Table Of ContentA RELATIONAL AXIOMATIC FRAMEWORK
FOR THE FOUNDATIONS OF MATHEMATICS
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1
LIDIAOBOJSKA
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n ABSTRACT. WeproposeaRelationalCalculusbasedontheconceptofunaryrelation.In
thisRelationalCalculusdifferentaxiomaticsystemsconvergetoamodelcalledDynamic
a
GenerativeSystemwithSymmetry(DGSS).InDGSSwedefinetheconceptsofrelational
J
setandfunctionandprovethatextensionalityandthesubstitutionpropertyofequalityare
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theoremsofDGSS.
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AsafirstexemplificationofDGSS,weconstructamodelofnaturalnumberswithout
relyingonPeano’sAxioms. Eventually, somenewclarifications regardingthenatureof
]
M thenumberzeroaregiven.
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a 1. INTRODUCTION
m
KurtGo¨del[7]thoughtthatsettheoreticantinomiesareperhapsnotproblemsconnected
[
tosettheoryitself buttothe theoryofconcepts. Inthispaperwe wouldlike to focuson
1 theconceptofunaryrelation.Inmathematicalliteraturethetermunaryrelationisapplied
v
tosubsetsofagivenset[11].
0
Thuswecanthinkofmathematicalobjectsasiftheywereeithersetsorrelations(unary,
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3 binary,etc.).Forexample,afunction f foragivenargumentxcanbeseenasasetconsist-
4 ingoforderedpairs(x,f(x))orasa binaryrelationsuchthatforeveryargumentxthere
. existsexactlyonetermywith f(x)=y. Additionally,beginningwiththeworkofChurch,
1
0 Turingandothers[2],theconceptoffunctionsasalgorithmswasdeveloped.
0 In 1996E. De Giorgiintroducedthe conceptof FundamentalRelation [3], [5] which
1 describes the phenomenon of autoreference. Generally, Fundamental Relations change
:
v theperspectiveofhowwe considerentities. In a certainsense, a fundamentalrelationis
i a relation abstraction describing the entities related and the relation between them. For
X
thisreasonthe presenttheorymightbe thoughtofasa kindofa CombinatoryLogic[1],
r
a but it also calls to mind Mereology [8], the theory of part-whole relationships. In the
contextofFundamentalRelations,aunaryrelationmakessenseifandonlyifthereexists
afundamentalbinaryrelationwhichdescribesitsbehavior.
FundamentalRelationspermitustointroducea specific kindofunaryrelation,which
exhibitsdynamicbehavior.AsaresultaRelationalCalculuscanbedefinedbytheintuitive
conceptofbinaryrelation.
Startingwithasimplesystemoftwoaxioms,whichdescribesakindofdynamiciden-
tity,wethenexpressthissysteminalgebraiclanguageandshowthatsomeequationalrules
andPeano’sAxiomsbecometheoremsofthissystem.
Date:18December2009.
2000MathematicsSubjectClassification. Primary03G27;Secondary03H05,18A15.
Keywordsandphrases. UnaryRelations; Relational Calculus; Non-standard Identity; Extensionality and
SubstitutionPropertyofEquality;Peano’sAxiomsofNaturalNumbers.
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2 LIDIAOBOJSKA
2. FUNDAMENTAL RELATIONS
AdoptingDeGiorgi’swayofconsideringrelations,whichheseesasfundamentalenti-
ties[3],werestrictourselvestorestateonlythosedefinitionsandaxiomswhicharestrictly
necessaryinthecontextofthisarticle. Moreover,wemodifythenotationinordertointro-
ducesomenewconcepts.
Letusbeginwithtwoprimitivenotions:qualityandbinaryrelations.
(1) Wewillsaythatifqisaquality(writtenasQq),thenqxmeansthattheobjectx
hasthequalityq.
(2) Giventwoobjectsx,yofanynatureandabinaryrelationr,wewillwriterx,yto
saythat“xandyareinrelationr”or“xisinrelationrwithy”.
Wecanproceed,asdoDeGiorgietal. [3],andintroducetheFundamentalRelationR
Q
whichisdefinedasabinaryrelation:
Axiom2.1. R isabinaryrelation.
Q
(1) IfR x,ythenQx.
Q
(2) IfQqthenR q,x≡qx.
Q
Thus,ifR isdefinedasabinaryrelation,itfollowsthatqcanbeconsideredaunary
Q
relation[9].
Definition2.1. Aunaryrelationisanyrelation∗suchthatR∗isabinaryrelation.
Tosimplifythenotationwewillwrite:(qx)insteadofR q,x(i.e. R q,x≡(qx)).
