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Home , Fuzzy set

A Development of Theory in

Petr Hajek and Zuzana Hanikova

Institute of Computer Science

Praha

Czech Republic

Abstract This pap er presents an axiomatic FST Fuzzy

Set Theory as a rstorder theory within the framework of fuzzy logic

in the style of In the classical ZFC we use a construction similar to

that of a Bo oleanvalued universeover an algebra of truth values of the

logic we useto show the nontriviali ty of FST We give the of

FST Finally we show that FST interprets ZF

Intro duction

If anything comes to p eoples minds on the term theory b eing used it

usually is the theory or theories handling fuzzy sets as realvalued functions on

a xed universe within the classical settheoretic universe However as fuzzy

or manyvalued logic has b een evolving into a formal axiomatic theory a few

exceptions from this general exp ectation have emerged these are formal set

theories within the resp ective manyvalued logics ie theories whose underlying

logic is governed by a manyvalued semantics

We rely esp ecially on those works which develop a theory in the language and

style of the classical ZermeloFraenkel set theory ZF We have b een inspired

by a series of pap ers developing a theory generalizing ZF in a formally weaker

logicintuitionistic and later its strengthening commonly referred to

as Godel logic some results and pro ofs carry over to our system

For an imp ortant example the of foundation together with a very weak

fragment of ZF implies the law of the excluded middle which yields the full

classical logic b oth in Godel logic and in the logic we use in this pap er and

thus the theory develop ed b ecomes crisp For this reason we start with building

a noncrisp universe in which we verify our axioms

However all these pap ers fall short in tackling one of the distinguishing traits

