aarjaneiro/planetmath huggingface dataset:
Chapter 4 Notes

Chapter 4 Notes

The fact that the space of continuous maps equipped with

quasi-inverses

has the wrong

homotopy type

to be the “space of

homotopy equivalences

” is well-known in

algebraic topology

.
In that

context

, the “space of homotopy equivalences”

(

A

≃

B

)

is usually defined simply as the

subspace

of the

function space

(

A

→

B

)

consisting of the

functions

that are homotopy equivalences.
In

type theory

, this would correspond most closely to

∑

(

f

:

A

→

B

)

∥

𝗊𝗂𝗇𝗏

(

f

)

∥

; see

http://planetmath.org/node/87824

Exercise 3.11.

The first definition of

equivalence

given in

homotopy type theory

was the one that we have called

𝗂𝗌𝖢𝗈𝗇𝗍𝗋

⁢

(

f

)

, which was due to Voevodsky.
The possibility of the other definitions was subsequently observed by various people.
The basic

theorems

about adjoint equivalences such as Lemma 4.2.2 (

http://planetmath.org/42halfadjointequivalences#Thmprelem1

),Theorem 4.2.3 (

http://planetmath.org/42halfadjointequivalences#Thmprethm1

) are adaptations of standard facts in higher category theory and

homotopy theory

.
Using bi-invertibility as a definition of equivalences was suggested by André Joyal.

The properties of equivalences discussed in §4.6 (

http://planetmath.org/46surjectionsandembeddings

),§4.7 (

http://planetmath.org/47closurepropertiesofequivalences

) are well-known in homotopy theory.
Most of them were first proven in type theory by Voevodsky.

The fact that every function is

equivalent

to a

fibration

is a standard fact in homotopy theory.
The notion of object classifier
in

(

∞

,

1

)

-

category
theory

(the

categorical

analogue of Theorem 4.8.3 (

http://planetmath.org/48theobjectclassifier#Thmprethm1

)) is due to Rezk (see

[

2

]

,

[

1

]

).

Finally, the fact that univalence implies function extensionality (§4.9 (

http://planetmath.org/49univalenceimpliesfunctionextensionality

)) is due to Voevodsky.
Our proof is a simplification of his.

References

1

Jacob Lurie,

Higher topos theory

Princeton University Press,2009

2

Charles Rezk. Toposes and

homotopy

toposes.

\url

http://www.math.uiuc.edu/ rezk/homotopy-topos-sketch.pdf, 2005.

Title

Chapter 4 Notes

\metatable

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example of quantifier

example of quantifier

there are some examples and

theorems

about logical

quantifiers

in the Word Document below .
you can download it:

http://www.freewebs.com/hkkass

or

http://www.hkkass.blogspot.com/

I include extracts of this Document below:

Definition

: a

property

is something like

x

>

0

or

x

=

0

in which

x

is a

variable

in some set. Such a

formula

is shown by

p

⁢

(

x

)

,

q

⁢

(

x

)

,etc. if x is fixed then

p

⁢

(

x

)

is a

proposition

, i.e. it is a true or a false

sentence

.

Example 1: let

p

⁢

(

x

)

be the property

0

<

x

where x is a

real number

.

p

⁢

(

1

)

is true and

p

⁢

(

0

)

is false.

Example 2: a property can have two or more variables. Let

p

⁢

(

x

,

y

)

be

x

=

y

. in this case

p

⁢

(

1

,

1

)

is true but

p

⁢

(

0

,

1

)

is false because

0

is not equal to

1

.

Definition: let

p

⁢

(

x

)

be a property on the set

X

, i.e.

p

⁢

(

x

)

is a property and

x

varies in the set

X

.
a) The symbol

(

∀

x

∈

X

)

(

p

(

x

)

)

means for every

x

in the set

X

the proposition

p

⁢

(

x

)

is true.
b) The symbol

(

∃

x

∈

X

)

(

p

(

x

)

)

means there is some

x

in the set

X

for which the proposition

p

⁢

(

x

)

is true.
If

X

=

∅

, i.e. if the set

X

is empty,

(

∀

x

∈

X

)

(

p

(

x

)

)

is defined to be true and

(

∃

x

∈

X

)

(

p

(

x

)

)

is defined to be false.

Example 1:

(

∀

x

∈

ℝ

)

(

x

=

0

or

x

>

0

or

x

<

0

)

is a true proposition.

Example 2:

(

∃

x

∈

ℝ

)

(

x

2

+

1

=

0

)

is false, because no real number

satisfies

x

2

+

10

=

0

.

