aarjaneiro/planetmath huggingface dataset:
labelled state transition system

labelled state transition system

A

state transition system

(TS) is a pair

(

S

,

→

)

, where

→

is a

binary relation

on

S

. Elements of

S

are called

states

and elements of

→

are called

transitions

. When

(

p

,

q

)

∈

→

, we write

p

→

q

.

A

labelled state transition system

(LTS) is a triple

(

S

,

Σ

,

→

)

where

→

⊆

S

×

Σ

×

S

. Elements of

Σ

are called

labels

of the system. If

(

p

,

α

,

q

)

∈

→

, we write

p

→

α

q

. In this way, an LTS can be thought of as an STS where each transition is labelled (possibly by more than one label). On the other hand, any TS is just a LTS with one label and every transition has the same label.

Examples of LTS are mostly found in theoretical computer science, such as

automata

and

Turing machines

. An automaton, for example, is just an LTS

(

S

,

→

)

with two distinguished

mutually disjoint

sets (subsets of

S

) of states: a set of

start states

, and a set of

final

or

accept states

.

Given an LTS

(

S

,

→

)

, if we take the

reflexive transitive closure

of

→

, then we get a binary relation

⇒

on

S

called the

reachability

relation

. In other words, for any

p

∈

S

,

p

⇒

p

, and for any

q

∈

S

, if

p

⇒

q

, then there is a

finite sequence

p

=

p

1

,

…

⁢

p

n

+

1

=

q

, such that

p

i

→

p

i

+

1

for

i

=

1

,

…

⁢

n

. When

p

⇒

q

, we say that

q

is

reachable

from

p

.

For an LTS

(

S

,

→

)

with a set

I

of start states, a state is said to be

inaccessible

if it is not reachable from any start state. If

S

is finite, one can use the one of the search techniques to find all inaccessible states of

S

. Here’s a basic breadth-first search

algorithm

:

1.

Initially, circle all start states as

accessible

. If there are no start states, skip to step 5.

2.

Then for each circled

p

, circle all

q

such that

p

→

q

.

3.

Then cross out each

p

after all

q

such

p

→

q

have been circled.

4.

Repeat the last two steps for each circled

q

. Do this until there are no more circled states.

5.

Finally, the states in

S

are either crossed out or unmarked, and inaccessible states are precisely the unmarked states.

An LTS is said to be

deterministic

if

→

is a

partial function

. Otherwise, it is

non-deterministic

.

Title

labelled state transition system

Canonical name

LabelledStateTransitionSystem

Date of creation

2013-03-22 19:23:20

Last modified on

2013-03-22 19:23:20

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

17

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B45

Classification

msc 68Q85

Classification

msc 68Q10

Synonym

labeled state transition system

Synonym

TS

Synonym

LTS

Synonym

STS

Synonym

LSTS

Related topic

SemiThueSystem

Defines

state transition system

Defines

reachability

Defines

reachable

Defines

inaccessible state

Generated on Thu Feb 8 19:49:52 2018 by

LaTeXML
modal logic B

modal logic B

The

modal logic

B

(for Brouwerian) is the smallest

normal modal logic

containing the following schemas:

•

(T)

□

⁢

A

→

A

, and

•

(B)

A

→

□

⋄

A

.

In this entry (

http://planetmath.org/ModalLogicT

), we show that T is valid in a frame iff the frame is

reflexive

.

Proposition 1

.

B is valid in a frame

F

iff

F

is

symmetric

.

Proof.

First, suppose B is valid in a frame

ℱ

, and

w

⁢

R

⁢

u

. Let

M

be a model based on

ℱ

, with

V

⁢

(

p

)

=

{

w

}

,

p

a

propositional variable

. Since

w

∈

V

⁢

(

p

)

,

⊧

w

p

, and

⊧

w

p

→

□

⋄

p

by

assumption

,

⊧

v

⋄

p

for all

v

such that

w

⁢

R

⁢

v

. In particular,

⊧

u

⋄

p

, which means there is a

t

such that

u

⁢

R

⁢

t

and

⊧

t

p

. But this means that

t

∈

V

⁢

(

p

)

, so

t

=

w

, whence

u

⁢

R

⁢

w

, and

R

is symmetric.

