which belongs to

U

is

∩

Y

n

. Therefore,

U

is

σ

-complete.

Title

complete ultrafilter and partitions

Canonical name

CompleteUltrafilterAndPartitions

Date of creation

2013-03-22 18:55:52

Last modified on

2013-03-22 18:55:52

Owner

yesitis (13730)

Last modified by

yesitis (13730)

Numerical id

4

Author

yesitis (13730)

Entry type

Definition

Classification

msc 03E02

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