aarjaneiro/planetmath huggingface dataset:
inverse statement

inverse statement

Let a statement be of the form of an

implication

If

p

, then

q

i.e. (

http://planetmath.org/Ie

), it has a certain

premise

p

and a

conclusion

q

. The statement in which one has negated the conclusion and the premise,

If

¬

⁢

p

, then

¬

⁢

q

is the

inverse

(or

inverse statement

) of the first. Note that the following constructions yield the same statement:

•

the inverse of the original statement;

•

the

contrapositive

of the

converse

of the original statement;

•

the converse of the contrapositive of the original statement.

Therefore, just as an implication and its contrapositive are

logically equivalent

(proven here (

http://planetmath.org/SomethingRelatedToContrapositive

)), the converse of the original statement and the inverse of the original statement are also logically equivalent.

The phrase “inverse

theorem

” is in usage; however, it is nothing akin to the phrase “converse theorem (

http://planetmath.org/ConverseTheorem

)”. In the phrase “inverse theorem”, the word “inverse” typically refers to a

multiplicative inverse

. An example of this usage is the binomial inverse theorem (

http://planetmath.org/BinomialInverseTheorem

).

Title

inverse statement

Canonical name

InverseStatement

Date of creation

2013-03-22 17:20:00

Last modified on

2013-03-22 17:20:00

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

10

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 03B05

Synonym

inverse

Related topic

Converse

Related topic

SomethingRelatedToContrapositive

Related topic

ConverseTheorem

Generated on Thu Feb 8 19:23:27 2018 by

LaTeXML
logically equivalent

logically equivalent

Two

formulas

A

and

B

are said to be

logically equivalent

(typically shortened to

equivalent

) when

A

is true if and only if

B

is true (that is,

A

implies

B

and

B

implies

A

):

⊧

A

↔

B

.

This is sometimes abbreviated as

A

⇔

B

.

For example, for any integer

z

, the statement “

z

is

positive

” is equivalent to “

z

is not negative and

z

≠

0

”.

More generally, one says that a formula

A

is a logical consequence of a set

Γ

of formulas, written

Γ

⊧

A

if whenever every formula in

Γ

is true, so is

A

. If

Γ

is a

singleton

consisting of formula

B

, we also write

B

⊧

A

.

Using this, one sees that

⊧

A

↔

B

iff

A

⊧

B

⁢

and

⁢

B

⊧

A

.

To see this: if

⊧

A

↔

B

, then

A

→

B

and

B

→

A

are both true, which means that if

A

is true so is

B

and that if

B

is true so is

A

, or

A

⊧

B

and

B

⊧

A

. The

argument

can be reversed.

Remark

. Some authors call the above notion semantical equivalence or tautological equivalence, rather than logical equivalence. In their view, logical equivalence is a syntactic notion:

A

and

B

are logically equivalent whenever

A

is

deducible

from

B

and

B

is deducible from

A

in some

deductive system

.

Title

logically equivalent

Canonical

name

LogicallyEquivalent

Date of creation

2013-03-22 13:17:00

Last modified on

2013-03-22 13:17:00

Owner

sleske (997)

Last modified by

sleske (997)

Numerical id

10

Author

sleske (997)

Entry type

Definition

Classification

msc 03B05

Synonym

tautologically equivalent

Synonym

semantically equivalent

Synonym

tautological equivalence

Synonym

semantical equivalence

Synonym

tautological consequence

Synonym

semantical consequence

Related topic

Biconditional

Defines

logical equivalence

Defines

logical consequence

Generated on Thu Feb 8 19:23:31 2018 by

LaTeXML
necessary and sufficient

necessary and sufficient

The statement “

p

is

necessary

for

q

” “

q

implies (

http://planetmath.org/

Implication

)

p

”.

The statement “

p

is

sufficient

for

q

” “

p

implies (

http://planetmath.org/Implication

)

q

”.

The statement “

p

is

necessary and sufficent

for

q

” “

p

if and only if (

http://planetmath.org/Iff

)

q

”.

For an example of how these terms are used in mathematics, see the entry on

complete ultrametric fields

.

Biconditional

statements are often proven by breaking them into two implications and proving them separately. Often, the terms

necessity

and

sufficiency

are used to indicate which implication is being proven. For an example of this usage, see the entry called

relationship between totatives and divisors

.