Conversely

, let

ℱ

be a symmetric frame,

M

a model based on

ℱ

, and

w

a world in

M

. Suppose

⊧

w

A

. If

⊧̸

w

□

⋄

A

, then there is a

u

such that

w

⁢

R

⁢

u

, with

⊧̸

u

⋄

A

. This mean for no

t

with

u

⁢

R

⁢

t

, we have

⊧

t

A

. Since

R

is symmetric,

u

⁢

R

⁢

w

, so

⊧̸

w

A

, a

contradiction

. Therefore,

⊧

w

□

⋄

A

, and

⊧

w

A

→

□

⋄

A

as a result.
∎

As a result,

Proposition 2

.

B

is sound in the class of symmetric frames.

Proof.

Since any

theorem

in

B

is

deducible

from a

finite sequence

consisting of

tautologies

, which are valid in any frame,

instances

of B, which are valid in symmetric frames by the

proposition

above, and applications of

modus ponens

and

necessitation

, both of which

preserve

validity

in any frame, whence the result.
∎

In

addition

, using the

canonical model

of

B

, we have

Proposition 3

.

B

is

complete

in the class of reflexive, symmetric frames.

Proof.

Since

B

contains T, its canonical frame

ℱ

𝐁

is reflexive. We next show that any

consistent

normal logic

Λ

containing the schema B is symmetric. Suppose

w

⁢

R

Λ

⁢

u

. We want to show that

u

⁢

R

Λ

⁢

w

, or that

Δ

u

:=

{

B

∣

□

⁢

B

∈

u

}

⊆

w

. It is then enough to show that if

A

∉

w

, then

A

∉

Δ

u

. If

A

∉

w

,

¬

⁢

A

∈

w

because

w

is maximal, or

□

⋄

¬

⁢

A

∈

w

by modus ponens on B, or

□

⁢

¬

⁢

□

⁢

A

∈

w

by the

substitution theorem

on

A

↔

¬

⁢

¬

⁢

A

, or

¬

⁢

□

⁢

A

∈

Δ

w

by the

definition

of

Δ

w

, or

¬

⁢

□

⁢

A

∈

u

since

w

⁢

R

Λ

⁢

u

, or

□

⁢

A

∉

u

, since

u

is maximal, or

A

∉

Δ

u

by the definition of

Δ

u

. So

R

Λ

is symmetric, and

R

𝐁

is both reflexive and symmetric.
∎

Title

modal logic B

Canonical

name

ModalLogicB

Date of creation

2013-03-22 19:34:11

Last modified on

2013-03-22 19:34:11

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B45

Defines

B

Generated on Thu Feb 8 19:50:25 2018 by

LaTeXML
modal logic D

modal logic D

The

modal logic

D

(for deontic) is the smallest

normal modal logic

containing the schema D:

□

⁢

A

→

⋄

A

A

binary relation

R

on

W

is

serial

if for any

w

∈

W

, there is a

u

∈

W

such that

w

⁢

R

⁢

u

. In other words,

R

is

first order

definable

:

∀

w

⁢

∃

u

⁢

(

w

⁢

R

⁢

u

)

.

The

Kripke frames

corresponding to

D

are serial, in the following sense:

Proposition 1

.

D is valid in a frame

F

iff

F

is serial.

Proof.

First, assume

D

valid in a frame

ℱ

:=

(

W

,

R

)

, and

w

∈

W

. Let

M

be a model based on

ℱ

, with

V

⁢

(

p

)

=

{

u

∣

w

⁢

R

⁢

u

}

. Then

⊧

w

□

⁢

p

, so that

⊧

w

⋄

p

. This means there is a

v

such that

w

⁢

R

⁢

v

, and hence

R

is serial.

Conversely

, let

ℱ

be a serial frame,

M

a model based on

ℱ

, and

w

a world in

M

. Then there is a

u

such that

w

⁢

R

⁢

u

. Suppose

⊧

w

□

⁢

A

. Then for all

v

such that

w

⁢

R

⁢

v

, we have

⊧

v

A

. In particular,

⊧

u

A

. Therefore,

⊧

w

⋄

A

, whence

⊧

w

□

⁢

A

→

⋄

A

.
∎

As a result,

Proposition 2

.

D

is sound in the class of serial frames.

Proof.

Since any

theorem

in

D

is

deducible

from a

finite sequence

consisting of

tautologies

, which are valid in any frame,

instances

of D, which are valid in serial frames by the

proposition

above, and applications of

modus ponens

and

necessitation

, both of which

preserve

validity

in any frame, whence the result.
∎

In

addition

, using the

canonical model

of

D

, we have

Proposition 3

.

D

is

complete

in the class of serial frames.

Proof.

