aarjaneiro/planetmath huggingface dataset:
transitive relation

transitive relation

A

relation

ℛ

on a set

A

is

transitive

if and only if

∀

x

,

y

,

z

∈

A

,

(

x

⁢

ℛ

⁢

y

∧

y

⁢

ℛ

⁢

z

)

→

(

x

⁢

ℛ

⁢

z

)

.

For example, the “is a subset of” relation

⊆

on any set of sets is transitive.
The “less than” relation

<

on the set of real numbers
is also transitive.

The “is not equal to” relation

≠

on the set of integers is not transitive,
because

1

≠

2

and

2

≠

1

does not imply

1

≠

1

.

Title

transitive relation

Canonical name

TransitiveRelation

Date of creation

2013-03-22 12:15:52

Last modified on

2013-03-22 12:15:52

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

14

Author

yark (2760)

Entry type

Definition

Classification

msc 03E20

Related topic

Reflexive

Related topic

Symmetric

Related topic

Antisymmetric

Defines

transitivity

Defines

transitive

Generated on Thu Feb 8 20:15:51 2018 by

LaTeXML
two simple facts about well-founded relations

two simple facts about well-founded relations

The following are two simple facts about

well-founded relation

R

on

X

:

1.

For each

x

∈

X

,

x

⁢

R̸

⁢

x

. (See the entry

R-minimal element

.)

2.

The requirement for

symmetry

is absent, i.e., for each

x

,

y

∈

X

, either

x

⁢

R

⁢

y

or

y

⁢

R

⁢

x

, but not both.

Justifications for these two facts are simple. For 1, consider the

subclass

{

x

}

. Then

{

x

}

has an

R

-

minimal

element

, which can only be

x

itself. For 2, consider

{

x

,

y

}

. It has an

R

-

minimal

element, which is either

x

or

y

, not both.

Fact 1 is provided here for easy reference. Keeping these two facts in mind is helpful when dealing with (proving) basic

theorems

about the

relation

.

Title

two simple facts about well-founded relations

Canonical name

TwoSimpleFactsAboutWellfoundedRelations

Date of creation

2013-03-22 18:25:43

Last modified on

2013-03-22 18:25:43

Owner

yesitis (13730)

Last modified by

yesitis (13730)

Numerical id

10

Author

yesitis (13730)

Entry type

Feature

Classification

msc 03E20

Generated on Thu Feb 8 20:15:53 2018 by

LaTeXML
well-founded relation

well-founded relation

A

binary relation

R

on a class (

http://planetmath.org/Class

)

X

is

well-founded

if and only if

•

each nonempty

subclass

of

X

contains an

R

-

minimal element

and,

•

for each

x

∈

X

,

{

y

∣

y

⁢

R

⁢

x

}

is a set.

The notion of a well-founded relation is a

generalization

of that of a

well-ordering

relation

: proof by induction and definition by

recursion

may be carried out over well-founded relations.

Title

well-founded relation

Canonical name

WellfoundedRelation

Date of creation

2013-03-22 17:24:31

Last modified on

2013-03-22 17:24:31

Owner

ratboy (4018)

Last modified by

ratboy (4018)

Numerical id

9

Author

ratboy (4018)

Entry type

Definition

Classification

msc 03E20

Related topic

Relation

Related topic

RMinimalElement

Defines

well-founded

Generated on Thu Feb 8 20:15:55 2018 by

LaTeXML
axiom of countable choice

axiom of countable choice

The

Axiom of Countable Choice

(CC) is a weak form of the

Axiom of Choice

(

http://planetmath.org/AxiomOfChoice

).
It states that every

countable set

of

nonempty sets

has a

choice function

.

(that is, the

Zermelo-Fraenkel axioms

together with the Axiom of Countable Choice) suffices to prove that the union of countably many countable sets is

countable

. It also suffices to prove that every

infinite set

has a

countably infinite

subset, and that a set

X

is

infinite

if and only if there is a

bijection

between

X

and a

proper subset

of

X

.

Title

axiom of countable choice

Canonical

name

AxiomOfCountableChoice

Date of creation

2013-03-22 14:46:23

Last modified on

2013-03-22 14:46:23

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

14

Author

yark (2760)

Entry type

Definition

Classification

msc 03E25

Synonym

countable axiom of choice

Synonym

countable AC

Defines

countable choice

Generated on Thu Feb 8 20:15:56 2018 by

LaTeXML
choice function

choice function

A

choice function

on a set

S

is a

function

f

with domain

S

such that

f

⁢

(

x

)

∈

x

for all

x

∈

S

.