Example 3:

(

∀

x

∈

ℝ

)

(

x

<

y

)

is a property.

y

varies in

ℝ

. As a result

(

∀

x

∈

ℝ

)

(

∀

y

∈

ℝ

)

(

x

<

y

)

is a proposition, i.e. it is a true or a false sentence. In fact

(

∀

x

∈

ℝ

)

(

∀

y

∈

ℝ

)

(

x

<

y

)

is false but

(

∀

x

∈

ℝ

)

(

∀

y

∈

(

x

,

∞

)

(

x

<

y

)

is true; here

(

x

,

∞

)

is the

interval

containing real numbers greater than

x

.

some theorems:

for proofs of the following theorems see the address above

Theorem 1: if

(

∀

x

∈

A

)

(

p

(

x

)

)

and

(

∀

x

∈

A

)

(

p

(

x

)

→

q

(

x

)

)

then

(

∀

x

∈

A

)

(

q

(

x

)

)

.

Theorem 2: suppose

{

a

}

is a

singleton

, i.e. a set with only one

element

. We have
”

(

∀

x

∈

{

a

}

)

(

p

(

x

)

)

” is

equivalent

to

p

⁢

(

a

)

.

Theorem 22: if

(

∃

y

∈

B

)

(

∀

x

∈

A

)

(

r

(

x

,

y

)

)

then

(

∀

x

∈

A

)

(

∃

y

∈

B

)

(

r

(

x

,

y

)

)

.

here

r

⁢

(

x

,

y

)

is a property on

A

×

B

.

Title

example of quantifier

Canonical

name

ExampleOfQuantifier

Date of creation

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Numerical id

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Author

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Entry type

Example

Classification

msc 03B15

Classification

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fuzzy logics of living systems

fuzzy logics of living systems

0.1

Fuzzy logics of living organisms.

Living organisms or biosystems can be represented as

super-complex systems

(

http://planetmath.org/ComplexSystemsBiology

) with dynamics
that is not reducible to that of their

components

, such as molecules and atoms. It is an
empirically accepted fact that living organisms exhibit a wide degree of ‘

biological variability

’:
genetic/epigenetic and also phenotypic/metabolic within the same species; their behavior
and dynamics thus exhibit a type of ‘fuzziness’ (refs.

[

2

,

3

]

) that unlike Zadeh’s

fuzzy sets

characteristic

(

[

7

,

8

]

) is neither random nor always following a (

symmetric

) Gaussian distribution.
It has been proposed that the operational logics underlying
super-complex systems dynamics (

http://planetmath.org/FundamentalDiagramsInCategoricalTheoryOfLevels

) are

L

⁢

M

n

many-valued logics

(

http://planetmath.org/AlgebraicCategoryOfLMnLogicAlgebras

) for both genetic
and neural networks (refs.

[

3

,

6

]

).

[Entry under construction]

References

1

Georgescu, G. 2006, N-valued Logics and Łukasiewicz-Moisil

Algebras

,

Axiomathes

,

16

(1-2): 123-136.

2

Baianu, I.C. and M. Marinescu: 1968,

Organismic Supercategories

:
Towards a

Unitary

Theory of Systems.

Bulletin of

Mathematical Biophysics

30

, 148-159.

3

Baianu, I.C.: 1977, A Logical Model of Genetic Activities in Łukasiewicz Algebras: The Non-linear Theory.

Bulletin of Mathematical Biology

,

39

: 249-258.

4

Baianu, I. C.: 1986–1987a, Computer Models and Automata Theory in Biology and Medicine., in M. Witten (ed.),

Mathematical Models in Medicine

, vol. 7., Ch.11 Pergamon Press, New York, 1513 -1577; URLs:

http://doe.cern.ch//archive/electronic/other/ext/ext-2004-072.pdf

CERN Preprint No. EXT-2004-072 ,
and

http://en.scientificcommons.org/1857371

html Abstract.

5

Baianu, I. C.: 1987b, Molecular Models of Genetic and Organismic

Structures

, in

Proceed.

Relational Biology

Symp.

Argentina;

http://doc.cern.ch//archive/electronic/other/ext/ext-2004-067.pdf

CERN Preprint No.EXT-2004-067 .

6

Baianu, I.C.: 2004. Łukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamic Models (2004). Eprint: w. Cogprints at Sussex Univ.

7

Zadeh, L.A., Fuzzy Sets,

Information and Control

, 8 (1965) 338ÃÆÃâÃÂ­-353.

8

Zadeh L. A., The

concept

of a linguistic variable and its application to approximate reasoning I, II, III,

Information Sciences

, vol. 8, 9(1975), pp. 199-275, 301-357, 43-80.