We show that the canonical frame

ℱ

𝐃

is serial. Let

w

be any maximally consistent set containing

D

. For any

A

∈

Δ

w

:=

{

B

∣

□

⁢

B

∈

w

}

, we have

□

⁢

A

∈

w

, so that

⋄

A

∈

w

by modus ponens on D. This means that

□

⁢

¬

⁢

A

∉

w

since

w

is maximal. As a result,

¬

⁢

A

∉

Δ

w

, showing that

Δ

w

is

consistent

, and hence can be enlarged to a maximally consistent set

u

. As a result,

A

∈

u

, whence

w

⁢

R

𝐃

⁢

u

.
∎

D

is a subsystem of

T

, for any

reflexive relation

is serial. As a result, any theorem of

D

is valid in any serial frame, and therefore in any

reflexive

frame in particular, and as a result a theorem of

T

by the completeness of

T

in reflexive frames.

Title

modal logic D

Canonical

name

ModalLogicD

Date of creation

2013-03-22 19:33:54

Last modified on

2013-03-22 19:33:54

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B45

Related topic

ModalLogicT

Defines

D

Defines

serial

Generated on Thu Feb 8 19:50:35 2018 by

LaTeXML
modal logic T

modal logic T

The

modal logic

T

is the smallest

normal modal logic

containing the schema T:

□

⁢

A

→

A

A

Kripke frame

(

W

,

R

)

is

reflexive

if

R

is reflexive on

W

.

Proposition 1

.

T is valid in a frame

F

iff

F

is reflexive.

Proof.

First, suppose

ℱ

is not reflexive, say,

(

w

,

w

)

∉

R

. Let

M

be a model based on

ℱ

such that

V

⁢

(

p

)

=

{

u

∣

w

⁢

R

⁢

u

}

, where

p

is a

propositional variable

. By the construction of

V

⁢

(

p

)

, we see that for all

u

such that

w

⁢

R

⁢

u

, we have

⊧

u

p

, so

⊧

w

□

⁢

p

. But since

w

∉

V

⁢

(

p

)

,

⊧̸

w

p

. This means that

⊧̸

w

□

⁢

p

→

p

.

Conversely

, let

ℱ

be a reflexive frame, and

M

any model based on

ℱ

, with

w

a world in

M

. Suppose

⊧

w

□

⁢

A

. Then for all

u

such that

w

⁢

R

⁢

u

,

⊧

u

A

. Since

w

⁢

R

⁢

w

, we get

⊧

w

A

. Therefore,

⊧

w

□

⁢

A

→

A

.
∎

As a result,

Proposition 2

.

T

is sound in the class of reflexive frames.

Proof.

Since any

theorem

in

T

is

deducible

from a

finite sequence

consisting of

tautologies

, which are valid in any frame,

instances

of T, which are valid in reflexive frames by the

proposition

above, and applications of

modus ponens

and

necessitation

, both of which

preserve

validity

in any frame, whence the result.
∎

In

addition

, using the

canonical model

of

T

, we have

Proposition 3

.

T

is

complete

in the class of reflexive frames.

Proof.

We show that the canonical frame

ℱ

𝐓

is reflexive. For any maximally consistent set

w

, if

A

∈

Δ

w

:=

{

B

∣

□

⁢

B

∈

w

}

, then

□

⁢

A

∈

w

. Since

T

contains

□

⁢

A

→

A

, we get that

A

∈

w

by modus ponens and the fact that

w

is

closed under

modus ponens. Therefore

w

⁢

R

𝐓

⁢

w

, or

R

𝐓

is reflexive.
∎

T

properly extends the modal system

D

, for

□

⁢

A

→

A

is not valid in any non-reflexive serial frame, such as the one

(

W

,

R

)

, where

W

=

{

u

,

w

}

and

R

=

{

(

u

,

u

)

,

(

w

,

u

)

}

: just let

V

⁢

(

p

)

=

{

w

}

. So

⊧

w

p

and

⊧̸

u

p

, or

⊧̸

w

□

⁢

p

. This means

⊧̸

w

□

⁢

p

→

p

.

Title

modal logic T

Canonical

name

ModalLogicT

Date of creation

2013-03-22 19:33:58

Last modified on

2013-03-22 19:33:58

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B45

Related topic

ModalLogicD

Defines

T

Generated on Thu Feb 8 19:51:08 2018 by

LaTeXML
normal modal logic

normal modal logic

The study of

modal logic

is based on the

concept

of a

logic

, which is a set

Λ

of wff’s satisfying the following:

•

contains all

tautologies

, and

•

is

closed under

modus ponens

.

