aarjaneiro/planetmath huggingface dataset:
intersection of sets

intersection of sets

Let

X

,

Y

be sets. The

intersection

of

X

and

Y

, denoted

X

∩

Y

is the set

X

∩

Y

=

{

z

:

z

∈

X

,

z

∈

Y

}

Title

intersection of sets

Canonical name

IntersectionOfSets

Date of creation

2013-03-22 14:01:50

Last modified on

2013-03-22 14:01:50

Owner

drini (3)

Last modified by

drini (3)

Numerical id

6

Author

drini (3)

Entry type

Definition

Classification

msc 03Exx

Generated on Thu Feb 8 20:22:29 2018 by

LaTeXML
natural deduction

natural deduction

Natural deduction

refers to related

proof systems

for several different kinds of logic, intended to be

similar

to the way people actually reason. Unlike many other proof systems, it has many rules and few axioms.

Sequents

in natural deduction have only one

formula

on the right side.

Typically the rules consist of one pair for each

connective

, one of which allows the

introduction

of that symbol and the other its elimination.

To give one example, the proof rules

→

I

and

→

E

are:

Γ

,

α

⇒

β

Γ

⇒

α

→

β

(

→

I

)

and

Γ

⇒

α

→

β

Σ

⇒

α

[

Γ

,

Σ

]

⇒

β

(

→

E

)

Title

natural deduction

Canonical name

NaturalDeduction

Date of creation

2013-03-22 13:05:14

Last modified on

2013-03-22 13:05:14

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Definition

Classification

msc 03F03

Related topic

GentzenSystem

Generated on Thu Feb 8 20:23:42 2018 by

LaTeXML
sequent

sequent

A

sequent

represents a formal step in a proof. Typically it consists of two lists of

formulas

, one representing the

premises

and one the

conclusions

. A typical sequent might be:

ϕ

,

ψ

⇒

α

,

β

where

ϕ

and

ψ

are the premises and

α

and

β

are the conclusions.

This claims that, from premises

ϕ

and

ψ

either

α

or

β

must be true. Note that

⇒

is not a symbol in the

language

, rather it is a symbol in the

metalanguage

used to discuss proofs. Also, notice the asymmetry: everything on the left must be true to conclude only one thing on the right. This does create a different kind of

symmetry

, since adding formulas to either side results in a weaker sequent, while removing them from either side gives a stronger one.

Some systems allow only one formula on the right.

Most

proof systems

provide ways to deduce one sequent from another. These rules are written with a list of sequents above and below a line. This rule indicates that if everything above the line is true, so is everything under the line. A typical rule is:

Γ

⇒

Σ

Γ

,

α

⇒

Σ

α

,

Γ

⇒

Σ

This indicates that if we can deduce

Σ

from

Γ

, we can also deduce it from

Γ

together with

α

.

Note that the capital

Greek letters

are usually used to denote a (possibly empty) list of formulas.

[

Γ

,

Σ

]

is used to denote the

contraction

of

Γ

and

Σ

, that is, the list of those formulas appearing in either

Γ

or

Σ

but with no repeats.

Title

sequent

Canonical

name

Sequent

Date of creation

2013-03-22 13:05:11

Last modified on

2013-03-22 13:05:11

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

10

Author

Henry (455)

Entry type

Definition

Classification

msc 03F03

Related topic

GentzenSystem

Defines

contraction

Defines

premise

Defines

conclusion

Generated on Thu Feb 8 20:24:00 2018 by

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sound,, complete

sound,, complete

If

T

⁢

h

and

P

⁢

r

are two sets of facts (in particular, a theory of some language and the set of things provable by some method) we say

P

⁢

r

is

sound

for

T

⁢

h

if

P

⁢

r

⊆

T

⁢

h

. Typically we have a theory and set of rules for constructing proofs, and we say the set of rules are sound (which theory is intended is usually clear from context) since everything they prove is true (in

T

⁢

h

).

If

T

⁢

h

⊆

P

⁢

r

we say

P

⁢

r

is

complete

for

T

⁢

h

. Again, we usually have a theory and a set of rules for constructing proofs, and say that the set of rules is complete since everything true (in

T

⁢

h

) can be proven.