Title

fuzzy logics of living systems

Canonical name

FuzzyLogicsOfLivingSystems

Date of creation

2013-03-22 18:23:58

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Owner

bci1 (20947)

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Numerical id

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Author

bci1 (20947)

Entry type

Feature

Classification

msc 03B15

Classification

msc 03B10

Synonym

biological variability

Synonym

logics of variable supercomplex systems

Related topic

FuzzyLogic2

Defines

operational logic of super-complex systems

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game-theoretical quantifier

game-theoretical quantifier

A

Henkin

or

branching

quantifier

is a multi-variable quantifier in which the selection of variables depends only on some, but not all, of the other quantified variables. For

instance

the simplest Henkin quantifier can be written:

∀

x

⁢

∃

y

∀

a

⁢

∃

b

⁢

ϕ

⁢

(

x

,

y

,

a

,

b

)

This quantifier,

inexpressible

in ordinary

first order logic

, can best be understood by its

skolemization

. The

formula

above is

equivalent

to

∀

x

⁢

∀

a

⁢

ϕ

⁢

(

x

,

f

⁢

(

y

)

,

a

,

g

⁢

(

a

)

)

. Critically, the selection of

y

depends only on

x

while the selection of

b

depends only on

a

. For instance, given a value for

a

, a value of

b

must be chosen which is

compatible

with every possible value of

x

, while given any

x

, the value of

y

chosen must be compatible with every value of

a

.

Logics with this quantifier are stronger than first order logic, lying between first and

second order logic

in strength. For instance the Henkin quantifier can be used to define the

Rescher quantifier

, and by

extension

Härtig’s quantifer:

∀

x

⁢

∃

y

∀

a

⁢

∃

b

[

(

x

=

a

↔

y

=

b

)

∧

ϕ

(

x

)

→

ψ

(

y

)

]

↔

R

x

y

ϕ

(

x

)

ψ

(

y

)

To see that this is true, observe that this essentially requires that the

Skolem functions

f

⁢

(

x

)

=

y

and

g

⁢

(

a

)

=

b

the same, and moreover that they are

injective

. Then for each

x

satisfying

ϕ

⁢

(

x

)

, there is a different

f

⁢

(

x

)

satisfying

ψ

(

(

f

(

x

)

)

.

This

concept

can be generalized to the

game-theoretical quantifiers

. This concept comes from interpreting a formula as a game between a “Prover” and “Refuter.” A

theorem

is provable whenever the Prover has a winning

strategy

; at each

∧

the Refuter chooses which side they will play (so the Prover must be prepared to win on either) while each

∨

is a choice for the Prover. At a

¬

, the players switch roles. Then

∀

represents a choice for the Refuter and

∃

for the Prover.

Classical first order logic, then, adds the requirement that the games have

perfect information

. The game-theoretical quantifers remove this requirement, so for instance the Henkin quantifier, which would be written

∀

x

⁢

∃

y

⁢

∀

a

⁢

∃

/

∀

x

b

⁢

ϕ

⁢

(

x

,

y

,

a

,

b

)

states that when the Prover makes a choice for

b

, it is made without knowledge of what was chosen at

x

.

Title

game-theoretical quantifier

Canonical name

GametheoreticalQuantifier

Date of creation

2013-03-22 12:59:19

Last modified on

2013-03-22 12:59:19

Owner

Henry (455)

Last modified by

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Author

Henry (455)

Entry type

Definition

Classification

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Related topic

Quantifier

Defines

Henkin quantifier

Defines

Henkin

Defines

branching quantifier

Defines

branching

Defines

game-theoretic quantifier

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Härtig’s quantifier

Härtig’s quantifier

Härtig’s

quantifier

is a quantifier which takes two

variables

and two

formulas

, written

I

⁢

x

⁢

y

⁢

ϕ

⁢

(

x

)

⁢

ψ

⁢

(

y

)

. It asserts that

|

{

x

∣

ϕ

⁢

(

x

)

}

|

=

|

{

y

∣

ψ

⁢

(

y

)

}

|

. That is, the

cardinality

of the values of

x

which make

ϕ

is the same as the cardinality of the values which make

ψ

⁢

(

x

)

true. Viewed as a

generalized quantifier

,

I

is a

⟨

2

⟩

quantifier.

