An equational reengineering of set theories

y z

Andrea Formisano Eugenio Omo deo

AugustNovember

Abstract

New successes in dealing with set theories by means of stateoftheart theorem

provers may ensue from terse and concise axiomatizations such as can b e moulded in

the framework of the fully equational TarskiGivant map calculus In this pap er we

carry out this task in detail setting the ground for a number of exp eriments

Key words Set theory relation algebras rstorder theoremproving algebraic logic

Introduction

Like other mature elds of mathematics Set Theory deserves sustained eorts that bring

to light richer and richer decidable fragments of it general inference rules for reasoning

in it eective pro of strategies based on its domainknowledge and so forth

Advances in this sp ecialized area of automated reasoning tend in spite of their steadi

ness to b e slow compared to the overall progress in the eld Many exp eriments with set

theories have hence b een carried out with standard theoremproving systems Still to day

such exp eriments p ose considerable stress on stateoftheart theorem provers or demand

man to give much guidance to pro of assistants they therefore constitute ideal b enchmarks

Moreover in view of the p ervasiveness of Set Theory they are likely when successful in

something tough to have a strong echo amidst computer scientists and mathematicians

Even for those who are striving to develop something entirely ad hoc in the challenging

arena of set theories it is imp ortant to assess what can to day b e achieved by unsp ecialized

pro of metho ds and where the contextspecic b ottlenecks of Set Theory precisely reside

In its most p opular rstorder version namely the ZermeloFraenkelSkolem axiomatic

system ZF set theory very much like Peano arithmetic presents an immediate obstacle

it do es not admit a nite axiomatization This is why the von NeumannGodelBernays

theory GB of sets and classes is sometimes preferred to it as a basis for exp erimentation

Various authors eg have b een able to retain the traits of ZF

by resorting to higherorder features of sp ecic theoremprovers such as Isab elle

In this pap er we will pursue a minimalist approach to prop ose a purely equational

formulation of b oth ZF and nite set theory Our approach heavily relies on but we

go into much ner detail with the axioms ending in such a concise formulation as to oer

a go o d starting p oint for exp erimentation with Otter say or with a more markedly

equational theoremprover Our formulation of the axioms is based on the formalism

L of which is equational and devoid of variables but somewhat out of standards

Work partially supp orted by the CNR of Italy co ordinated pro ject SETA and by MURST

Tecniche sp eciali p er la sp ecica lanalisi la verica la sintesi e la trasformazione di programmi

y

University La Sapienza of Rome Dept of Computer Science formisandimiuniudit

z

University of LAquila Dept of Pure and Applied Mathematics omodeounivaqit

Luckily L can easily b e emulated through a rstorder system simply by treating the

metavariables that o ccur in the schematic formulation of its axioms b oth the logical

ones and those endowed with a genuinely settheoretic content as if they were rstorder

variables In practice this means treating ZF as if it were an extension of the theory of

relation algebras we can express it through a nite number of axioms b ecause variables

are not supp osed to range over sets but over the dyadic relations on the universe of sets

Taken in its entirety Set Theory oers a panorama of alternatives cf px that

is it consists of axiomatic systems not equivalent and sometimes antithetic cf

to one another This is why we will not pro duce the axioms of just one theory and will

also touch the theme of individuals ultimate entities entering in the formation of sets

Future work will expand the material of this pap er into a to olkit for assembling set theories

of all kindsafter we have singled out through exp eriments formulations of the axioms

that work decidedly b etter than others

Syntax and semantics of L

L is a ground equational language where one can state prop erties of dyadic relations

maps as we will call them over an unsp ecied yet xed domain U of discourse In

this pap er the map whose prop erties we intend to sp ecify is the membership relation

over the class U of all sets The language L consists of map equalities QR where Q

and R are map expressions

Denition Map expressions are al l terms of the fol lowing signature

1

symbol l  n y

degree

priority

1

Of these n y will b e used as leftasso ciative inx op erators as a p ostx

as a line topping its argument 2 op erator and

For an interpretation of L one must x along with a nonempty U a subset of

2

U U U Then each map expression P comes to designate a sp ecic map P and

Def

accordingly any equality QR b etween map expressions turns out to b e either true or

false on the basis of the following evaluation rules

2

l U  fa a a in U g

Def Def Def

QR f a b Q a b R g

Def

2

Q R f a b U a b Q if and only if a b R g

Def

2

QR f a b U there are cs in U for which a c Q and c b R g

Def

1

Q f b a a b Q g

Def

Of the op erators and constants in the signature of L only a few deserve b eing re