A choice function on

S

simply picks one

element

from each member of

S

. So in order for

S

to have a choice function, every member of

S

must be a

nonempty set

. The

Axiom of Choice

(

http://planetmath.org/AxiomOfChoice

) (AC) states that every set of nonempty sets does have a choice function.

Without AC the situation is more complicated, but we can still show that some sets have a choice function. Here are some examples:

•

If

S

is a

finite set

of nonempty sets, then we can construct a choice function on

S

by picking one element from each member of

S

. This requires only finitely many choices, so we don’t need to use AC.

•

If every member of

S

is a

well-ordered

nonempty set, then we can pick the least element of each member of

S

. In this case we may be making infinitely many choices, but we have a rule for making the choices, so AC is not needed. The distinction between “well-ordered” and “

well-orderable

” is important here: if the members of

S

were merely well-orderable, we would first have to choose a well-ordering of each member, and this might require infinitely many arbitrary choices, and therefore AC.

•

If every member of

S

is a nonempty set, and the union

∪

S

is well-orderable, then we can choose a well-ordering for this union, and this induces a well-ordering on every member of

S

, so we can now proceed as in the previous example. In this case we were able to well-order every member of

S

by making just one choice, so AC wasn’t needed. (This example shows that the

Well-Ordering Principle

, which states that every set is well-orderable, implies AC. The

converse

is also true, but less

trivial

— see the proof (

http://planetmath.org/ProofOfZermelosWellOrderingTheorem

).)

Title

choice function

Canonical name

ChoiceFunction

Date of creation

2013-03-22 14:46:26

Last modified on

2013-03-22 14:46:26

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

11

Author

yark (2760)

Entry type

Definition

Classification

msc 03E25

Related topic

AxiomOfChoice

Related topic

AxiomOfCountableChoice

Related topic

HausdorffParadox

Related topic

ProofOfHausdorffParadox

Related topic

OneToOneFunctionFromOntoFunction

Generated on Thu Feb 8 20:16:11 2018 by

LaTeXML
definition of well ordered set, a variant

definition of well ordered set, a variant

A

well-ordered set

is normally defined as a

totally

ordered set

in which every nonempty subset has a least member, as the parent object does.

It is possible to define well-ordered sets as follows:

a well-ordered set

X

is a

partially ordered set

in which every nonempty subset of

X

has a least member.

To justify the alternative, we prove that every partially ordered set

X

in which every nonempty subset has a least member is total:

let

x

∈

X

and

y

∈

X

,

x

≠

y

. Now,

{

x

,

y

}

has a least member, a fortiori,

x

,

y

are

comparable

. Hence,

X

is

totally ordered

.

The alternative has the benefit of being a stronger statement in the sense that

(

p

⁢

a

⁢

r

⁢

t

⁢

i

⁢

a

⁢

l

⁢

o

⁢

r

⁢

d

⁢

e

⁢

r

)

⟹

(

t

⁢

o

⁢

t

⁢

a

⁢

l

⁢

o

⁢

r

⁢

d

⁢

e

⁢

r

)

given that every nonempty subset has a least member.

References

1

Schechter, E.,

Handbook of Analysis and Its Foundations

, 1st ed., Academic Press, 1997.

2

Jech, T.,

Set Theory

, 3rd millennium ed., Springer, 2002.

Title

definition of well ordered set, a variant

Canonical name

DefinitionOfWellOrderedSetAVariant

Date of creation

2013-03-22 18:04:47

Last modified on

2013-03-22 18:04:47

Owner

yesitis (13730)

Last modified by

yesitis (13730)

Numerical id

9

Author

yesitis (13730)

Entry type

Derivation

Classification

msc 03E25

Classification

msc 06A05

Related topic

SomethingRelatedToNaturalNumber

Related topic

NaturalNumbersAreWellOrdered

Generated on Thu Feb 8 20:16:27 2018 by

LaTeXML
equivalence of Kuratowski’s lemma and Zorn’s lemma

equivalence of Kuratowski’s lemma and Zorn’s lemma

In this entry, we prove the equivalence of Kuratowski’s lemma and Zorn’s lemma, thereby establishing the equivalence of Kuratowski’s lemma and the

axiom of choice

. The proof, of course, only uses ZF axioms.