Title

sound,, complete

Canonical

name

SoundComplete

Date of creation

2013-03-22 13:02:33

Last modified on

2013-03-22 13:02:33

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

7

Author

Henry (455)

Entry type

Definition

Classification

msc 03F03

Defines

sound

Defines

complete

Generated on Thu Feb 8 20:24:02 2018 by

LaTeXML
𝒩⁢𝒥⁢p

𝒩

⁢

𝒥

⁢

p

𝒩

⁢

𝒥

⁢

p

is a

natural deduction

proof system

for intuitionistic

propositional logic

. Its only axiom is

α

⇒

α

for any atomic

α

. Its rules are:

Γ

⇒

α

Γ

⇒

α

∨

β

Γ

⇒

β

∨

α

⁢

c

⁢

c

⁢

(

∨

I

)

Γ

⇒

α

Σ

,

α

0

⇒

ϕ

Π

,

β

0

⇒

ϕ

[

Γ

,

Σ

,

Π

]

⇒

ϕ

⁢

(

∨

E

)

The syntax

α

0

indicates that the rule also holds if that

formula

is omitted.

Γ

⇒

α

Σ

⇒

β

[

Γ

,

Σ

]

⇒

α

∧

β

⁢

(

∧

I

)

Γ

⇒

α

∧

β

Γ

⇒

α

Γ

⇒

β

⁢

(

∧

E

)

Γ

,

α

⇒

β

Γ

⇒

α

→

β

(

→

I

)

Γ

⇒

α

→

β

Σ

⇒

α

[

Γ

,

Σ

]

⇒

β

(

→

E

)

Γ

⇒

⊥

Γ

⇒

α

⁢

(

⊥

i

)

,

where

⁢

α

⁢

is atomic

Title

𝒩

⁢

𝒥

⁢

p

Canonical name

mathcalNJp

Date of creation

2013-03-22 13:05:17

Last modified on

2013-03-22 13:05:17

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

7

Author

Henry (455)

Entry type

Definition

Classification

msc 03F03

Synonym

NJp

Generated on Thu Feb 8 20:24:04 2018 by

LaTeXML
𝒩⁢𝒦⁢p

𝒩

⁢

𝒦

⁢

p

𝒩

⁢

𝒦

⁢

p

is a

natural deduction

proof system

for classical

propositional logic

. It is identical to NJp except that it replaces the rule

⊥

i

with the rule:

Γ

,

¬

⁢

α

⇒

⊥

Γ

⇒

α

⁢

(

⊥

c

)

,

where

⁢

α

⁢

is atomic

.

Title

𝒩

⁢

𝒦

⁢

p

Canonical name

mathcalNKp

Date of creation

2013-03-22 13:05:23

Last modified on

2013-03-22 13:05:23

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Definition

Classification

msc 03F03

Synonym

NKp

Generated on Thu Feb 8 20:24:06 2018 by

LaTeXML
circular reasoning

circular reasoning

Circular reasoning

is an attempted proof of a statement that uses at least one of the following two things:

•

the statement that is to be proven

•

a fact that relies on the statement that is to be proven

Such proofs are not valid.

As an example, below is a faulty proof that the

well-ordering principle

implies the

axiom of choice

(

http://planetmath.org/WellOrderingPrincipleImpliesAxiomOfChoice

). The step where circular reasoning is used is surrounded by brackets [ ].

Let

C

be a

collection

of

nonempty sets

. By the well-ordering principle, each

S

∈

C

is

well-ordered

. [For each

S

∈

C

, let

<

S

denote the well-ordering of

S

.] Let

m

S

denote the least member of each

S

∈

C

with respect to

<

S

. Then a

choice function

f

:

C

→

⋃

S

∈

C

S

can be defined by

f

⁢

(

S

)

=

m

S

.

The step surrounded by brackets is faulty because it actually uses the axiom of choice, which is what is to be proven. In the step, for each

S

∈

C

, an

ordering

is chosen. This cannot be done in general without appealing to the axiom of choice.

Title

circular reasoning

Canonical name

CircularReasoning

Date of creation

2013-03-22 16:06:32

Last modified on

2013-03-22 16:06:32

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

15

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 03F07

Synonym

circular argument

Generated on Thu Feb 8 20:24:08 2018 by

LaTeXML
contradiction

contradiction

A

contradiction

occurs when the statements

p

and

¬

⁢

p

are shown to be true simultaneously. This

concept

appears most often in a

proof by contradiction

(also known as

reductio ad absurdum

), which is proving a statement by supposing its

negation

is true and logically deducing an absurd statement. That is, in attempting to prove

q

, one may assume

¬

⁢

q

and attempt to obtain a statement of the form

¬

⁢

r

, where

r

is a statement that is assumed or known to be true.