garded as primitive constructs indeed we choose to regard as derived constructs the ones

for which we gave no evaluation rule as well as others that we will tacitly add to the

signature

P P l

P yQ P Q

Def

Def

P n Q P Q

funPart P P n P 

Def

Def

P nQ P Q etc

Def

The interpretation of L obviously extends to the new constructs eg

2

P yQ f a b U for all c in U either a c P or c b Q g

Def

funPart P fa b P a c P for any c bg

Def

so that funPart P P will mean P is a partial function very much like Fun P to b e

seen b elow

Through abbreviating denitions we can also dene shortening notation for map equal

ities that follow certain patterns eg

1

Fun P P P n Total P P ll

Def Def

so that Total P states that for all a in U there is at least one pair a b in P

One often strives to sp ecify the class C of interpretations that are of interest in some

application through a collection of equalities that must b e true in every of C The

task we are undertaking here is of this nature our aim is to capture through simple map

equalities the interpretations of that comply with

standard ZermeloFraenkel theory on the one hand

a theory of nite sets ultimately based on individuals on the other hand

In part the game consists in expressing in L common settheoretic notions To start

with something obvious

1

Def Def Def

where each stands for one of l

0 1 n 0 1 n i

Def

To see something slightly more sophisticated

Examples With resp ect to an interpretation one says that a intersects b if a and

b have some element in common ie there is a c for which c a and c b A map

2

expression P such that P f a b U a intersects b g is

Likewise one can dene in L the relation a includes b ie no element of b fails to

The expression  translates the relation a b elong to a by the map expression

is strictly included in b and so on

Let a splits b mean that every element of a intersects b and that no two elements of a

intersect each other These conditions translate into the map expression dened as follows

splits y (  )l

Def

2

Secondly the reconstruction of set theory within L consists in restating ordinary

axioms and subsequently theorems through map equalities

Example One of the many ways of stating the muchdebated axiom of choice under

adequately strong remaining axioms is by claiming that when a splits some b there is

a c which is also split by a and which does not strictly include any other set split by a

Formally

splitslsplits n splits  Ch Total

where the second and third o ccurrence of splits could b e replaced by y

1

To relate the original version of this axiom in with ours notice that a set a splits

S

some b if and only if a consists of pairwise disjoint sets and accordingly a splits a

Moreover an inclusionminimal c split by a must have a singleton intersection with each d

in a otherwise of two elements in c d either one could b e removed from c conversely

S

if c is included in a and has a singleton intersection with each d in a then none of its

elements e can b e removed else c n feg would no longer intersect the d in a to which e

b elongs 2

In the third place we are to prove theorems ab out sets by equational reasoning moving

from the equational sp ecication of the set axioms To discuss this p oint we must refer to

an inferential apparatus for L we hence delay this discussion to much later cf Sec

For alternative versions of this axiom cf p

Extensionality subset sumset and p owerset axioms

Two derived constructs and r will b e of great help in stating the prop erties of mem

b ership simply

P r P P nP P

Def Def

Plainly a Q b and ar R b will hold in an interpretation if and only if resp ectively

all those c in U for which aQ c holds are elements of b in the sense that c b

the elements of b are precisely those c in U for which aR c holds

Our rst axiom extensionality states that sets are the same whose elements are

the same

E r 

A useful strenghtening of this axiom is the scheme Fun r P where P ranges over

all map expressions

Two rather elementary p ostulates the powerset axiom and the sumset axiom

state that for any set a there is a set whose elements comprise all sets included in a and

there is one whose elements comprise all elements of elements of a

Pow Total( )

Un Total( )

A customary strenghtening of the sumset axiom is the transitive embedding axiom

stating that every a b elongs to a set b which is transitively closed wrt membership in

the sense sp ecied by trans here b elow

T Total trans where trans 

Def

The subset axioms enable one to extract from any given a the set b consisting of

those elements of a that meet a condition sp ecied by means of a predicate expression

P In this form still overly nave this separation principle could b e stated as simply as

Total( r P ) This would suce taking as P to ensure the existence of a nul l set

devoid of elements We need the following more general form of separation whence the

previous one is obtained by taking  as Q

S Total( r( funPart Q P ) )

Example Plainly funPart is the map holding b etween c and d in U i c fdg ie

d is the sole element of c moreover funPart funPart is the map holding b etween

a and d i there is exactly one singleton c in a and d is the element of that particular c