Title

necessary and sufficient

Canonical name

NecessaryAndSufficient

Date of creation

2013-03-22 16:07:31

Last modified on

2013-03-22 16:07:31

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

12

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 03B05

Classification

msc 03F07

Related topic

UniversalAssumption

Related topic

SufficientConditionOfPolynomialCongruence

Defines

necessary

Defines

necessity

Defines

sufficient

Defines

sufficiency

Generated on Thu Feb 8 19:23:45 2018 by

LaTeXML
negation

negation

In logics and mathematics,

negation

(from Latin

negare

‘to deny’) is the unary operation “

¬

” which swaps the truth value of any operand to the truth value.  So, if the statement

P

is true then its negated statement

¬

⁢

P

is false, and vice versa.

Note 1.

The negated statement

¬

⁢

P

(by Heyting) has been denoted also with

-

P

(Peano),

∼

P

(Russell),

P

¯

(Hilbert) and

N

⁢

P

(by the

Polish notation

).

Note 2.

¬

⁢

P

may be expressed by

implication

as

P

→

⋏

where

⋏

means any

contradictory statement

.

Note 3.

The negation of

logical or

and

logical and

give the results

¬

⁡

(

P

∨

Q

)

≡

¬

⁢

P

∧

¬

⁢

Q

,

¬

⁡

(

P

∧

Q

)

≡

¬

⁢

P

∨

¬

⁢

Q

.

Analogical results concern the

quantifier

statements:

¬

⁡

(

∃

x

)

⁢

P

⁢

(

x

)

≡

(

∀

x

)

⁢

¬

⁢

P

⁢

(

x

)

,

¬

⁡

(

∀

x

)

⁢

P

⁢

(

x

)

≡

(

∃

x

)

⁢

¬

⁢

P

⁢

(

x

)

.

These all are known as de Morgan’s laws.

Note 4.

Many mathematical

relation

statements, expressed with such special

relation symbols

as

=

,

⊆

,

∈

,

≅

,

∥

,

∣

,  are negated by using in the symbol an additional cross line:

≠

,

⊈

,

∉

,

≇

,

∦

,

∤

.

Title

negation

Canonical name

Negation

Date of creation

2015-04-25 17:44:13

Last modified on

2015-04-25 17:44:13

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Definition

Classification

msc 03B05

Synonym

logical not

Related topic

SetMembership

Generated on Thu Feb 8 19:23:47 2018 by

LaTeXML
proof that contrapositive statement is logically equivalent to original statement

proof that contrapositive statement is logically equivalent to original statement

You can see that the

contrapositive

of an

implication

is true by considering
the following:

The statement

p

⇒

q

is

logically equivalent

to

¬

⁢

p

∨

q

which can also be written as

p

¯

∨

q

.

By the same token, the contrapositive statement

q

¯

⇒

p

¯

is logically equivalent to

¬

⁢

q

¯

∨

p

¯

which, using

double negation

on

q

, becomes

q

∨

p

¯

.

This, of course, is the same logical statement.

Title

proof that contrapositive statement is logically equivalent to original statement

Canonical name

ProofThatContrapositiveStatementIsLogicallyEquivalentToOriginalStatement

Date of creation

2013-03-22 13:42:10

Last modified on

2013-03-22 13:42:10

Owner

sprocketboy (2515)

Last modified by

sprocketboy (2515)

Numerical id

10

Author

sprocketboy (2515)

Entry type

Proof

Classification

msc 03B05

Related topic

Inverse7

Related topic

Inverse6

Generated on Thu Feb 8 19:23:49 2018 by

LaTeXML
propositional logic

propositional logic

A

propositional logic

is a logic in which the only objects are

propositions

, that is, objects which themselves have truth values. Variables represent propositions, and there are no

relations

,

functions

, or

quantifiers

except for the

constants

T

and

⊥

(representing true and false respectively). The

connectives

are typically

¬

,

∧

,

∨

, and

→

(representing

negation

,

conjunction

,

disjunction

, and

implication

), however this set is

redundant

, and other choices can be used (

T

and

⊥

can also be considered

0

-ary connectives).