Closely related is the

Rescher quantifier

, which also takes two variables and two formulas, is written

J

⁢

x

⁢

y

⁢

ϕ

⁢

(

x

)

⁢

ψ

⁢

(

y

)

, and asserts that

|

{

x

∣

ϕ

(

x

)

}

|

≤

|

{

y

∣

ψ

(

y

)

|

. The Rescher quantifier is sometimes defined instead to be a

similar

but different quantifier,

J

⁢

x

⁢

ϕ

⁢

(

x

)

↔

|

{

x

∣

ϕ

⁢

(

x

)

}

|

>

|

{

x

∣

¬

⁢

ϕ

⁢

(

x

)

}

|

. The first

definition

is a

⟨

2

⟩

quantifier while the second is a

⟨

1

⟩

quantifier.

Another similar quantifier is Chang’s quantifier

Q

C

, a

⟨

1

⟩

quantifier defined by

Q

M

C

=

{

X

⊆

M

∣

|

X

|

=

|

M

|

}

. That is,

Q

C

⁢

x

⁢

ϕ

⁢

(

x

)

is true if the number of

x

satisfying

ϕ

has the same cardinality as the

universe

; for finite models this is the same as

∀

, but for

infinite

ones it is not.

Title

Härtig’s quantifier

Canonical

name

HartigsQuantifier

Date of creation

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Entry type

Definition

Classification

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Related topic

Quantifier

Defines

Rescher quantifier

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interior point of set in topological space

interior point of set in topological space

include definition of interior point of set in

topological space

.

Title

interior point of set in topological space

Canonical name

InteriorPointOfSetInTopologicalSpace

Date of creation

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Introduction

Introduction

Homotopy type theory

is a new branch of mathematics that combines aspects of several different fields in a surprising way. It is based on a recently discovered

connection

between

homotopy theory

and

type theory

.
Homotopy theory is an outgrowth of

algebraic topology

and

homological algebra

, with relationships to higher category theory; while type theory is a branch of mathematical logic and theoretical computer science.
Although the connections between the two are currently the focus of intense investigation, it is increasingly clear that they are just the beginning of a subject that will take more time and more hard work to fully understand.
It touches on topics as seemingly distant as the homotopy groups of spheres, the algorithms for type checking, and the definition of weak

∞

-

groupoids

.

Homotopy type theory also brings new ideas into the very

foundation of mathematics

.
On the one hand, there is Voevodsky’s subtle and beautiful

univalence axiom

.
The univalence axiom implies, in particular, that

isomorphic

structures

can be identified, a principle that mathematicians have been happily using on workdays, despite its incompatibility with the “official” doctrines of conventional

foundations

.
On the other hand, we have

higher inductive types

, which provide direct, logical descriptions of some of the basic spaces and constructions of homotopy theory: spheres, cylinders, truncations,

localizations

, etc.
Both ideas are impossible to capture directly in classical set-theoretic foundations, but when combined in homotopy type theory, they permit an entirely new kind of “logic of

homotopy types

”.

This suggests a new conception of foundations of mathematics, with intrinsic homotopical content, an “

invariant

” conception of the objects of mathematics — and convenient

machine

implementations, which can serve as a practical aid to the working mathematician.
This is the

Univalent Foundations

program.
The present book is intended as a first systematic exposition of the basics of univalent foundations, and a

collection

of examples of this new style of reasoning — but without requiring the reader to know or learn any

formal logic

, or to use any computer proof assistant.

We emphasize that homotopy type theory is a young field, and univalent foundations is very much a work in progress.
This book should be regarded as a “snapshot” of the state of the field at the time it was written, rather than a polished exposition of an established edifice.
As we will discuss briefly later, there are many aspects of homotopy type theory that are not yet fully understood — but as of this writing, its broad outlines seem clear enough.
The ultimate theory will probably not look exactly like the one described in this book, but it will surely be

at least

as capable and powerful; we therefore believe that univalent foundations will eventually become a viable alternative to

set theory

as the “implicit foundation” for the unformalized mathematics done by most mathematicians.

Title

Introduction

\metatable

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logic

logic

Generally, by logic, people mean

first order logic

, a formal set of rules for building mathematical statements out of symbols like

¬

(

negation

) and

→

(

implication

) along with

quantifiers

like

∀

(for every) and

∃

(there exists).

More generally, a

logic

is any set of rules for forming

sentences

(the logic’s

syntax

) together with rules for assigning truth values to them (the logic’s

semantics

). Normally it includes a (possibly empty) set of

types

T

(also called

sorts

), which represent the different kinds of objects that the theory discusses (typical examples might be sets, numbers, or sets of numbers). In

addition

it specifies particular quantifiers,

connectives

, and

variables

. Particular theories in the logic can then add

relations

and

functions

to fully specify a

logical language

.