Thus the instance Total( r( funPart funPart ) ) of S states that to every

set a there corresp onds a set b which is null unless there is exactly one singleton c fdg in

a and which in the latter case consists of all elements of d that do not b elong to a 2

Pairing and niteness axioms

A list of maps are said to b e conjugated quasiprojection if they are

0 1 n

partial functions and they are collectively surjective in the sense that for any list

a a of entities in U there is a b in U such that b a for i n We

0 n i i

assume in what follows that   are map expressions designating a pair of conjugated

0 1

quasipro jections It is immaterial whether they are added as primitive constants to L

or they are map expressions suitably chosen so as to reect one of the various notions of

ordered pair available around and sub ject to axioms that are adequate to ensure that the

desired conditions namely

1

Pair   l Fun  Fun  l

1 0 1

0

hold cf pp Notice that the clause Pair of this pairing axiom will

4

b ecome sup eruous when the replacement axiom scheme will enter into play cf

pp

Examples A use of the  s is that they enable one to represent settheoretic functions

b

yields a value by means of entities f of U such that no two elements b c of f for which 

0

c Symbolically we can dene the class of these singlevalued sets as b  have 

0 0

1

 where     sval 

0

Def Def

0

Cantors classical theorem that the p owerset of a set has more elements than the set

itself can b e phrased cf p as follows for every set a and for every function

f there is a subset b of a which is not hit by the function f restricted to the set a in

2

question A rendering of this theorem in L could b e Total( funPart P ) but

this would not faithfully reect the idea that the theorem concerns settheoretic functions

rather than functions funPart P of L The typical use of  and  is illustrated by

0 1

a more faithful translation which exploits the p ossibility to enco de the pair a f by an

c f c a and  entity c with 

1 0

1

 (     ) )  as b efore Total(

0 0 1 1

0

The latter states that to every c there corresp onds a b such that

c includes b if it exists 

0

d for any d in f such that c f b oth exist then b  c a and  if 

1 1 0

d e 2 d e exists and b elongs to a and no d in f other than d fullls  

0 0

A standard technique used to derive statements of the form Total( r R ) which

are often very useful is by breaking r R into an equivalent expression of the form

1 1

P   r   Q where Total P is easier to prove Exploiting the

0 1

0 1

same graph representation of map expressions utilized in this situation can b e depicted

as follows

v

Qk

Q

 Q P

0

Q

Q

Q

 r(  3\ Q )

0 1

1

v n m

z

r( R )

The desired totality of r R will then follow in view of Pair and of S Pair

1 2

For example by means of the instantiation P trans Q  of this pro of scheme we

obtain Total( r  ) where r  designates the singleton op eration a fag on U then

 l   l r  ) Q   we by taking P ( 

0 1 1 0 1 0

obtain the totality of r   which designates the adjunction op eration a b

0 1

a fbg Similarly one gets the totality of r r r   of any r R

0 1

such that b oth R nQ and Total( Q ) are known for some Q etc Even the full S

could b e derived with this approach from its restrained version Total( r(   P ) )

0 1

Under the set axioms E Pow S Pair introduced so far it is reasonable to

characterize a set a as b eing nite if and only if every set b of which a is an element has an

element which is minimal wrt inclusion cf p Intuitively sp eaking in fact the

set formed by all innite cs in the p owerset a of a has no minimal elements when a is

This was one of the rst ma jor theorems whose pro of was automatically found by a theorem prover

cf This achievement originally to ok place in the framework of typed lambdacalculus

innite b ecause every such c remains innite after a singleelement removal Conversely

if a b elongs to some b which has no minimal elements then the intersection of b with a

has no minimal elements either and hence a is innite In conclusion to instruct a theory

concerned exclusively with nite sets one can adopt the following finiteness axiom

F nite where nite ( l(  y )y )