A model for propositional logic is just a

truth function

ν

on a set of variables. Such a truth function can be easily extended to a truth function

ν

¯

on all

formulas

which contain only the variables

ν

is defined on by adding

recursive

clauses for the usual definitions of connectives. For

instance

ν

¯

⁢

(

α

∧

β

)

=

T

iff

ν

¯

⁢

(

α

)

=

ν

¯

⁢

(

β

)

=

T

.

Then we say

ν

⊧

ϕ

if

ν

¯

⁢

(

ϕ

)

=

T

, and we say

⊧

ϕ

if for every

ν

such that

ν

¯

⁢

(

ϕ

)

is defined,

ν

⊧

ϕ

(and say that

ϕ

is a

tautology

).

Propositional logic is

decidable

: there is an easy way to determine whether a

sentence

is a tautology. It can be done using

truth tables

, since a truth table for a particular formula can be easily produced, and the formula is a tautology if every assignment of truth values makes it true. It is not known whether this method is efficient: the

equivalent

problem of whether a formula is

satisfiable

(that is, whether its negation is a tautology) is a canonical example of an

𝒩

⁢

𝒫

-

complete

problem.

Title

propositional logic

Canonical name

PropositionalLogic

Date of creation

2013-03-22 13:04:01

Last modified on

2013-03-22 13:04:01

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

6

Author

Henry (455)

Entry type

Definition

Classification

msc 03B05

Related topic

Implication

Related topic

Biconditional

Related topic

Conjunction

Related topic

Disjunction

Related topic

PropositionalCalculus

Related topic

ExclusiveOr

Related topic

InterpretationOfPropositions

Defines

proposition

Generated on Thu Feb 8 19:23:51 2018 by

LaTeXML
tautology

tautology

A

tautology

is a statement (or form) which is true solely on account of its logical form rather than because of the meaning of the terms employed.

In

propositional logic

, a tautology is a statement which is true regardless of the truth value of the substatements of which it is composed. An example would be

It is a sunny day and on a sunny day the rain does not fall so the rain does not fall.

This statement is true no matter what the truth value of the statements of which it is comprised, “It is a sunny day” and “The rain does not fall” may be. More generally, any statement of the form

(P and (if P then Q)) implies Q

or, in symbols,

(

P

∧

(

P

→

Q

)

)

→

Q

is true no matter what statements one may substitute for

P

and for

Q

. (They may even contain

connectives

themselves. An

expression

of the sort appearing above in which one can obtain tautologies by substituting arbitrary statements for the

variables

which appear in the expression is known as a

tautologous form

.)

To test a statement or form to see if it is a tautology, one may construct a

truth table

. If it turns out that one obtains “T” in every column, then the statement is a tautology.

In predicate logic, a tautology is a statement which is true no matter what choice one makes for the

predicates

which appear in that statement. A simple example would be

If all cats are black then there exists a black cat.

This is of the general form

(

∀

x

)

(

A

(

x

)

→

B

(

x

)

)

→

(

∃

x

)

(

A

(

x

)

→

B

(

x

)

)

with

A

being “is a cat” and

B

being “is black”.

In the notation of Peano, tautology is denoted by “

⋎

”. (“v”
comes from the Latin “verum” = ’true’.)

Title

tautology

Canonical

name

Tautology

Date of creation

2013-03-22 15:27:35

Last modified on

2013-03-22 15:27:35

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

13

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 03B05

Classification

msc 03B10

Synonym

tautological

Synonym

tautologically

Related topic

ContradictoryStatement

Generated on Thu Feb 8 19:24:52 2018 by

LaTeXML
transitive

transitive

The

transitive property

of logic is

(

a

⇒

b

)

∧

(

b

⇒

c

)

⇒

(

a

⇒

c

)

Where

⇒

is the

conditional truth function

. From this we can derive that

(

a

=

b

)

∧

(

b

=

c

)

⇒

(

a

=

c

)

Title

transitive

Canonical name

Transitive1

Date of creation

2013-03-22 12:14:16

Last modified on

2013-03-22 12:14:16

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

5

Author

akrowne (2)

Entry type

Definition

Classification

msc 03B05

Synonym

transitive property

Generated on Thu Feb 8 19:24:54 2018 by

LaTeXML
truth function

truth function

A

truth function

is a

function

that returns one of two values, one of which is interpreted as “true,” and the other which is interpreted as “false”. Typically either “T” and “F” are used, or “1” and “0”, respectively. Using the latter, we can write

f

:

{

0

,

1

}

n

→

{

0

,

1

}

defines a truth function

f

. That is,

f

is a

mapping

from any number (

n

) of true/false (0 or 1) values to a single value, which is 0 or 1.