Proposition 1

.

Kuratowski’s lemma implies Zorn’s lemma.

Proof.

Suppose

P

is a poset such that every chain has an

upper bound

. Let

C

be a chain in

P

. By Kuratowski’s lemma,

C

can be extended to a maximal chain

C

′

, which, by

assumption

, has an upper bound

a

∈

P

. Suppose for some

b

∈

P

,

a

≤

b

. If

b

≠

a

, then

b

≰

a

, or

b

∉

C

′

, which means

C

′

∪

{

b

}

is a chain extending

C

′

, thus extending

C

. But this means that

C

′

is not maximal, a

contradiction

. This shows that

b

=

a

, or that

a

is a

maximal element

in

P

, proving Zorn’s lemma.
∎

Proposition 2

.

Zorn’s lemma implies Kuratowski’s lemma.

Proof.

Suppose

P

is a poset and

C

a chain in

P

. We assume that

P

≠

∅

. Let

𝒫

be the set of all chains in

P

extending

C

. Partially order

𝒫

by

inclusion

so that

𝒫

is a poset. Let

𝒞

be a chain in

𝒫

. Let

C

′

=

⋃

𝒞

. We want to prove that

C

′

is an upper bound of

𝒞

in

𝒫

.

1.

C

′

is a chain in

P

. If

x

,

y

∈

C

′

, then

x

∈

D

and

y

∈

E

for some

D

,

E

∈

𝒞

. Since

𝒞

is a chain in

𝒫

,

D

⊆

E

or

E

⊆

D

, which implies that

x

,

y

belong to the same chain (either

D

or

E

) in

P

. So

x

≤

y

or

y

≤

x

. This shows that

C

′

is a chain in

P

.

2.

C

′

is in

𝒫

. If

a

∈

C

, then

a

∈

D

for every

D

∈

𝒞

, and therefore

a

∈

C

′

, showing that

C

′

extends

C

, or

C

′

∈

𝒫

.

3.

C

′

is an upper bound of

𝒞

. Pick any

x

∈

D

, for an arbitrary

D

∈

𝒞

. Then

x

∈

⋃

𝒞

=

C

′

, so

D

⊆

C

′

. Since

D

is arbitrary,

C

′

is an upper bound of

𝒞

.

By Zorn’s lemma,

𝒫

has a maximal element

M

. Then

M

is a maximal chain in

P

extending

C

, for if there is a chain

N

in

P

such that

M

⊆

N

, then

N

∈

𝒫

and

M

would no longer be maximal.
∎

Title

equivalence of Kuratowski’s lemma and Zorn’s lemma

Canonical name

EquivalenceOfKuratowskisLemmaAndZornsLemma

Date of creation

2013-03-22 18:44:23

Last modified on

2013-03-22 18:44:23

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Proof

Classification

msc 03E25

Generated on Thu Feb 8 20:16:41 2018 by

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equivalence of Zorn’s lemma and the axiom of choice

equivalence of Zorn’s lemma and the axiom of choice

Let

X

be a set partially ordered by

<

such that each chain has an

upper bound

. Define

p

⁢

(

x

)

=

{

y

∈

X

∣

x

<

y

}

∈

P

⁢

(

X

)

.
Let

p

⁢

(

X

)

=

{

p

⁢

(

x

)

∣

x

∈

X

}

. If

p

⁢

(

x

)

=

∅

then it follows that

x

is maximal.

Suppose no

p

⁢

(

x

)

=

∅

. Then by the

axiom of choice

(

http://planetmath.org/AxiomOfChoice

) there is a

choice function

f

on

p

⁢

(

X

)

, and since for each

p

⁢

(

x

)

we have

f

⁢

(

p

⁢

(

x

)

)

∈

p

⁢

(

x

)

, it follows that

x

<

f

⁢

(

p

⁢

(

x

)

)

. Define

f

α

⁢

(

p

⁢

(

x

)

)

for all

ordinals

α

by

transfinite induction

:

f

0

⁢

(

p

⁢

(

x

)

)

=

x

f

α

+

1

⁢

(

p

⁢

(

x

)

)

=

f

⁢

(

p

⁢

(

f

α

⁢

(

p

⁢

(

x

)

)

)

)

And for a

limit ordinal

α

, let

f

α

⁢

(

p

⁢

(

x

)

)

be an upper bound of

f

i

⁢

(

p

⁢

(

x

)

)

for

i

<

α

.