of manyvalued logic which is the general nonidemp otence of the conjunction

conjunction is idemp otent in Godel logic The nonidemp otence of conjunction

aects the resulting theory considerably cf in coping with some of the

diculties we appreciated an elegant solution found in The author works

over the socalled phase spaces as algebras of truth values and builds using an

analogy of the construction of a Bo oleanvalued universe over a phasespace

instead of a Bo olean algebra a with class op erations evaluating formulas

in the language f g in which he veries the chosen axioms of his set

theory Having observed that the standard Luk asiewicz algebra enriched with

the op erator is a particular phase space we have studied this pap er thinking

of a more general approach employing linearly ordered BLalgebras with

this should form a common generalization of the approach of and and

that of

This pap er is an extension of it brings in a simplied denition of the

initial universe and an inner mo del of ZF in FST Regarding the nature of this

pap er we omit most of the logical background which is to b e found mainly in

as well as some technical details and some pro ofs

The authors acknowledge gratefully a partial supp ort from the grant IAA

Grant Agency of the Academy of Sciences Czech Republic

Prerequisites

Denition BL is a formal logical system with basic connectives

and dened connectives and where is is

is and is

and with two quantiers and The axioms are as

follows

A

A

A

A

Aa

Ab

A

A

xx t t substitutable for x in

t xx t substitutable for x in

x x x not free in

x x x not free in

x x x not free in

The deduction rules of BL are mo dus p onens generalization and fg

Denition Let C b e an arbitrary schematic extension of BL A C algebra

is a BLalgebra L in which all the axioms of C are Ltautologies

Theorem Strong completeness Let C be a schematic extension of BL

let T be a theory over C and let be a formula of the language of T Then T

proves i holds in any safe model M of T over any linearly ordered C algebra

The Initial Universe

Consider the classical ZFC Fix C as a schematic extension of BL and x

a constant L for an arbitrary linearly ordered complete C algebra write L

L as usual L denotes b oth the algebra and its supp ort In

ZFC let us make the following construction in analogy to the construction of

a Bo oleanvalued universe over a complete Bo olean algebra we build the class

L

V by ordinal induction Dene L L fg

L

V fg

L L

V ff Fnc f D f V Rf L g

for any ordinal and for limit ordinals

L L

V V

Here Fnc x is a unary predicate stating that x is a function and D x and Rx

are unary functions assigning to x its domain and range resp ectively

Note that functions taking the value on any of their domain are

not considered as elements of the universe

Finally we put

L L

V V

2On

L L

Observe that for V V

L

We dene two binary functions from V into L assigning to any x y

L

V the values jjx y jj and jjx y jj representing the truth values of the two

predicates and

jjx y jj y x if x D y otherwise

jjx y jj if x y otherwise

We now use induction on the complexity of formulas to dene for any formula

x x the free variables in b eing x x a corresp onding nary

n n

L n

function from V into L The induction steps admit the following cases we

just write for short

is then jjjj

is then jjjj jj jj jjjj

is then jjjj jj jj jjjj

is then jjjj jj jj jjjj

is then jjjj jj jj jjjj

is then jjjj jj jj

V

xu is x then jjjj

L

u2V

W

is x then jjjj xu

L

u2V

Here xu is the result of substituting u for the variable x in We use the

symb ols for b oth logical connectives and op erations in L

L

Denition Let b e a closed formula We say that is valid in V i jj jj

is provable in ZFC

As usual we take closures of formulas containing free variables when considering

their validity

L

Lemma ZFC proves for u V

ux jjx ujj for x D u jju ujj

L

ZFC proves for u v w V

ijju v jj jjv w jj jju w jj

iijju v jj jjv w jj jju w jj

iiijju v jj jjv w jj jju w jj

Proof immediate

i immediate

ii if v w then jjv w jj so jju v jj jju v jj jjv w jj jju

w jj jjv w jj jju w jj otherwise jjv w jj in which case the statement

holds trivially

iii analogously ut

L

Lemma Substitution For any formula ZFC proves u v V jju

v jj jjujj jjv jj

L

Lemma Bounded quantiers ZFC proves x V

W

xy jjy jj i jjy xy jj jjy y x y jj

y 2D x

V

xy jjy jj ii jjy xy jj jjy y x y jj

y 2D x

W W

xy jjy xjj jjy jj Proof i jjy y x y jj

L

y 2D x y 2V

jjy jj since jjy xjj is nonzero only if y D x and in that case it is xy

ii analogously ut

Corollary jjx y jj jju xu y jj jjuu x u y jj

The Theory FST

We intro duce the theory FST in the language fg we take to b e a logical

symb ol with the usual axioms imp osing reexivity symmetry transitivity and

congruence wrt on the corresp onding relation The underlying logic of FST is

a schematic extension C p ossibly void of BL when proving theorems within

FST we only rely on the logical axioms of BL thus any schematic extension

will for example interpret ZF On the other hand for a given extension C the

L

universe V for L a C algebrawill yield an interpretation of FST over C

The reader will have noticed that ours is a crisp equality This was imp osed

by the following fact

Lemma A theory with comprehension for open formulas and pairing or

singletons over a logic which proves the propositional formula

proves x y x y x y

Proof Given x y let z b e s t u z u fxgu x i e u z u x

Since x x we have x z If y x then y z by congruence but then

y x thus we have proved y x y x thus by assumption on the

logic x y x y ut

Thus eg in L ukasiewicz logic and in pro duct logic we get a crisp Also

under the usual formulation of extensionality crispness of implies crispness of

in a theory with pairings and unions cf We b orrow the elegant solution

from a mo dication of extensionality uses the op erator which invalidates

the pro of of crispness of from the crispness of

Denition FST is a rst order theory in the language fg with the follow

ing axioms

i extensionality xy x y x y y x

ii xy y x

iii pair xy z uu z u x u y

iv xz uu z y u y y x

v weak p ower xz uu z u x

vi innityz z x z x fxg z

vii separation xz uu z u x u x for any formula not con

taining z as a free variable

viii collection xz u xv u v u xv z u v for any for

mula not containing z as a free variable

ix induction xy xy x xx for any formula

x supp ort xz Crispz x z

The s in the axioms of weak p ower and induction are intro duced to

weaken the statements the s after an existential quantiers is used to guar

antee that an element with the p ostulated prop erty exists in every mo del of the