The last condition means: if

A

and

A

→

B

are in

Λ

, so is

B

in

Λ

.

A

normal modal logic

is a modal logic

Λ

that includes the

law of distribution

K (after Kripke):

□

(

A

→

B

)

→

(

□

A

→

□

B

)

as an

axiom schema

, and obeying the

rule of

necessitation

R

⁢

N

:

from

⊢

A

, we may infer

⊢

□

⁢

A

: if

A

∈

Λ

, then

□

⁢

A

∈

Λ

.

Normal modal logics are the most widely studied modal logics. The smallest normal modal logic is called

K

. Other normal modal logics are built from

K

by attaching wff’s as axiom schemas. Below is a list of schemas used to form some of the most common normal modal logics:

•

4:

□

⁢

A

→

□

⁢

□

⁢

A

•

5:

◇

⁢

A

→

□

⁢

◇

⁢

A

•

D:

□

⁢

A

→

◇

⁢

A

•

T:

□

⁢

A

→

A

•

B:

A

→

□

⁢

◇

⁢

A

•

C:

□

⁢

(

A

∧

□

⁢

B

)

→

□

⁢

(

A

∧

B

)

•

M:

□

⁢

(

A

∧

B

)

→

□

⁢

A

∧

□

⁢

B

•

G:

◇

⁢

□

⁢

A

→

□

⁢

◇

⁢

A

•

L:

□

(

A

∧

□

A

→

B

)

∨

□

(

B

∧

□

B

→

A

)

•

W:

□

(

□

A

→

A

)

→

□

A

For example, the normal modal logic

D

is the smallest normal modal logic containing

D

as its axiom schema.

Notation

. The smallest normal modal logic containing schemas

Σ

1

,

…

,

Σ

n

is typically denoted

K

𝚺

𝟏

⁢

⋯

⁢

𝚺

𝐧

.

It is easy to see that

K

𝚺

𝟏

⁢

⋯

⁢

𝚺

𝐧

can be built from the “bottom up”: call a

finite sequence

of wff’s a

deduction

if each wff is either a tautology, an

instance

of

Σ

i

for some

i

, or as a result of an application of modus ponens or necessitation on earlier wff’s in the sequence. A wff is

deducible

from if it is the last member of some deduction. Let

Λ

k

be the set of all wff’s deducible from deductions of lengths at most

k

. Then

K

𝚺

𝟏

⁢

⋯

⁢

𝚺

𝐧

=

⋃

i

=

1

∞

Λ

i

Below are some of the most common normal modal logics:

Title

normal modal logic

Canonical name

NormalModalLogic

Date of creation

2013-03-22 19:33:38

Last modified on

2013-03-22 19:33:38

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

16

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B45

Related topic

DisjunctionProperty

Defines

law of distribution

Defines

necessitation

Defines

K

Defines

logic

\@unrecurse

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

LaTeXML
universe of discourse

universe of discourse

\PMlinkescapephrase

field

\PMlinkescapephrase

Field

\PMlinkescapephrase

limit

\PMlinkescapephrase

Limit

\PMlinkescapephrase

object

\PMlinkescapephrase

Object

\PMlinkescapephrase

objects

\PMlinkescapephrase

Objects

Boole (1854) defines

universe of discourse

in the following manner:

In every discourse, whether of the mind conversing with its own thoughts, or of the

individual

in his intercourse with others, there is an assumed or expressed limit within which the subjects of its

operation

are confined.

…

Now, whatever may be the extent of the field within which all the objects of our discourse are found, that field may properly be termed the universe of discourse. (Boole 1854/1958, p. 42).

1

References

•

Boole, George (1854/1958),

An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities

, Macmillan Publishers, 1854. Reprinted with corrections, Dover Publications, New York, NY, 1958.

2

External links

•

http://www.helsinki.fi/science/commens/dictionary.html

Commens Dictionary of Peirce’s Terms:

http://www.helsinki.fi/science/commens/terms/logicaluniv.html

Logical

Universe

.

http://www.helsinki.fi/science/commens/terms/universedisc.html

Universe of Discourse.

Peirce

attributes

the term “universe of discourse” to De Morgan (1846).