Proofs by contradiction can become confusing. This is especially the case when such proofs are nested; i.e. (

http://planetmath.org/Ie

), a proof by contradiction occurs within a proof by contradiction. Some mathematicians prefer to use a direct proof whenever possible, as such are easier to follow in general. A small minority of mathematicians go so far as to reject proof by contradiction as a valid proof technique. It should be pointed out that something good can be said for proof by contradiction: If one wants to prove a statement of the form

p

⟹

q

, using the technique of proof by contradiction gives an additional

hypothesis

with which to work.

Title

contradiction

Canonical

name

Contradiction

Date of creation

2013-03-22 16:02:48

Last modified on

2013-03-22 16:02:48

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

9

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 03F07

Classification

msc 03B05

Related topic

ContradictoryStatement

Defines

proof by contradiction

Defines

reductio ad absurdum

Generated on Thu Feb 8 20:24:09 2018 by

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existential theorem

existential theorem

An

existential theorem

is a

theorem

which exists.

In general, there are two ways to prove an existential theorem. The most convincing method is a

constructive proof

, and another common method is an existential proof. The reason that a constructive proof is most convincing is that, after reading such a proof, readers can actually get their hands on the mathematical in question. In some cases, however, constructing the mathematical is difficult, if not impossible. In this case, an existential proof may be the only method for proving an existential theorem. An example of this is the

primitive element theorem

.

Title

existential theorem

Canonical

name

ExistentialTheorem

Date of creation

2013-03-22 17:08:57

Last modified on

2013-03-22 17:08:57

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

5

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 03F07

Classification

msc 00A35

Synonym

existence theorem

Related topic

TechniquesInMathematicalProofs

Generated on Thu Feb 8 20:24:12 2018 by

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universal assumption

universal assumption

Given a

biconditional

statement, a

universal assumption

is an

assumption

that can be used in a proof of both the necessity and sufficiency directions.

For an example of a universal assumption, please see the entry

continuous functions

on the

extended real numbers

.

Title

universal assumption

Canonical name

UniversalAssumption

Date of creation

2013-03-22 17:11:47

Last modified on

2013-03-22 17:11:47

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

5

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 03F07

Classification

msc 03B05

Synonym

universal hypothesis

Related topic

NecessaryAndSufficient

Generated on Thu Feb 8 20:24:26 2018 by

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additively indecomposable

additively indecomposable

An

ordinal

α

is called

additively indecomposable

if it is not

0

and for any

β

,

γ

<

α

, we have

β

+

γ

<

α

.
The set of additively indecomposable ordinals is denoted

ℍ

.

Obviously

1

∈

ℍ

, since

0

+

0

<

1

.
No finite ordinal other than

1

is in

ℍ

.
Also,

ω

∈

ℍ

, since the sum of two finite ordinals is still finite.
More generally, every

infinite

cardinal

is in

ℍ

.

ℍ

is closed and

unbounded

, so the enumerating function of

ℍ

is normal.
In fact,

f

ℍ

⁢

(

α

)

=

ω

α

.

The

derivative

f

ℍ

′

⁢

(

α

)

is written

ϵ

α

.
Ordinals of this form (that is,

fixed points

of

f

ℍ

) are called

epsilon numbers

.
The number

ϵ

0

=

ω

ω

ω

⋅

⋅

⋅

is therefore the first fixed point of the series

ω

,

ω

ω

,

ω

ω

ω

,

…

Title

additively indecomposable

Canonical

name

AdditivelyIndecomposable

Date of creation

2013-03-22 13:29:04

Last modified on

2013-03-22 13:29:04

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

11

Author

mathcam (2727)

Entry type

Definition

Classification

msc 03F15

Classification

msc 03E10

Related topic

OrdinalArithmetic

Defines

epsilon number

Defines

epsilon zero

Generated on Thu Feb 8 20:24:28 2018 by

LaTeXML
classes of ordinals and enumerating functions

classes of ordinals and enumerating functions

A

class of

ordinals

is just a

subclass

of the class (

http://planetmath.org/Class

)

𝐎𝐧

of all ordinals. For every class of ordinals

M

there is an

enumerating function

f

M

defined by

transfinite recursion

:

f

M

⁢

(

α

)

=

min

⁡

{

x

∈

M

∣

f

⁢

(

β

)

<

x

⁢

for all

⁢

β

<

α

}

,

and we define the

order type

of

M

by

otype

⁡

(

M

)

=

dom

⁡

(

f

)

. The possible values for this value are either

𝐎𝐧

or some ordinal

α

. The above

function

simply lists the

elements of

M

in order. Note that it is not necessarily defined for all ordinals, although it is defined for a

segment

of the ordinals. If

α

<

β

then

f

M

⁢

(

α

)

<

f

M

⁢

(

β

)

, so

f

M

is an order isomorphism between

otype

⁡

(

M

)

and

M

.