Title

truth function

Canonical name

TruthFunction

Date of creation

2013-03-22 11:53:03

Last modified on

2013-03-22 11:53:03

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

8

Author

akrowne (2)

Entry type

Definition

Classification

msc 03B05

Generated on Thu Feb 8 19:24:56 2018 by

LaTeXML
arithmetical hierarchy

arithmetical hierarchy

The

arithmetical hierarchy

is a hierarchy of either (depending on the

context

)

formulas

or

relations

. The relations of a particular level of the hierarchy are exactly the relations defined by the formulas of that level, so the two uses are essentially the same.

The first level consists of formulas with only

bounded quantifiers

, the corresponding relations are also called the

Primitive Recursive

relations (this definition is

equivalent

to the definition from computer science). This level is called any of

Δ

0

0

,

Σ

0

0

and

Π

0

0

, depending on context.

A formula

ϕ

is

Σ

n

0

if there is some

Δ

0

0

formula

ψ

such that

ϕ

can be written:

ϕ

⁢

(

k

→

)

=

∃

x

1

⁢

∀

x

2

⁢

⋯

⁢

Q

⁢

x

n

⁢

ψ

⁢

(

k

→

,

x

→

)

where

⁢

Q

⁢

is either

⁢

∀

or

⁢

∃

, whichever maintains the pattern of alternating quantifiers

The

Σ

1

0

relations are the same as the

Recursively Enumerable

relations.

Similarly,

ϕ

is a

Π

n

0

relation if there is some

Δ

0

0

formula

ψ

such that:

ϕ

⁢

(

k

→

)

=

∀

x

1

⁢

∃

x

2

⁢

⋯

⁢

Q

⁢

x

n

⁢

ψ

⁢

(

k

→

,

x

→

)

where

⁢

Q

⁢

is either

⁢

∀

or

⁢

∃

, whichever maintains the pattern of alternating quantifiers

A formula is

Δ

n

0

if it is both

Σ

n

0

and

Π

n

0

. Since each

Σ

n

0

formula is just the

negation

of a

Π

n

0

formula and vice-versa, the

Σ

n

0

relations are the

complements

of the

Π

n

0

relations.

The relations in

Δ

1

0

=

Σ

1

0

∩

Π

1

0

are the Recursive relations.

Higher levels on the hierarchy correspond to broader and broader classes of relations. A formula or relation which is

Σ

n

0

(or, equivalently,

Π

n

0

) for some integer

n

is called

arithmetical

.

The superscript

0

is often omitted when it is not necessary to distinguish from the

analytic hierarchy

.

Functions

can be described as being in one of the levels of the hierarchy if the graph of the function is in that level.

Title

arithmetical hierarchy

Canonical name

ArithmeticalHierarchy

Date of creation

2013-03-22 12:55:11

Last modified on

2013-03-22 12:55:11

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

19

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B10

Synonym

arithmetic hierarchy

Synonym

arithmetic

Synonym

arithmetical

Synonym

arithmetic formula

Synonym

arithmetical formulas

Related topic

AnalyticHierarchy

Defines

sigma n

Defines

sigma-n

Defines

pi n

Defines

pi-n

Defines

delta n

Defines

delta-n

Defines

recursive

Defines

recursively enumerable

Defines

delta-0

Defines

delta 0

Defines

delta-1

Defines

delta 1

Defines

arithmetical

Generated on Thu Feb 8 19:25:43 2018 by

LaTeXML
arithmetical hierarchy is a proper hierarchy

arithmetical hierarchy is a proper hierarchy

By

definition

, we have

Δ

n

=

Π

n

∩

Σ

n

. In

addition

,

Σ

n

∪

Π

n

⊆

Δ

n

+

1

.

This is proved by

vacuous quantification

. If

R

is

equivalent

to

ϕ

⁢

(

n

→

)

then

R

is equivalent to

∀

x

⁢

ϕ

⁢

(

n

→

)

and

∃

x

⁢

ϕ

⁢

(

n

→

)

, where

x

is some

variable

that does not

occur free

in

ϕ

.