theory which do es not follow from the semantics of the quantier

As usual in the formulation of some axioms we use the functions of empty

set and union which can b e intro duced using the appropriate axioms

as in classical ZF and also the usual denition of For a detailed treatment of

intro ducing functions in BL see The function intro duced by the weak p ower

set axiom will b e denoted WP x for a set x We use the following denition of

crispness of a set

Denition Crisp x uu x u x

Theorem Let be a closed formula provable in FST Then ZFC proves

L

jjjj in V

L

Having proved this theorem we shall take the lib erty to call V a model of

FST More p edantically we have constructed an interpretation of FST over C

in ZFC We omit all validity pro ofs for logical axioms and inference rules The

validity of congruence axioms for has b een veried in Lemma It remains to

verify the settheoretic axioms

L

Lemma The settheoretical axioms ix hold in V

L

Proof extensionality For xed x y V either x y and then the axiom

holds or x y and jjx y jj then either D x D y and w l o g there

is a z D x s t xz y z so jjy xjj or w l o g there is a

z D x s t z D y and then jjz xjj is nonzero while jjz y jj is zero thus

jjx y jj

L

empty set There is only one candidate for the role of an empty set in V

L

Indeed for an arbitrary x we get jjx jj since no and this is the in V

x can b e in the domain of taken as a function

L L

pair For xed x y V there is a z V such that D z fx y g and

L

z x z y The set z has the desired prop erties for arbitrary u V

either u D z then either u x or u y and jju z jj z u or

u D z and then jju z jj jju xjj jju y jj

S

L

union For a xed x V dene auxiliary D x fD v v D xg

W

Dene z s t D z fu D x v u xv g with a nilp otent

v 2D x

t the union of a nonempty set may well b e empty and for u D z set

W

L

z u v u xv Then for an arbitrary u V if u D z then

v 2D x

W

jju z jj z u v u xv jjy u y xjj If u D z then

v 2D x

jju z jj and also jjy xu y jj by denition of D z

L L

weak p ower For a xed x V dene z st D z fu V D u

L

D x uv xv for v D ug and z u for u D z For u V

either u D z and then jju z jj z u and also by denition of D z

jju xjj jju xjj or u D z thus jju z jj and by denition

of D z either D u D x or for some v D u uv xv and in either

W

case jjv uv xjj uv jjv xjj

v 2D u

L

innity Dene a function z with D z V and z u for u D z

L L

Then jj z jj and since for x V we have x fxg V also jjx

z x fxg z jj

L

separation For a xed x V and a given dene z s t D z fu D x

xu jjujj g and for u D z set z u xu jjujj Obviously this

L

denition of z demonstrates the validity of separation in V

collection in

L

induction Fix a formula and supp ose the axiom do es not hold in V Then

since is twovalued it must b e the case that jjxy xy xjj

L

and jjxxjj thus there is a successor ordinal s t x V x

L

y V y Supp ose rst but jjy y

jj thus the antecedent would b e Supp ose now

L

and x V is s t x From the condition that jjy xy

xjj and since jjy xy jj we get jjxjj a contradiction

L

supp ort For a xed x V take z such that D z D x and u D z

V

z u Then jjCrispz jj and xu jju z jj ut

u2D x

An Interpretation of ZF in FST

Within FST we shall dene a class ie we shall give a formula with one free

variable in the language of FST of hereditarily crisp sets which will b e proved

an inner mo del of ZF in FST

We start with a bunch of technical statements

Lemma BL x x

Proof The righttoleft implication is easy We give a BL pro of of the con

verse one an analogy to the pro of of the Barcan formula in S Let stand

for

i

In BL thus thus

ii

In BL thus This gives

Thus the following two are provable in BL

iii

iv

Next v

since

vi x x is a consequence of v

Finally x x x x x ut

Lemma BL

Proof Takes place inside BL Implication lefttoright

In the last

thus the last formula implies we formula in the chain