For an ordinal

κ

, we say

M

is

κ

-

closed

if for any

N

⊆

M

such that

|

N

|

<

κ

, also

sup

⁡

N

∈

M

.

We say

M

is

κ

-

unbounded

if for any

α

<

κ

there is some

β

∈

M

such that

α

<

β

.

We say a function

f

:

M

→

𝐎𝐧

is

κ

-

continuous

if

M

is

κ

-closed and

f

⁢

(

sup

⁡

N

)

=

sup

⁡

{

f

⁢

(

α

)

∣

α

∈

N

}

A function is

κ

-normal

if it is order preserving (

α

<

β

implies

f

⁢

(

α

)

<

f

⁢

(

β

)

) and continuous. In particular, the enumerating function of a

κ

-closed class is always

κ

-normal.

All these definitions can be easily extended to all ordinals: a class is

closed

(resp.

unbounded

) if it is

κ

-closed (unbounded) for all

κ

. A function is

continuous

(resp.

normal

) if it is

κ

-continuous (normal) for all

κ

.

Title

classes of ordinals and enumerating functions

Canonical name

ClassesOfOrdinalsAndEnumeratingFunctions

Date of creation

2013-03-22 13:28:55

Last modified on

2013-03-22 13:28:55

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

14

Author

mathcam (2727)

Entry type

Definition

Classification

msc 03F15

Classification

msc 03E10

Defines

order type

Defines

enumerating function

Defines

closed

Defines

kappa-closed

Defines

continuous

Defines

kappa-continuous

Defines

continuous function

Defines

kappa-continuous function

Defines

closed class

Defines

kappa-closed class

Defines

normal function

Defines

kappa-normal function

Defines

normal

Defines

kappa-normal

Defines

unbounded

Defines

unbounded clas

Generated on Thu Feb 8 20:24:30 2018 by

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Elementary Functional Arithmetic

Elementary Functional Arithmetic

Elementary Functional Arithmetic

, or EFA, is a weak theory of

arithmetic

created by removing

induction

from

Peano Arithmetic

. Because it lacks induction, axioms defining

exponentiation

must be added.

•

∀

x

(

x

′

≠

0

)

(

0

is the first number)

•

∀

x

,

y

(

x

′

=

y

′

→

x

=

y

)

(the successor function is

one-to-one

)

•

∀

x

(

x

+

0

=

x

)

(

0

is the

additive identity

)

•

∀

x

,

y

(

x

+

y

′

=

(

x

+

y

)

′

)

(

addition

is the repeated application of the successor function)

•

∀

x

(

x

⋅

0

=

0

)

•

∀

x

,

y

(

x

⋅

(

y

′

)

=

x

⋅

y

+

x

(

multiplication

is repeated addition)

•

∀

x

⁢

(

¬

⁡

(

x

<

0

)

)

(

0

is the smallest number)

•

∀

x

,

y

(

x

<

y

′

↔

x

<

y

∨

x

=

y

)

•

∀

x

(

x

0

=

1

)

•

∀

x

(

x

y

′

=

x

y

⋅

x

)

Title

Elementary Functional Arithmetic

Canonical

name

ElementaryFunctionalArithmetic

Date of creation

2013-03-22 12:56:39

Last modified on

2013-03-22 12:56:39

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Definition

Classification

msc 03F30

Synonym

EFA

Related topic

PeanoArithmetic

Generated on Thu Feb 8 20:24:33 2018 by

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PA

PA

(PA) is the

restriction

of Peano’s axioms to a

first order theory

of . The only change is that the

induction axiom

is replaced by

induction

restricted to

arithmetic formulas

:

ϕ

(

0

)

∧

∀

x

(

ϕ

(

x

)

→

ϕ

(

x

′

)

)

→

∀

x

ϕ

(

x

)

)

where

ϕ

is arithmetical

Note that this replaces the single,

second-order

,

axiom of induction

with a

countably infinite

schema of axioms.