This construction can go on forever, for any ordinal. Then we can easily construct an

injective function

from

O

⁢

r

⁢

d

to

X

by

g

⁢

(

α

)

=

f

α

⁢

(

p

⁢

(

x

)

)

for an arbitrary

x

∈

X

. This must be injective, since

α

<

β

implies

g

⁢

(

α

)

<

g

⁢

(

β

)

. But that requires that

X

be a

proper class

, in

contradiction

to the fact that it is a set. So there can be no such choice function, and there must be a

maximal element

of

X

.

For the reverse, assume Zorn’s lemma and let

C

be any set of non-empty sets. Consider the set of

functions

F

=

{

f

∣

∀

a

∈

dom

(

f

)

(

a

∈

C

∧

f

(

a

)

∈

a

)

}

partially ordered by

inclusion

. Then the union of any chain in

F

is also a member of

F

(since the union of a chain of functions is always a function). By Zorn’s lemma,

F

has a maximal element

f

, and since any function with domain smaller than

C

can be easily expanded,

dom

⁡

(

f

)

=

C

, and so

f

is a choice function for

C

.

Title

equivalence of Zorn’s lemma and the axiom of choice

Canonical name

EquivalenceOfZornsLemmaAndTheAxiomOfChoice

Date of creation

2013-03-22 12:59:04

Last modified on

2013-03-22 12:59:04

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

10

Author

Henry (455)

Entry type

Proof

Classification

msc 03E25

Generated on Thu Feb 8 20:16:44 2018 by

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Hausdorff paradox

Hausdorff paradox

Let

S

2

be the

unit sphere

in the

Euclidean space

ℝ

3

. Then
it is possible to take “half” and “a third” of

S

2

such that
both of these parts are essentially

congruent

(we give a formal
version in a minute). This sounds

paradoxical

:
wouldn’t that mean that half of the sphere’s area is equal to only a
third? The “

paradox

” resolves itself if one takes into account that
one can choose non-measurable subsets of the sphere which ostensively are “half” and a “third” of it, using

geometric congruence

as means of comparison.

Let us now formally state the

Theorem

.

Theorem

(Hausdorff paradox

[

H

]

).

There exists a

disjoint

of the unit sphere

S

2

in the
Euclidean space

R

3

into four subsets

A

,

B

,

C

,

D

, such that the
following conditions are met:

1.

Any two of the sets

A

,

B

,

C

and

B

∪

C

are congruent.

2.

D

is

countable

.

A crucial ingredient to the proof is the

http://

planetmath

.org/node/310

axiom of choice

, so the
sets

A

,

B

and

C

are not

constructible

. The theorem itself is a
crucial ingredient to the proof of the so-called

Banach-Tarski
paradox

.

References

H

F. Hausdorff

, Bemerkung über den Inhalt von
Punktmengen,

Math. Ann.

75, 428–433, (1915),

http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?did=D28919http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?did=D28919

(in German).

Title

Hausdorff paradox

Canonical name

HausdorffParadox

Date of creation

2013-03-22 15:16:12

Last modified on

2013-03-22 15:16:12

Owner

GrafZahl (9234)

Last modified by

GrafZahl (9234)

Numerical id

9

Author

GrafZahl (9234)

Entry type

Theorem

Classification

msc 03E25

Classification

msc 51M04

Related topic

ChoiceFunction

Related topic

BanachTarskiParadox

Related topic

ProofofBanachTarskiParadox

Generated on Thu Feb 8 20:16:46 2018 by

LaTeXML
Hausdorff’s maximum principle

Hausdorff’s maximum principle

Theorem

Let

X

be a

partially ordered set

. Then there exists a maximal

totally
ordered

subset of

X

.

The

Hausdorff

’s

maximum principle

is one of the many theorems

equivalent

to the

axiom of choice

(

http://planetmath.org/AxiomOfChoice

).
The below proof uses Zorn’s lemma, which
is also equivalent to the
.

Proof.

Let

S

be the set of all totally ordered subsets of

X

.

S

is not empty, since the

empty set

is an element of

S

.