Title

logic

Canonical

name

Logic

Date of creation

2013-03-22 13:00:09

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Owner

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Last modified by

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Author

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Entry type

Definition

Classification

msc 03B15

Classification

msc 03B10

Related topic

FuzzySubset

Defines

syntax

Defines

semantics

Defines

type

Defines

sort

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point-free geometry

point-free geometry

Point-free geometry

is based on the idea that in geometry it is not necessary to assume as a primitive the notion of point. Instead, we can start from the notion of a ”region” in the space. The points are defined by suitable

abstraction processes

, i.e. order-reversing

sequences

of regions. Firstly, such a question was analized by A. N. Whitehead in the books ”An Inquiry Concerning the Principles of Natural Knowledge” and ”The

concept

of Nature”. In these books the ”inclusion relation” is the only primitive (”mereology” is the correct collocation). Successively in ”Process and Reality” Whitehead proposed a new approach in which the ”connection

relation

” and the one of ”oval” is considered which are topological and affine in nature, respectively. Notice that Whitehead’s

analysis

was philosophical but that several authors translated this analysis into systems of axioms for a formal mathematical treatment (for further information see [1]). Successively several approaches to point-free geometry where proposed metrical in nature. The primitives are the

inclusion

and either the

diameter

([4] and [5]) or the diameter and the

minimum distance

between regions [2]. A totally different approach to point-free geometry is proposed in a very interesting book by H. J. Schmidt. We quote also the deep and extensive investigation of several authors in point-free

topology

[3].

References

1. G. Gerla, Pointless geometries, in Handbook of

Incidence Geometry

, F. Buekenhout and W. Kantor (eds) 1994 North-Holland.

2.G. Gerla, Pointless

Metric spaces

. J. Symb. Logic 55, 207-219, 1990.

3. S. Papert, An abstract theory of

topological spaces

, Proc. Cambridge Philos. Soc. 60, 197-203, 1964.

4. A. Pultr, Diameters in locales: How bad they can be. Comment. Math. Univ. Carolinae 29, 731-742, 1988.

5. F. Previale, Su una caratterizzazione reticolare del concetto di spazio metrico, Atti d. Acad. Sc. di Torino, Cl Sc. Fis., 100, 766-779, 1966.

6. H. J. Schmidt, Axiomatic

Characterization

of Physical Geometry, Lecture Notes in Physics, Springer-Verlag, Berlin Heidelberg, 1979.

7. A. N. Whitehead, An Inquiry Concerning the Principles of Natural Knowledge, Camb. Univ. Press, Cambridge, 1919.

8. A. N. Whitehead, The concept of Nature, Camb. Univ. Press, Cambridge, 1920.

9. A. N. Whitehead, Process and Reality, The Macmillan Co., New York 1929.

Title

point-free geometry

Canonical name

PointfreeGeometry

Date of creation

2013-03-22 16:33:35

Last modified on

2013-03-22 16:33:35

Owner

ggerla (15808)

Last modified by

ggerla (15808)

Numerical id

15

Author

ggerla (15808)

Entry type

Topic

Classification

msc 03B15

Classification

msc 03A05

Synonym

pointless geometry

Related topic

Geometry

Generated on Thu Feb 8 19:45:43 2018 by

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second order logic

second order logic

Second order logic

refers to logics with two (or three) types where one type consists of the objects of interest and the second is either sets of those objects or

functions

on those objects (or both, in the three type case). For

instance

,

second order arithmetic

has two types: the numbers and the sets of numbers.

Formally, second order logic usually has:

•

the standard

quantifiers

(four of them, since each type needs its own

universal

and

existential quantifiers

)

•

the standard

connectives

•

the

relation

=

with its normal

semantics

•

if the second type represents sets, a relation

∈

where the first

argument

is of the first type and the second argument is the second type

•

if the second type represents functions, a binary function which takes one argument of each type and results in an object of the first type, representing function application

Specific second order logics may deviate from this definition slightly. In particular, there are some

first order logics

with additional quantifiers whose strength is

comparable

to that of second order logic. Some mathematicians have argued that these should be considered second order logics, despite not precisely

matching

the definition above.

Some people, chiefly Quine, have raised philisophical objections to second order logic, centering on the question of whether models require fixing some set of sets or functions as the “actual” sets or functions for the purposes of that model.

Title

second order logic

Canonical name

SecondOrderLogic

Date of creation

2013-03-22 13:00:17

Last modified on

2013-03-22 13:00:17

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

8

Author

Henry (455)

Entry type

Definition

Classification

msc 03B15

Synonym

second-order logic

Synonym

second order

Synonym

second-order

Related topic

IFLogic

Defines

second order language

Defines

second-order language

Defines

second order theory

Defines

second-order theory

Generated on Thu Feb 8 19:45:53 2018 by

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