Appropriate axioms defining

+

,

⋅

, and

<

are included. A full list of the axioms of PA looks like this (although the exact list of axioms varies somewhat from source to source):

•

∀

x

(

x

′

≠

0

)

(

0

is the first number)

•

∀

x

,

y

(

x

′

=

y

′

→

x

=

y

)

(the successor function is

one-to-one

)

•

∀

x

(

x

+

0

=

x

)

(

0

is the

additive identity

)

•

∀

x

,

y

(

x

+

y

′

=

(

x

+

y

)

′

)

(

addition

is the repeated application of the successor function)

•

∀

x

(

x

⋅

0

=

0

)

•

∀

x

,

y

(

x

⋅

(

y

′

)

=

x

⋅

y

+

x

)

(

multiplication

is repeated addition)

•

∀

x

⁢

(

¬

⁡

(

x

<

0

)

)

(

0

is the smallest number)

•

∀

x

,

y

(

x

<

y

′

↔

x

<

y

∨

x

=

y

)

•

ϕ

(

0

)

∧

∀

x

(

ϕ

(

x

)

→

ϕ

(

x

′

)

)

→

∀

x

ϕ

(

x

)

)

where

ϕ

is arithmetical

Title

PA

Canonical

name

PA

Date of creation

2013-03-22 12:56:36

Last modified on

2013-03-22 12:56:36

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

10

Author

Henry (455)

Entry type

Definition

Classification

msc 03F30

Synonym

Peano arithmetic

Synonym

first order Peano arithmetic

Related topic

PeanoArithmetic

Generated on Thu Feb 8 20:24:35 2018 by

LaTeXML
Peano arithmetic

Peano arithmetic

Peano’s axioms

are a

definition

of the set of natural numbers, denoted

ℕ

. From these axioms

Peano arithmetic

on

natural numbers

can be derived.

1.

0

∈

ℕ

(0 is a natural number)

2.

For each

x

∈

ℕ

, there exists exactly one

x

′

∈

ℕ

, called the

successor

of

x

3.

x

′

≠

0

(0 is not the successor of any natural number)

4.

x

=

y

if and only if

x

′

=

y

′

.

5.

(

axiom of induction

) If

M

⊆

ℕ

and

0

∈

M

and

x

∈

M

implies

x

′

∈

M

, then

M

=

ℕ

.

The

successor

of

x

is sometimes denoted

S

⁢

x

instead of

x

′

. We then have

1

=

S

⁢

0

,

2

=

S

⁢

1

=

S

⁢

S

⁢

0

, and so on.

Peano arithmetic consists of statements derived via these axioms. For

instance

, from these axioms we can define

addition

and

multiplication

on natural numbers.
Addition is defined as

x

+

1

=

x

′

for all

⁢

x

∈

ℕ

x

+

y

′

=

(

x

+

y

)

′

for all

⁢

x

,

y

∈

ℕ

Addition defined in this manner can then be proven to be both

associative

and

commutative

.

Multiplication is

x

⋅

1

=

x

for all

⁢

x

∈

ℕ

x

⋅

y

′

=

x

⋅

y

+

x

for all

⁢

x

,

y

∈

ℕ

This definition of multiplication can also be proven to be both associative and commutative, and it can also be shown to be distributive over addition.

Title

Peano arithmetic

Canonical

name

PeanoArithmetic

Date of creation

2013-03-22 12:32:42

Last modified on

2013-03-22 12:32:42

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

8

Author

alozano (2414)

Entry type

Axiom

Classification

msc 03F30

Related topic

NaturalNumber

Related topic

PressburgerArithmetic

Related topic

ElementaryFunctionalArithmetic

Related topic

PeanoArithmeticFirstOrder

Defines

Peano’s axioms

Defines

successor

Defines

axiom of induction

Generated on Thu Feb 8 20:24:37 2018 by

LaTeXML
A⁢C⁢A_0

A

⁢

C

⁢

A

0

A

⁢

C

⁢

A

0

is a weakened form of

second order arithmetic

. Its axioms include the axioms of PA together with

arithmetic

comprehension

.

Title

A

⁢

C

⁢

A

0

Canonical

name

ACA0

Date of creation

2013-03-22 12:57:03

Last modified on

2013-03-22 12:57:03

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

4

Author

Henry (455)

Entry type

Definition

Classification

msc 03F35

Synonym

ACA

Synonym

ACA0

Generated on Thu Feb 8 20:24:39 2018 by

LaTeXML
comprehension axiom

comprehension axiom

The

axiom of comprehension

(CA) states that every

formula

defines a set. That is,

∃

X

∀

x

(

x

∈

X

↔

ϕ

(

x

)

)

for any formula

ϕ

where

X

does not occur free in

ϕ

The specification and

separation

are sometimes used in place of comprehension, particularly for weakened forms of the axiom (see below).