get which generalizes to the desired implication

Conversely thus also i

Moreover thus ii

Since BL proves we get

the righttoleft implication by putting together i and ii

ut

Recall the denition of a crisp set We get

Corollary xCrispx uu x u x

Note that Crispx uu x u x uu x u x so

crispness is a crisp prop erty

Lemma Crispx Crispx

We write for

Lemma BL

Proof and so

ut

Lemma BL xy Crisp xCrisp y x y x y

Proof Crisp xCrispy u x u y

u xu x u y u x u y ut

Denition Hereditarily crisp HCT x Crispx u

xCrispuu x

The formula HCT x denes a class and we adopt the habit of writing

x HCT instead of HCT x and approaching classes in a similar way we

approach sets A crisp class C is a class for which xx C x C or

equivalently xx C x C

Lemma Crispness of HCT x HCT x HCT

Proof x HCT Crispxuu x Crisp uu x

Crispxu u xu x Crispuu x

Crispxuu x Crispuu x

x HCT ut

0

Denition Hereditarily crisp set Hx Crispx x HCTx

0

x

Lemma Crispness of H x H x H

Proof By denition of H we are to prove

Crisp xy HCTx y Crispxy HCTx y

Since Crispx Crispx it suces to prove

Crisp xy HCTx y y HCTx y

we may use the presumption Crisp x in each of the two implications since it

is idemp otent

First Crisp xy HCTx y y HCTx y by Lemma

and Lemma Now generalize y Crisp xy HCTx y y

HCTx y hence Crispxy y HCTx y y y HCTx

y and the succedent implies y y HCTx y ut

We show that FST proves H to b e an inner mo del of ZF In more detail for

H

a formula in the language of ZF dene inductively as follows

H

for atomic

H

for

H H H

for

H H H

for

H H

x H for x

H H

x H for x

H

Theorem Let be a theorem of ZF Then FST

To prove this theorem we rst show that the law of the excluded middle

LEM holds in H since BL together with LEM yield classical logic we will

have proved that all logical axioms of ZF are provable when relativized to H

Then we prove the Hrelativized versions of all the axioms of the ZF set theory

Lemma Let x x be a ZFformula whose free variables are among

n

H H

x x Then FST proves x H x H x x x x

n n n n

Proof We consider a formula with at most one free variable x the mo d

ication for multiple free variables b eing easy The formula to b e proved in

H H

FST is xx H x x it suces to prove xx H

H H

x x and by Lemma and Lemma it suces to prove xx

H H

H x x

The pro of pro ceeds by induction on the complexity of

Atomic subformulas is a crisp predicate for we have to prove x

y x y assuming x y H In fact y H implies Crispy which entails

xx y x y

Conjunction for a subformula x y x z of we assume one free

variable in common and one distinct free variable for each subformula and we

henceforth omit their explicit listings for legibilitys sake assume x y H

H H H H

and x z H Then x H y z H

H H H H

and since x H is idemp otent this completes the

induction step for conjunction

Implication for a subformula x y x z of assume x y H

H H H H H H

and x z H Thus x z H

H H H

and since x y H implies crispness of we get x y H

H H H H H H H

Thus x y z H

H

The universal quantier for a subformula y x y of the induction hy

H H

p othesis is x y H x y x y Generalize in y x H

H H

y y H x y x y now since for a crisp BL proves

we may mo dify the succedent to

H H

y y H x y y H x y and distributing y we have

H H

proved x H y H x y y H x y To ip the and the

y H in the succedent use Lemma

The existential quantier for a subformula y x y of the induction

H H

hyp othesis is x y H x y x y Generalize in y x

H H H

H y y H x y x y now y H x y

H H H

x y y H y H x y y H x y y

H H H

H x y y H x y We get x H y y H x y

H H

y y H x y and the last succedent implies y y H x y

ut

Denition WPC x fu WP x Crispug

Lemma xx H Crisp x x H

0 0 0

Proof Fix an x x H is by denition u xu u HCT u u Thus

0 0 0

v u xu v u HCT u u Fix v and separate v fu v u

0 0 0

v u HCT u xu ug Note that u xu v u HCT u u