Partial order

S

by

inclusion

. Let

τ

be a chain (of

elements

) in

S

. Being each totally ordered, the union of all these elements of

τ

is again a totally ordered subset of

X

, and hence an element of

S

, as is easily verified. This shows that

S

, ordered by inclusion, is inductive. The result now follows from Zorn’s lemma.
∎

Title

Hausdorff’s maximum principle

Canonical name

HausdorffsMaximumPrinciple

Date of creation

2013-03-22 13:04:42

Last modified on

2013-03-22 13:04:42

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 03E25

Synonym

maximum principle

Synonym

Hausdorff maximality theorem

Related topic

ZornsLemma

Related topic

AxiomOfChoice

Related topic

ZermelosWellOrderingTheorem

Related topic

ZornsLemmaAndTheWellOrderingTheoremEquivalenceOfHaudorffsMaximumPrinciple

Related topic

EveryVectorSpaceHasABasis

Related topic

MaximalityPrinciple

Generated on Thu Feb 8 20:16:48 2018 by

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Kuratowski’s lemma

Kuratowski’s lemma

Any chain in an

ordered set

is

contained

in a maximal chain.

This

proposition

is

equivalent

to the

axiom of choice

.

Title

Kuratowski’s lemma

Canonical name

KuratowskisLemma

Date of creation

2013-03-22 13:13:39

Last modified on

2013-03-22 13:13:39

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

6

Author

Koro (127)

Entry type

Theorem

Classification

msc 03E25

Related topic

AxiomOfChoice

Related topic

TukeysLemma

Related topic

ZermelosWellOrderingTheorem

Related topic

ZornsLemma

Related topic

ZermelosPostulate

Related topic

EveryVectorSpaceHasABasis

Generated on Thu Feb 8 20:16:50 2018 by

LaTeXML
one-to-one function from onto function

one-to-one function from onto function

Theorem

.

Given an onto

function

from a set

A

to a set

B

, there exists a one-to-one function from

B

to

A

.

Proof.

Suppose

f

:

A

→

B

is onto, and define

ℱ

=

{

f

-

1

⁢

(

{

b

}

)

:

b

∈

B

}

; that is,

ℱ

is the set containing the

pre-image

of each singleton subset of

B

. Since

f

is onto, no

element of

ℱ

is empty, and since

f

is a function, the elements of

ℱ

are

mutually disjoint

, for if

a

∈

f

-

1

⁢

(

{

b

1

}

)

and

a

∈

f

-

1

⁢

(

{

b

2

}

)

, we have

f

⁢

(

a

)

=

b

1

and

f

⁢

(

a

)

=

b

2

, whence

b

1

=

b

2

. Let

𝒞

:

ℱ

→

⋃

ℱ

be a

choice function

, noting that

⋃

ℱ

=

A

, and define

g

:

B

→

A

by

g

⁢

(

b

)

=

𝒞

⁢

(

f

-

1

⁢

(

{

b

}

)

)

. To see that

g

is

one-to-one

, let

b

1

,

b

2

∈

B

, and suppose that

g

⁢

(

b

1

)

=

g

⁢

(

b

2

)

. This gives

𝒞

⁢

(

f

-

1

⁢

(

{

b

1

}

)

)

=

𝒞

⁢

(

f

-

1

⁢

(

{

b

2

}

)

)

, but since the elements of

ℱ

are

disjoint

, this implies that

f

-

1

⁢

(

{

b

1

}

)

=

f

-

1

⁢

(

{

b

2

}

)

, and thus

b

1

=

b

2

. So

g

is a one-to-one function from

B

to

A

.
∎

Title

one-to-one function from onto function

Canonical name

OnetooneFunctionFromOntoFunction

Date of creation

2013-03-22 16:26:55

Last modified on

2013-03-22 16:26:55

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Definition

Classification

msc 03E25

Related topic

function

Related topic

ChoiceFunction

Related topic

AxiomOfChoice

Related topic

set

Related topic

onto

Related topic

SchroederBernsteinTheorem

Related topic

AnInjectionBetweenTwoFiniteSetsOfTheSameCardinalityIsBijective

Related topic

ASurjectionBetweenTwoFiniteSetsOfTheSameCardinalityIsBijective

Related topic

Set

Related topic

Surjective

Generated on Thu Feb 8 20:16:53 2018 by

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principle of finite induction

principle of finite induction

The

principle of finite induction

, also known as

mathematical induction

, is commonly formulated in two ways. Both are

equivalent

. The first formulation is known as

weak

induction

. It asserts that if a statement

P

⁢

(

n

)

holds for

n

=

0

and if

P

⁢

(

n

)