In theories which make no distinction between objects and sets (such as ZF), this formulation leads to Russell’s

paradox

, however in stratified theories this is not a problem (for example

second order arithmetic

includes the axiom of comprehension).

This axiom can be restricted in various ways. One possibility is to restrict it to forming subsets of sets:

∀

Y

∃

X

∀

x

(

x

∈

X

↔

x

∈

Y

∧

ϕ

(

x

)

)

for any formula

ϕ

where

X

does not occur free in

ϕ

This formulation (used in ZF

set theory

) is sometimes called the Aussonderungsaxiom.

Another way is to restrict

ϕ

to some family

F

, giving the axiom F-CA. For

instance

the axiom

Σ

1

0

-CA is:

∃

X

∀

x

(

x

∈

X

↔

ϕ

(

x

)

)

where

ϕ

is

Σ

1

0

and

X

does not occur free in

ϕ

A third form (usually called separation) uses two formulas, and guarantees only that those satisfying one are included while those satisfying the other are excluded. The unrestricted form is the same as unrestricted

collection

, but, for instance,

Σ

1

0

separation:

∀

x

¬

(

ϕ

(

x

)

∧

ψ

(

x

)

)

→

∃

X

∀

x

(

(

ϕ

(

x

)

→

x

∈

X

)

∧

(

ψ

(

x

)

→

x

∉

X

)

)

where

⁢

ϕ

⁢

and

⁢

ψ

⁢

are

⁢

Σ

1

0

⁢

and

⁢

X

⁢

does not occur free in

⁢

ϕ

⁢

or

⁢

ψ

is weaker than

Σ

1

0

-CA.

Title

comprehension axiom

Canonical name

ComprehensionAxiom

Date of creation

2013-03-22 12:56:54

Last modified on

2013-03-22 12:56:54

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

10

Author

Henry (455)

Entry type

Definition

Classification

msc 03F35

Synonym

CA

Synonym

-CA

Synonym

comprehension

Synonym

comprehension axiom

Synonym

axiom of comprehension

Synonym

separation

Synonym

separation axiom

Synonym

axiom of separation

Synonym

specification

Synonym

specification axiom

Synonym

axiom of specification

Synonym

Aussonderungsaxiom

Generated on Thu Feb 8 20:24:41 2018 by

LaTeXML
induction axiom

induction axiom

An

induction axiom

specifies that a theory includes

induction

, possibly restricted to specific

formulas

. IND is the general

axiom of induction

:

ϕ

(

0

)

∧

∀

x

(

ϕ

(

x

)

→

ϕ

(

x

+

1

)

)

→

∀

x

ϕ

(

x

)

for any formula

ϕ

If

ϕ

is restricted to some family of formulas

F

then the axiom is called F-IND, or F induction. For example the axiom

Σ

1

0

-IND is:

ϕ

(

0

)

∧

∀

x

(

ϕ

(

x

)

→

ϕ

(

x

+

1

)

)

→

∀

x

ϕ

(

x

)

where

ϕ

is

Σ

1

0

Title

induction axiom

Canonical name

InductionAxiom

Date of creation

2013-03-22 12:56:51

Last modified on

2013-03-22 12:56:51

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

7

Author

Henry (455)

Entry type

Definition

Classification

msc 03F35

Synonym

IND

Synonym

-IND

Synonym

axiom of induction

Generated on Thu Feb 8 20:24:43 2018 by

LaTeXML
R⁢C⁢A_0

R

⁢

C

⁢

A

0

R

⁢

C

⁢

A

0

is a weakened form of

second order arithmetic

. It consists of the axioms of PA other than

induction

, together with

Σ

1

0

-IND and

Δ

1

0

-CA.

Title

R

⁢

C

⁢

A

0

Canonical name

RCA0

Date of creation

2013-03-22 12:57:00

Last modified on

2013-03-22 12:57:00

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

4

Author

Henry (455)

Entry type

Definition

Classification

msc 03F35

Synonym

RCA

Synonym

RCA0

Generated on Thu Feb 8 20:24:45 2018 by

LaTeXML