b ecause x is crisp v is a crisp set and all its elements are crisp all of them

S S

are in HCT hence v is crisp its elements are crisp since a v implies

S S

b HCT a b v thus a is crisp v is also transitive b a v implies

S

y b a y v and since y HCT is transitive b y v and b v Now

S S

consider WPC v the crisp elements of the weak p ower set of v this is a

S S

crisp set and it is a of WPC v v which is a crisp transitive set of

S S

crisp elements thus in HCT thus WPC v H Since v WPC v we

S S

0 0

get u xu WPC v u u thus u xu WPC v this means

S

x WPC v HCT so x H ut

We consider ZF with the following axioms empty set pair union p ower set

innity separation collection extensionality induction The exact sp elling of

these axioms is given separately when proving in FST their Hed versions

Theorem For being any of the abovementioned axioms of ZF FST proves

H

Proof empty set z uu z

The Htranslation which reads z H u H u z is provable since

HCT

pair x y z uu z u x u y

The Htranslation x y H z H u H u z u x u y is

absolute the set fx y g is crisp since is crisp to show that it is a subset of

0 0

a set which is in HCT consider x HCT a witness for x H and y HCT

0 0

a witness for y H Then fx y g x y is a crisp transitive set with crisp

elements hence in HCT and thus fx y g is in H

union xz uu z y u y x

The Htranslation x H z H u H u z y H u y x

S

0

is absolute let x HCT witness x H Then x is a crisp set with crisp

S S S

0 0

elements since x x and x x HCT which witnesses x H

p ower set xz uu z u x

The H translation reads x H z H u H u z y H y u

0

y x Let x b e a witness for x H then for x H it holds that u

H H

H u WPC x u x u x WPC x is crisp and WPC x

0 0

WPC x x which is a transitive crisp set with crisp elements thus in HCT

and a witness for WPC x H

separation xz uu z u xu for a ZFformula not containing z

freely

0

The Htranslation is absolute let x HCT witness x H and set z fu

H 0 0

x ug then z is a crisp set and z x x HCT ie x is a witness for

z H

innity z z u z u fug z

It suces to prove that there is a set z H st z u z u fug z

Let z b e any set satisfying the axiom of innity in FST and dene z fx

z x z Crispxg Then z is a subset of z and z and u z

u fug z Now let z fx z x z g ie a transitive subset of z

Obviously z let us prove x z x fxg z that is by denition of z

x z y xy z x fxg z aa x a x a z We

know x z x fxg z Also x z y x y z y x y

x y z Note that we made multiple use of the presumption x z we

may do that b ecause the formula is crisp thus x z x z Finally z is

a crisp transitive set with crisp elements so z HCT

extensionality xy x y z z x z y

The Htranslation x y H x y z H z x z y follows from

extensionality in FST by H b eing transitive and by the crispness of its elements

the s may b e left out

collection ux uy x y v x uy v x y for not contain

ing v freely

H

The Htranslation reads u H x H x u y H x y v

H

H x H x u y H y v x y Fix u H and a ZFformula

we want to nd a corresp onding v H st the ab ove is true Dene v by

H

collection in FST for u and the formula y H x y separate v fx v

x v x H g Then v H is a crisp set and since u is crisp satises the

collection axiom By Lemma v H

induction xy xy x xx for any ZFformula

H H

The Htranslation is x H y H y x y x x

H

H x Fix a ZFformula and consider the instance of induction in FST

H

for the formula x H x

H H H

xy xy H y x H x xx H x

This formula is our aim except for the s Let us denote A the antecedent and S

the succedent in the implication with the s omitted Since S S it remains

H

to b e proved that A A Let us further denote y H y x y with

I First it is obvious that

H

x H y y x y and

x H y y x Thus

H

x H y y xy x y hence

H

x H y y x y and

H

x H y y x y thus x H I Since x H x H

we get

H

A x H I I x hence by use of Lemma

H

A x H I x so

H

x which is A A ut A x H I

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