More significant is the proof that all containments are proper. First, let

n

≥

1

and

U

be

universal

for

2

-ary

Σ

n

relations

. Then

D

⁢

(

x

)

↔

U

⁢

(

x

,

x

)

is obviously

Σ

n

. But suppose

D

∈

Δ

n

. Then

D

∈

P

⁢

i

n

, so

¬

⁢

D

∈

Σ

n

. Since

U

is universal, ther is some

e

such that

¬

⁢

D

⁢

(

x

)

↔

U

⁢

(

e

,

x

)

, and therefore

¬

⁢

D

⁢

(

e

)

↔

U

⁢

(

e

,

e

)

↔

¬

⁢

U

⁢

(

e

,

e

)

. This is clearly a

contradiction

, so

D

∈

Σ

n

∖

Δ

n

and

¬

⁢

D

∈

Π

n

∖

Δ

n

.

In addition the

recursive

join of

D

and

¬

⁢

D

, defined by

D

⊕

¬

D

(

x

)

↔

(

∃

y

<

x

[

x

=

2

⋅

y

]

∧

D

(

x

)

)

∨

(

¬

∃

y

<

x

[

x

=

2

⋅

y

]

∧

¬

D

(

x

)

)

Clearly both

D

and

¬

⁢

D

can be recovered from

D

⊕

¬

⁢

D

, so it is

contained

in neither

Σ

n

nor

Π

n

. However the definition above has only

unbounded

quantifiers

except for those in

D

and

¬

⁢

D

, so

D

⊕

¬

⁢

D

⁢

(

x

)

∈

Δ

n

+

1

∖

Σ

n

∪

Π

n

Title

arithmetical hierarchy is a proper hierarchy

Canonical

name

ArithmeticalHierarchyIsAProperHierarchy

Date of creation

2013-03-22 12:55:14

Last modified on

2013-03-22 12:55:14

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

6

Author

Henry (455)

Entry type

Result

Classification

msc 03B10

Generated on Thu Feb 8 19:25:45 2018 by

LaTeXML
Boolean

Boolean

1

Boolean

Boolean

refers to that which can take on the values “true” or “false,” or that which concerns truth and falsity. For example “

Boolean variable

,” “Boolean logic,” “Boolean statement,” etc.

“Boolean” is named for George Boole, the 19th century mathematician.

References

1

George Boole at MacTutor:

http://www-gap.dcs.st-and.ac.uk/ history/Mathematicians/Boole.html

http://www-gap.dcs.st-and.ac.uk/ history/Mathematicians/Boole.html

2

George Boole at Wikipedia:

http://en.wikipedia.org/wiki/George_Boole

http://en.wikipedia.org/wiki/George_Boole

Title

Boolean

Canonical name

Boolean

Date of creation

2013-03-22 11:54:43

Last modified on

2013-03-22 11:54:43

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

11

Author

akrowne (2)

Entry type

Definition

Classification

msc 03B10

Classification

msc 55Q35

Classification

msc 55Q05

Classification

msc 20L05

Classification

msc 18D05

Classification

msc 18-00

Generated on Thu Feb 8 19:26:12 2018 by

LaTeXML
creating an infinite model

creating an infinite model

From the

syntactic compactness theorem for first order logic

, we get this nice (and useful) result:

Let

T

be a theory of

first-order logic

. If

T

has finite models of unboundedly large sizes, then

T

also has an

infinite

model.

Proof.

Define the

propositions

Φ

n

≡

∃

x

1

⋯

∃

x

n

.

(

x

1

≠

x

2

)

∧

⋯

∧

(

x

1

≠

x

n

)

∧

(

x

2

≠

x

3

)

∧

⋯

∧

(

x

n

-

1

≠

x

n

)

¯

(

Φ

n

says “

there exist (at least)

n

different

elements

in the world

”). Note that

⋯

⊢

Φ

n

⊢

⋯

⊢

Φ

2

⊢

Φ

1

.

Define a new theory

𝐓

∞

=

𝐓

∪

{

Φ

1

,

Φ

2

,

…

}

.

For any

finite

subset

𝐓

′

⊂

𝐓

∞

, we claim that

𝐓

′

is

consistent

: Indeed,

𝐓

′

contains axioms of

T

, along with finitely many of

{

Φ

n

}

n

≥

1

. Let

Φ

m

correspond to the largest index appearing in

𝐓

′

. If

ℳ

m

⊧

𝐓

is a model of

T

with at least

m

elements (and by

hypothesis

, such as model exists), then

ℳ

m

⊧

𝐓

∪

{

Φ

m

}

⊢

𝐓

′

.