Q Q
Ingeneral, inthe expressionoftheform(xy), ( )willbe usedtoindicatethe relation
abstractionorFundamentalRelation(inthiscase binary)connectingthe unaryrelationx
and its argument y. Because unary relations underlie those binary relations in which at
leastoneoftheobjectsisdefinedintermsoftheother,unaryrelationswecanthinkofas
“qualifying”theirarguments.Inthisperspective,qualitiesareviewedasrelationalentities,
insofarastheycanonlybeconceivedofbeing“in”someobject(seeAxiom2.1).
3. RELATIONALCALCULUS
3.1. Basic Definitions. As presented in Section 2, at the basis of our relational system
isa primitivecalleda FundamentalRelation(). ThisFundamentalRelationis a kindof
relationabstraction which can act as a combinatoror as a pure relation. When we write
(xy),wecanreaditastheapplicationofxontoy,astheprocessxactingontheinputy,or
simplyastheconnectionbetweenxandy. xmightbeseenasaqualityinthesenseofDe
Giorgi,butcanalsobeanyarbitraryentityinrelationwithy. (xy)“creates”anewobject,
inthesensethat(xy)isawholewhichturnsouttoconsistoftworelatedentitiesxandy.
Thisinterpretationgivesagreatfreedom,aswewillseebelow. Thenotation(xy)allows
ustointroduceanykindofunaryrelationintheplaceofx,eventhoughthisconceptmight
notbeintuitive.
We definea logicwith unrestrictedquantification,bearinginmindthatquantifiersare
abbreviationsfor certain quantifier-free expressions. In the present system, well formed
formulasare those formed from atomic constants, i. e., the logical operators∀, ∃, ∧, ∨,
¬, =⇒, ⇐⇒ and =, parentheses [ ], { } and variables, possibly connected by means of
application().
Avariableisanyobjectwhichisabletoenterintorelationwithotherobjectsduetothe
applicationoperator().
ARELATIONALAXIOMATICFRAMEWORKFORTHEFOUNDATIONSOFMATHEMATICS 3
Thenatureoftheobjectsremainsexplicitlyopen,aslongastheseobjectsarecapableof
beinginrelationwithotherobjectsinaccordancewiththeAssociationRuletobedefined
inAxiom3.1. Theobjectsthemselvescanevenberelations.
Definition3.1. The languageof RelationalCalculus(RC) terms is builtfrom an infinite
numberofvariables: x,y,z,...usingtheapplicationoperator()asfollows:
(1) Ifxisavariable,thenxisanRCterm,
(2) ifx,yarevariables,then(xy)isanRCterm,
(3) ifxisavariableandMisanRCterm,then(Mx)and(xM)areRCterms.
Axiom3.1. ∀a,b,c:(abc)=((ab)c)=(a(bc))
Axiom3.1describesasimpleAssociationRuleusedinMathematics. Asstatedabove,
theapplicationoperatorallowsustoconsider(abc)asasingleobjectwithoutconstraining
theviewofitsinnerstructure.Hence,onecaneitherfind(ab)tobeinrelationwithcora
inrelationwith(bc).
Furthermore,wewillusetheclassicalDeductionRulesfor“=”:
Axiom3.2. ∀p,q,r:[p=p],
[p=q]=⇒[q=p],
{[p=q]∧[q=r]}=⇒[p=r]
3.2. DynamicIdentityTriple(DIT). TheDynamicIdentityTriple(DIT)[9]iscomposed
of three specific unary relations: DIT =[x,y,z] with x6=y, y6=z, z6=x, satisfying the
followingtwoaxioms:
Axiom3.3. (xy)=y
Axiom3.4. (zy)=x
Axiom3.3canbeunderstoodasaDistinctionRule(objectyisseparatedfromobjectx)
orasakindofrelationunderwhichyremainsinvariant.
Axiom3.4describestheprocessofreturningtox: yreturnstoxviaz.
Lemma3.1. (zx)=z
Proof. Assume(zx)6=z:
(zy)6=((zx)y)= (z(xy))= (zy)—contradiction
(Ax3.1) (Ax3.3)
(cid:3)
Proposition3.1. AnExtensionalRule∀p,q,x:[(px)=(qx)]=⇒[p=q]isatheoremof
DIT.