Title

universe of discourse

Canonical

name

UniverseOfDiscourse

Date of creation

2013-03-22 17:54:31

Last modified on

2013-03-22 17:54:31

Owner

Jon Awbrey (15246)

Last modified by

Jon Awbrey (15246)

Numerical id

7

Author

Jon Awbrey (15246)

Entry type

Definition

Classification

msc 03B48

Classification

msc 03B05

Classification

msc 03A05

Classification

msc 03G05

Synonym

universe

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

LaTeXML
more examples of Polish notation

more examples of Polish notation

The following examples, presented first in standard

infix notation

, converted to

Polish notation

, all use the same four operands but combined with different

operators

and parentheses. Operators are assumed to be binary.

(

1

+

2

)

×

(

3

+

4

)

in standard infix notation becomes

×

+

1 2

+

3

4

in Polish notation. Both

expressions

should evaluate to 21.

1

+

2

×

3

+

4

standard becomes

+

+

1

×

2 3 4

Polish notation. Both evaluate to 11.

1

+

2

×

(

3

+

4

)

turns to

+

1

×

2

+

3 4

. Evaluate to 15.

(

1

+

2

)

×

3

+

4

is

+

×

+

1 2 3 4

. Evaluate to 13.

Title

more examples of Polish notation

Canonical

name

MoreExamplesOfPolishNotation

Date of creation

2013-03-22 16:14:58

Last modified on

2013-03-22 16:14:58

Owner

Mravinci (12996)

Last modified by

Mravinci (12996)

Numerical id

5

Author

Mravinci (12996)

Entry type

Example

Classification

msc 03B70

Classification

msc 68N17

Generated on Thu Feb 8 19:53:16 2018 by

LaTeXML
more examples of reverse Polish notation

more examples of reverse Polish notation

The following examples, presented first in standard

infix notation

, converted to

reverse Polish notation

by using the

shunting yard algorithm

, all use the same four operands but combined with different

operators

and parentheses. Operators are assumed to be binary.

(

1

+

2

)

×

(

3

+

4

)

in standard infix notation becomes

1

2

+

3

4

+

×

in reverse Polish notation. Both

expressions

should evaluate to 21, each in the appropriate

calculator

.

1

+

2

×

3

+

4

standard becomes

1

2

3

×

4

+

RPN. Both evaluate to 11.

1

+

2

×

(

3

+

4

)

turns to

1

2

3

4

+

×

+

. Evaluate to 15.

(

1

+

2

)

×

3

+

4

is

1

2

+

3

×

4

+

. Evaluate to 13.

Title

more examples of reverse Polish notation

Canonical

name

MoreExamplesOfReversePolishNotation

Date of creation

2013-03-22 16:10:19

Last modified on

2013-03-22 16:10:19

Owner

Mravinci (12996)

Last modified by

Mravinci (12996)

Numerical id

5

Author

Mravinci (12996)

Entry type

Example

Classification

msc 03B70

Classification

msc 68N17

Generated on Thu Feb 8 19:53:18 2018 by

LaTeXML
praeclarum theorema

praeclarum theorema

The

praeclarum theorema

, or

splendid theorem

, is a

theorem

of

propositional calculus

that was noted and named by G.W. Leibniz, who stated and proved it in the following manner:

If

a

is

b

and

d

is

c

, then

a

⁢

d

will be

b

⁢

c

.

This is a fine theorem, which is proved in this way:

a

is

b

, therefore

a

⁢

d

is

b

⁢

d

(by what precedes),

d

is

c

, therefore

b

⁢

d

is

b

⁢

c

(again by what precedes),

a

⁢

d

is

b

⁢

d

, and

b

⁢

d

is

b

⁢

c

, therefore

a

⁢

d

is

b

⁢

c

. Q.E.D.

(Leibniz,

Logical Papers

, p. 41).

Expressed in contemporary logical notation, the praeclarum theorema (PT) may be written as follows:

(

(

a

⇒

b

)

∧

(

d

⇒

c

)

)

⇒

(

(

a

∧

d

)

⇒

(

b

∧

c

)

)

Representing

propositions

(

http://

planetmath

.org/PropositionalCalculus

) as logical graphs (

http://planetmath.org/LogicalGraph

) under the

existential interpretation

(

http://planetmath.org/LogicalGraphFormalDevelopment

), the praeclarum theorema is expressed by means of the following formal equation:

(1)

And here’s a neat proof of that nice theorem.

(2)

1

References

•

Leibniz, Gottfried W. (1679–1686 ?), “Addenda to the Specimen of the

Universal

Calculus”, pp. 40–46 in G.H.R. Parkinson (ed., trans., 1966),

Leibniz : Logical Papers

, Oxford University Press, London, UK.