So every finite subset of

𝐓

∞

is consistent; by the

compactness theorem

for first-order logic,

𝐓

∞

is consistent, and by Gödel’s

completeness theorem

for first-order logic it has a model

ℳ

. Then

ℳ

⊧

𝐓

∞

⊢

𝐓

, so

ℳ

is a model of

T

with infinitely many elements (

ℳ

⊧

Φ

n

for any

n

, so

ℳ

has at least

≥

n

elements for all

n

).
∎

Title

creating an infinite model

Canonical

name

CreatingAnInfiniteModel

Date of creation

2013-03-22 12:44:29

Last modified on

2013-03-22 12:44:29

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Example

Classification

msc 03B10

Classification

msc 03C07

Related topic

CompactnessTheoremForFirstOrderLogic

Related topic

GettingModelsIModelsConstructedFromConstants

Generated on Thu Feb 8 19:26:14 2018 by

LaTeXML
criterion for consistency of sets of formulas

criterion for consistency of sets of formulas

Let

L

be a

first order language

, and

Δ

⊆

L

be a set of

sentences

. Then

Δ

is

consistent

if and only if every finite subset of

Δ

is consistent.

Title

criterion for consistency of sets of formulas

Canonical name

CriterionForConsistencyOfSetsOfFormulas

Date of creation

2013-03-22 12:44:05

Last modified on

2013-03-22 12:44:05

Owner

jihemme (316)

Last modified by

jihemme (316)

Numerical id

5

Author

jihemme (316)

Entry type

Corollary

Classification

msc 03B10

Classification

msc 03C07

Generated on Thu Feb 8 19:26:19 2018 by

LaTeXML
example of Gödel numbering

example of Gödel numbering

We can define by

recursion

a

function

e

from

formulas

of

arithmetic

to numbers, and the corresponding Gödel numbering as the

inverse

.

The symbols of the

language

of arithmetic are

=

,

∀

,

¬

,

→

,

0

,

S

,

<

,

+

,

⋅

, the

variables

v

i

for any integer

i

, and

(

and

)

.

(

and

)

are only used to define the

order of operations

, and should be inferred where appropriate in the

definition

below.

We can define a function

e

by recursion as follows:

•

e

⁢

(

v

i

)

=

⟨

0

,

i

⟩

•

e

(

ϕ

=

ψ

)

=

⟨

1

,

e

(

ϕ

)

,

e

(

ψ

)

⟩

•

e

⁢

(

∀

v

i

⁢

ϕ

)

=

⟨

2

,

e

⁢

(

v

i

)

,

e

⁢

(

ϕ

)

⟩

•

e

⁢

(

¬

⁢

ϕ

)

=

⟨

3

,

e

⁢

(

ϕ

)

⟩

•

e

(

ϕ

→

ψ

)

=

⟨

4

,

e

(

ϕ

)

,

e

(

ψ

)

⟩

•

e

⁢

(

0

)

=

⟨

5

⟩

•

e

⁢

(

S

⁢

ϕ

)

=

⟨

6

,

e

⁢

(

ϕ

)

⟩

•

e

(

ϕ

<

ψ

)

=

⟨

7

,

e

(

ϕ

)

,

e

(

ψ

)

⟩

•

e

⁢

(

ϕ

+

ψ

)

=

⟨

8

,

e

⁢

(

ϕ

)

,

e

⁢

(

ψ

)

⟩

•

e

⁢

(

ϕ

⋅

ψ

)

=

⟨

9

,

e

⁢

(

ϕ

)

,

e

⁢

(

ψ

)

⟩

Clearly

e

-

1

is a Gödel numbering, with

⌜

⁢

ϕ

⁢

⌝

=

e

⁢

(

ϕ

)

.

Title

example of Gödel numbering

Canonical

name

ExampleOfGodelNumbering

Date of creation

2013-03-22 12:58:28

Last modified on

2013-03-22 12:58:28

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

7

Author

Henry (455)

Entry type

Example

Classification

msc 03B10

Generated on Thu Feb 8 19:26:45 2018 by

LaTeXML