⇒

P

⁢

(

n

+

1

)

, then

P

⁢

(

n

)

holds for all

natural numbers

n

. The case

n

=

0

is called the

base case

or

base step

and the

implication

P

⁢

(

n

)

⇒

P

⁢

(

n

+

1

)

is called the

inductive step

. In an inductive proof, one uses the term

induction hypothesis

or

inductive hypothesis

to refer back to the statement

P

⁢

(

n

)

when one is trying to prove

P

⁢

(

n

+

1

)

from it.

The second formulation is known as

strong

or

complete

induction. It asserts that if the implication

∀

n

(

(

∀

m

<

n

P

(

m

)

)

⇒

P

(

n

)

)

is true, then

P

⁢

(

n

)

is true for all natural numbers

n

. (Here, the

quantifiers

range over all natural numbers.) As we have formulated it, strong induction does not require a separate base case. Note that the implication

∀

n

(

(

∀

m

<

n

P

(

m

)

)

⇒

P

(

n

)

already entails

P

⁢

(

0

)

since the statement

∀

m

<

0

⁢

P

⁢

(

m

)

holds

vacuously

(there are no natural numbers less that zero).

A moment’s thought will show that the first formulation (weak induction) is equivalent to the following:

Let

S

be a set natural numbers such that

1.

0

belongs to

S

, and

2.

if

n

belongs to

S

, so does

n

+

1

.

Then

S

is the set of all natural numbers.

Similarly, strong induction can be stated:

If

S

is a set of natural numbers such that

n

belongs to

S

whenever all numbers less than

n

belong to

S

, then

S

is the set of all natural numbers.

The principle of finite induction can be derived from the fact that every

nonempty set

of natural numbers has a smallest

element

. This fact is known as the

well-ordering principle for natural numbers

. (Note that this is not the same thing as the

well-ordering principle

, which is equivalent to the

axiom of choice

and has nothing to do with induction.)

Title

principle of finite induction

Canonical

name

PrincipleOfFiniteInduction

Date of creation

2013-03-22 11:46:41

Last modified on

2013-03-22 11:46:41

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

24

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 03E25

Classification

msc 00-02

Related topic

TransfiniteInduction

Related topic

AnExampleOfMathematicalInduction

Related topic

Induction

Related topic

WellFoundedInduction

Defines

induction hypothesis

Defines

inductive hypothesis

Defines

base case

Defines

base step

Generated on Thu Feb 8 20:17:00 2018 by

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principle of finite induction proven from the well-ordering principle for natural numbers

principle of finite induction proven from the well-ordering principle for natural numbers

We give a proof for the “strong” formulation.

Let

S

be a set of natural numbers such that

n

belongs to

S

whenever all numbers less than

n

belong to

S

(i.e., assume

∀

n

(

∀

m

<

n

m

∈

S

)

⇒

n

∈

S

, where the

quantifiers

range over all

natural numbers

). For indirect proof, suppose that

S

is not the set of natural numbers

ℕ

. That is, the

complement

ℕ

∖

S

is nonempty. The

well-ordering principle for natural numbers

says that

ℕ

∖

S

has a smallest

element

; call it

a

. By

assumption

, the statement

(

∀

m

<

a

m

∈

S

)

⇒

a

∈

S

holds. Equivalently, the

contrapositive

statement

a

∈

ℕ

∖

S

⇒

∃

m

<

a

⁢

m

∈

ℕ

∖

S

holds. This gives a contradition since the element

a

is an element of

ℕ

∖

S

and is, moreover, the

smallest

element of

ℕ

∖

S

.

Title

principle of finite induction

proven from the well-ordering principle for natural numbers

Canonical

name

PrincipleOfFiniteInductionProvenFromTheWellorderingPrincipleForNaturalNumbers

Date of creation

2013-03-22 11:48:04

Last modified on

2013-03-22 11:48:04

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

16

Author

CWoo (3771)

Entry type

Proof

Classification

msc 03E25

Classification

msc 37H10

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