Proof. Weproveallpossiblecombinationsof p,qandx:
(1) [(xx)=(yx)]=⇒[x=y]:
x= (zy)= ((zx)y)= (z(xy))= (z(x(xy)))=
(Ax3.4) (Lm3.1) (Ax3.1) (Ax3.3) (Ax3.1)
(z((xx)y))=(z((yx)y))= (z(y(xy)))= (z(yy))=
(Ax3.1) (Ax3.3) (Ax3.1)
((zy)y)= (xy)= y
(Ax3.4) (Ax3.3)
(2) [(xx)=(zx)]=⇒[x=z]:
x= (zy)= ((zx)y)=((xx)y)= (x(xy))= (xy)= y
(Ax3.4) (Lm3.1) (Ax3.1) (Ax3.3) (Ax3.3)
x= (zy)=(zx)= z
(Ax3.4) (Lm3.1)
(3) [(yx)=(zx)]=⇒[y=z]:
x= (zy)= ((zx)y)=((yx)y)= (y(xy))= (yy)
(Ax3.4) (Lm3.1) (Ax3.1) (Ax3.3)
y= (xy)=((yy)y)= (y(yy))=(yx)=(zx)= z
(Ax3.3) (Ax3.1) (Lm3.1)
4 LIDIAOBOJSKA
(4) [(xy)=(yy)]=⇒[x=y]:
x= (zy)= (z(xy))=(z(yy))= ((zy)y)= (xy)= y
(Ax3.4) (Ax3.3) (Ax3.1) (Ax3.4) (Ax3.3)
(5) [(xy)=(zy)]=⇒[x=z]:
y= (xy)=(zy)= x
(Ax3.3) (Ax3.4)
x= (zy)=(zx)= z
(Ax3.4) (Lm3.1)
(6) [(yy)=(zy)]=⇒[y=z]:
y= (xy)= ((zy)y)= (z(yy))=(z(zy))= (zx)= z
(Ax3.3) (Ax3.4) (Ax3.1) (Ax3.4) (Lm3.1)
(7) [(xz)=(yz)]=⇒[x=y]:
(xz)= ((zy)z)= (z(yz))=(z(xz))= ((zx)z)= (zz)
(Ax3.4) (Ax3.1) (Ax3.1) (Lm3.1)
z= (zx)= (z(zy))= ((zz)y)=((xz)y)= (x(zy))=
(Lm3.1) (Ax3.4) (Ax3.1) (Ax3.1) (Ax3.4)
(xx)
z= (zx)=((xx)x)= (x(xx))=(xz)
(Lm3.1) (Ax3.1)
x= (zy)=((xz)y)= (x(zy))= (xx)=z
(Ax3.4) (Ax3.1) (Ax3.4)
x= (zy)=(xy)= y
(Ax3.4) (Ax3.3)
(8) [(xz)=(zz)]=⇒[x=z]:
z= (zx)= (z(zy))= ((zz)y)=((xz)y)= (x(zy))=
(Lm3.1) (Ax3.4) (Ax3.1) (Ax3.1) (Ax3.4)
(xx)
x= (zy)=((xx)y)= (x(xy))= (xy)= y
(Ax3.4) (Ax3.1) (Ax3.3) (Ax3.3)
x= (zy)=(zx)= z
(Ax3.4) (Lm3.1)
(9) [(yz)=(zz)]=⇒[y=z]:
x= (zy)= (z(xy))= ((zx)y)= ((z(zy))y)=
(Ax3.4) (Ax3.3) (Ax3.1) (Ax3.4) (Ax3.1)
(((zz)y)y)=(((yz)y)y)= ((y(zy))y)= ((yx)y)=
(Ax3.1) (Ax3.4) (Ax3.1)
(y(xy))= (yy)
(Ax3.3)
y= (xy)= ((zy)y)= (z(yy))=(zx)= z
(Ax3.3) (Ax3.4) (Ax3.1) (Lm3.1)
(cid:3)
Definition3.2. Wewillcallrabijectiverelationiff:
(1) ∀r,p,x:[(rx)=(px)]=⇒[r=p],
(2) ∀r,x,y:[(rx)=(ry)]=⇒[x=y].
Thus, by a bijective relation we intend a relation which obeysthe extensionalrule as
wellasthesubstitutionpropertyofequality.
Proposition3.2. x,y,zarebijectiverelations.
Proof. Aspect(1)ofDefinition3.2isassuredbyProposition3.1.