Def

Bringing individuals into set theory

Foundation and plenitude axioms

Taken together with the foundation axiom to b e seen b elow the axioms E Pow

T S Pair and F discussed ab ove constitute a fullblown theory of nite sets

However they do not say anything ab out individuals or urelements cf entities

that common sense places at the b ottom of the formation of sets These are not essential

for theoretical development but useful to mo del practical situations To avoid a revision

of E necessary if we wanted to treat individuals as entities devoid of elements but

dierent from the null set let us agree that individuals are selfsingletons a fag

cf pp Moreover to bring plenty of individuals into U at least as many

individuals as there are sets hence innitely many individuals we require that there are

individuals outside the sumset of any set Here comes the plenitude axiom

ur where ur r  Ur Total

Def

To develop a theory of pure sets one will p ostulate lack of individuals by adopting

the axiom ur instead of plenitude

When individuals are lacking the foundation or regularity axiom ensures that the

membership relation is wellfounded on U and can b e stated as follows when some b

belongs to a there is a c also belonging to a that does not intersect a On the surface this

statement has the same structure as the version of the axiom of choice seen at the end of

ln To reconcile this statement Sec in L it can hence b e rendered by Total

with individuals we can recast it as

R Total( lur yn n ur nlur )

which means unless every b in a is an individual there is a c in a such that every element

of a c is an individual and c itself is not an individual

As is wellknown cf p foundation helps one in making the denitions of basic

mathematical notions very simple In our framework we prop ose to adopt the following

3

denition of the class of natural numbers

nat  ( ( r  n  )y  l )

Def

which means a is a natural number if for every b in a fag other than the nul l set there

is a c in a such that b c fcg and b c

An innity axiom and the replacement axioms

Similarly under the foundation axiom the denition of ordinal numbers b ecomes

ord ( transn url )( y  y )

Def

where trans is the same as in T hence transn  holds and hence thanks to R

y  y requires that an ordinal b e totally ordered by membership

From this simple start one can rapidly reach the denition of imp ortant data structures eg ordered

and oriented nite trees

The existence of innite sets is often p ostulated by claiming that ord n nat is not empty

l ord nnat ll or equivalently Total( l ord n nat ) The following more essential

formulation of the infinity axiom based on and presupp osing R seems preferable

4

to us

1

I Total( l( n nn n ) )

What I means is There are distinct sets a a such that the sumset of either one is

0 1

included in the other neither one belongs to the other and for any pair c c with c in

0 1 0

5 6

a and c in a either c belongs to c or c belongs to c Of course this axiom is

0 1 1 0 1 1 0

antithetic to the axiom F seen earlier one can adopt either one but only one of the

two

In a theory with innite sets the replacement axiom scheme plays a fundamental

role Two simpleminded versions of it are

Total( ( funPart Q ) ) Total( ( r Q ) )

Both of these state under dierent conditions on a certain map P that to every c

there corresp onds a sup erset of a set of the form P c fu v P u for some v cg

Def

The former applies when P funPart Q designates a function the latter when r P

with P r Q designates a total map adopts a formulation of replacement

closer in spirit to the latter but it is the former that we generalize in what follows

Parameterless replacement like a parameterless subset axiom scheme would b e of

d represents the domain to little use Given an entity d of U we can think that 

0

d represents a list of parameters To which one wants to restrict a function and 

1

state replacement simply it is convenient to add to the conditions on the  s a new one

i

7

Sp ecically we imp ose that distinct entities never enco de the same pair

1 1

Pair     n 

0 1

0 1

5

The simplest formulation of replacement we could nd in L so far is

1 1

Repl Total( (     funPart Q ) )

0 1

0 1

This means To every pair d there corresponds a set comprising the images under the

e d  e  functional part of Q of al l pairs e that full l the conditions 

1 0 0

d 

1

Example To see that Pair can b e derived from Pair T S and Repl

4 123

lll Then by virtue one can argue as follows Thanks to S a null set f g exists

l ll Again through T of T a set to which this null set b elongs exists to o

we obtain a set c to which b oth of the preceding sets b elong l ll The

d c and latter c can b e combined with any given a to form a pair d fullling b oth 

0

d a by Pair Two uses of Repl referring to the singlevalued maps 

1

1

l   l  Q 

0 0

1

Def

with and resp ectively will complete the job Indeed the rst use of Repl

will form from d a set c comprising a and f g as elements while the second will form

a

Here like in the case of Ur which could have b een stated more simply as Total ur our

preference go es to a formulation whose imp ort is as little dep endent as p ossible from the remaining axioms

Notice that when c b elongs to a then c a hence there is a c in a n c so that c

b elongs to c Then c a and so on Starting wlog with c in a one nds distinct sets c c c

with c in a for and i

i

For the sake of completeness let us mention here that for a statement not relying on R the following

cumbersome expression should b e subtracted from the argument of Total in I

1 1 1 1 1 1

           