2

Readings

•

Sowa, John F. (2002), “Peirce’s

Rules of Inference

”,

http://www.jfsowa.com/peirce/infrules.htm

Online.

3

Resources

•

Dau, Frithjof (2008),

http://web.archive.org/web/20070706192257/http://dr-dau.net/pc.shtml

Computer Animated Proof of Leibniz’s Praeclarum Theorema.

•

Megill, Norman (2008),

http://us.metamath.org/mpegif/prth.html

Praeclarum Theorema @

http://us.metamath.org/mpegif/mmset.html

Metamath Proof Explorer.

Title

praeclarum theorema

Canonical name

PraeclarumTheorema

Date of creation

2013-03-22 17:47:37

Last modified on

2013-03-22 17:47:37

Owner

Jon Awbrey (15246)

Last modified by

Jon Awbrey (15246)

Numerical id

16

Author

Jon Awbrey (15246)

Entry type

Theorem

Classification

msc 03B70

Classification

msc 03B35

Classification

msc 03B22

Classification

msc 03B05

Classification

msc 03-03

Classification

msc 01A45

Synonym

splendid theorem

Related topic

OrderedGroup

Generated on Thu Feb 8 19:53:23 2018 by

LaTeXML
sole sufficient operator

sole sufficient operator

A

sole sufficient operator

or a

sole sufficient connective

is an

operator

that is

sufficient

by itself to define all of the operators in a specified set of operators.

In logical

contexts

this refers to a logical operator that suffices to define all of the

Boolean-valued functions

,

f

:

X

→

𝔹

, where

X

is an arbitrary set and where

𝔹

is a

generic

2-element set, typically

𝔹

=

{

0

,

1

}

=

{

false

,

true

}

, in particular, to define all of the finitary

Boolean functions

,

f

:

𝔹

k

→

𝔹

.

Title

sole sufficient operator

Canonical name

SoleSufficientOperator

Date of creation

2013-03-22 17:51:52

Last modified on

2013-03-22 17:51:52

Owner

Jon Awbrey (15246)

Last modified by

Jon Awbrey (15246)

Numerical id

6

Author

Jon Awbrey (15246)

Entry type

Definition

Classification

msc 03B70

Classification

msc 03B35

Classification

msc 03B22

Classification

msc 03B05

Synonym

sole sufficient connective

Related topic

Ampheck

Related topic

LogicalConnective

Generated on Thu Feb 8 19:53:25 2018 by

LaTeXML
Berry’s paradox

Berry’s paradox

We begin with calling a

positive integer

curious

if it can be defined
in the English

language

using no more than 1234 words. Since there
are finitely many English words, we see that there are only finitely
many curious positive integers.

Define

n

0

to be:

the least positive integer that is not
curious

.

n

0

has just been described in

8

≤

1234

words, therefore, it
is curious after all!

The

paradox

above is called Berry’s Paradox. Berry’s
Paradox suggests the advantage of separating the language used to formulate
mathematical statements or theory (the object language) from the language
used to discuss those statements or the theory (the

metalanguage

).

Berry’s Paradox can be avoided by the following reformulation:

1.

fix the object language, called

𝐄

∗

;

2.

declare

𝐄

∗

to be different from our
metalanguage, which is English here;

3.

define a curious positive integer to be one which can be
described in

𝐄

∗

using no more than 1234 words of the
language;

4.

define

n

0

to be the least positive integer that is not
curious.

In the reformulation, we have defined curious positive integers and

n

0

in English, which is not

𝐄

∗

. Thus, we have no
basis to conclude that

n

0

is curious, hence no

contradiction

arises.

Commonly,

𝐄

∗

is the

first order logic

. However, it is
not often necessarily the case, and

𝐄

∗

above could
have been English anyway. We only need to formally distinguish the
statements formulating the mathematics from the statements
discussing those formulations, i.e., declaring the two classes of
statements to be disjunct, perhaps by italicizing the former.
Nevertheless, such approach evidently involves more work and is
understandably hard to follow.

References

1

Schechter, E.,

Handbook of Analysis and Its Foundations

, 1st ed., Academic Press, 1997.

Title

Berry’s paradox

Canonical name

BerrysParadox

Date of creation

2013-03-22 18:06:03

Last modified on

2013-03-22 18:06:03

Owner

yesitis (13730)

Last modified by

yesitis (13730)

Numerical id

7

Author

yesitis (13730)

Entry type

Topic

Classification

msc 03B99

Generated on Thu Feb 8 19:53:29 2018 by

LaTeXML