Notice that Pair can b e sup erseded retaining Pair by the denitions

 y ) yl  funPart funPart  (

Def Def

from a pair d with comp onents c and b a set c comprising a and b as elements for any

a a

ab

given b

Notice that either use of Repl in the ab ove argument has exploited a single param

eter which was a and b resp ectively 2

Setting up exp eriments on a theoremprover

A map calculus ie an inferential apparatus for L is dened in pp along

the following lines

A certain number of equality schemes are chosen as logical axioms Each scheme com

prises innitely many map equalities P Q such that P Q holds in every interpre

tation syntactically it diers from an ordinary map equality in that metavariables

which stand for arbitrary map expressions may o ccur in it

Inference rules are singled out for deriving new map equalities V W from two equalities

P Q R S either assumed or derived earlier Of course V W must hold in

any interpretation fullling b oth P Q and R S The smallest collection

E of map equalities that comprises a given collection E of proper axioms together

with all instances of the logical axioms and which is closed wrt application of the

inference rules is regarded as the theory generated by E

A variant of this formalism which diers in the choice of the logical axioms b ecause

as primitive constructs has b een prop osed in and seem preferable to and

We omit the details here although we think that the choice of the logical axioms can

critically aect the p erformance of automatic deduction within our theories Ideally only

minor changes in the formulation E of the set axioms should b e necessary if the logical

axioms are prop erly chosen Similarly in the automation of GB one has to b estow some

care to the treatment of Bo olean constructs cf pp

To follow ortho doxly we should treat L as an autonomous formalism on a par

with rstorder predicate calculus This however would p ose us two problems we should

develop from scratch a theoremprover for L and we should cop e with the innitely

many instances of S and of Repl Luckily this is unnecessary if we treat as rstorder

variables the metavariables that o ccur in the logical axioms or in S Repl as well as

in induction schemes should any enter into play either as additional axioms or as theses to

b e proved Within the framework of rstorder logic the logical axioms lose their status

and b ecome just axioms on relation algebras conceptually forming a chapter of axiomatic

set theory interesting per se richer than Bo olean algebra and more fundamental and stable

than the rest of the axiomatic system

Attempts some of which rather challenging that one might carry out with any rst

order theorem prover have the following avor

Under the axioms E Pow T S Pair R ur and F prove Un

1234

8

n ln as theorems Repl Ch Fun r P and trans

Under E Un S Pair R and Ur prove that  ur natur

9

ord ur ord nord l yl ord l( (  y ) ) l etc

The last of these states that the null set b elongs to every transitively closed nonnull set

The last two of these state that elements of ordinals are ordinals and that every nonnull set of ordinals

has a minimum wrt

Under E Pow Un S Pair and Repl prove Pair Cantors

1235 4

r r  r   theorem and the totality of r r  r

0 1

r   r( funPart Q setPart P ) where setPart P P P l

0 1

Def

etc

We count on the opp ortunity to so on start a systematic series of exp eriments of this

nature for which we are inclined to using Otter The latter is not sp ecically oriented

to equational logic and it is conceivable that a system based on term rewriting might t

our needs b etter However we plan to p erform extensive exp erimentation with theories of

number and sets sp ecied in L and we are eager to compare the results of our exp eriments

with the work of others Otter is attractive in this resp ect b ecause it has b een the system

underlying exp eriments of the kind we have in mind as rep orted in Moreover

the fact that Otter encompasses full rstorder logic paves the way to combined reasoning

tactics that eg p erform resolution of l P ll against P

Conclusions

The language L may lo ok distasteful to reading but it ought to b e clear that techniques

for moving back and forth b etween rstorder logic and map logic exist and are partly

implemented cf moreover they can b e ameliorated and can easily b e extended

to meet the sp ecic needs of set theories Thanks to these the automatic crunching of

set axioms of the kind discussed in this pap er can b e hidden inside the backend of an

automated reasoner

Anyhow we think that it is worthwhile to riddle through exp eriments our exp ectation

that a basic machine reasoning layer designed for L may signicantly raise the degree of

automatizability of settheoretic pro ofs This exp ectation relies on the merely equational

character of L and on the go o d prop erties of the map constructs moreover when the

calculus of L gets emulated by means of rstorder predicate calculus we see an advantage

in the niteness of the axiomatization of ZF

Acknowledgements

We have grateful to Marco Temperini with whom we have enjoyed a number of discussions

on sub jects related to this pap er The idea is due to him that individual variables can

sup ersede metavariables in axiom schemes such as separation and replacement

The example at the end of Sec was suggested by Alb erto Policriti

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