Weproveaspect(2)foreverypossiblecombinationofr, pandx:
(1) [(xx)=(xy)]=⇒[x=y]:
x= (zy)= (z(xy))=(z(xx))= ((zx)x)= (zx)= z
(Ax3.4) (Ax3.3) (Ax3.1) (Lm3.1) (Lm3.1)
x= (zy)=(xy)= y
(Ax3.4) (Ax3.3)
(2) [(xx)=(xz)]=⇒[x=z]:
y= (xy)= (x(xy))= ((xx)y)=((xz)y)= (x(zy))=
(Ax3.3) (Ax3.3) (Ax3.1) (Ax3.1) (Ax3.4)
(xx)
x= (zy)=(z(xx))= ((zx)x)= (zx)= z
(Ax3.4) (Ax3.1) (Lm3.1) (Lm3.1)
(3) [(xy)=(xz)]=⇒[y=z]:
x= (zy)= (z(xy))=(z(xz))= ((zx)z)= (zz)
(Ax3.4) (Ax3.3) (Ax3.1) (Lm3.1)
y= (xy)=((zz)y)= (z(zy))= (zx)= z
(Ax3.3) (Ax3.1) (Ax3.4) (Lm3.1)
ARELATIONALAXIOMATICFRAMEWORKFORTHEFOUNDATIONSOFMATHEMATICS 5
(4) [(yx)=(yy)]=⇒[x=y]:
y= (xy)= ((zy)y)= (z(yy))=(z(yx))= ((zy)x)=
(Ax3.3) (Ax3.4) (Ax3.1) (Ax3.1) (Ax3.4)
(xx)
x= (zy)=(z(xx))= ((zx)x)= (zx)= z
(Ax3.4) (Ax3.1) (Lm3.1) (Lm3.1)
x= (zy)=(xy)= y
(Ax3.4) (Ax3.3)
(5) [(yx)=(yz)]=⇒[x=z]:
(yy)= (y(xy))= ((yx)y)=((yz)y)= (y(zy))= (yx)
(Ax3.3) (Ax3.1) (Ax3.1) (Ax3.4)
y= (xy)= ((zy)y)= (z(yy))=(z(yx))= ((zy)x)=
(Ax3.3) (Ax3.4) (Ax3.1) (Ax3.1) (Ax3.4)
(xx)
x= (zy)=(z(xx))= ((zx)x)= (zx)= z
(Ax3.3) (Ax3.1) (Lm3.1) (Lm3.1)
(6) [(yy)=(yz)]=⇒[y=z]:
y= (xy)= ((zy)y)= (z(yy))=(z(yz))= ((zy)z)=
(Ax3.3) (Ax3.4) (Ax3.1) (Ax3.1) (Ax3.4)
(xz)
x= (zy)=(z(xz))= ((zx)z)= (zz)
(Ax3.4) (Ax3.1) (Lm3.1)
y= (xy)=((zz)y)= (z(zy))= (zx)= z
(Ax3.3) (Ax3.1) (Ax3.4) (Lm3.1)
(7) [(zx)=(zy)]=⇒[x=y]:
x= (zy)= ((zx)y)=((zy)y)= (xy)= y
(Ax3.4) (Lm3.1) (Ax3.4) (Ax3.3)
(8) [(zx)=(zz)]=⇒[x=z]:
x= (zy)= ((zx)y)=((zz)y)= (z(zy))= (zx)= z
(Ax3.4) (Lm3.1) (Ax3.1) (Ax3.4) (Lm3.1)
(9) [(zy)=(zz)]=⇒[y=z]:
y= (xy)= ((zy)y)=((zz)y)= (z(zy))= (zx)= z
(Ax3.3) (Ax3.4) (Ax3.1) (Ax3.4) (Lm3.1)
(cid:3)
Note that x, y, z must be distinct. If they were not distinct, our model would either
collapse to a simple formulasimilar to a classical definition of identity, (xx)=x, or we
wouldseparatexfromy:
(1) Ifx=y,then[(xx)= x]and[(zx)= x];
(Ax3.3) (Ax3.4)
with[(xx)=(zx)]=⇒ [x=z]wegetx=y=zand(xx)=x.
(Pr3.1)
(2) Ifx=z,then[(xy)= x]and[(xy)= y],whichimplies[x=y];
(Ax3.4) (Ax3.3)
againwegetx=y=zand(xx)=x.
(3) Ify=z,then[(xy)= y]and[(yy)= x],weget
(Ax3.3) (Ax3.4)
x= (yy)= ((xy)y)= (x(yy))= (xx)and
(Ax3.4) (Ax3.3) (Ax3.1) (Ax3.4)
y= (xy)= ((yy)y)= (yyy),
(Ax3.3) (Ax3.4) (Ax3.1)
whichleadstotheeternalprogression
y=(yyy)= ((yyy)yy)= (yyyyy)=(yyyyyyy)... and
(Ax3.3) (Ax3.1)
x=(yy)= ((xy)y)= (xyy)= ((yy)yy)= (yyyy)=...
(Ax3.3) (Ax3.1) (Ax3.4) (Ax3.1)
Moreover, x, y, z are considered to be “co-essential”, in the sense that two relations
alone,xandyorxandzoryandz,cannotdefineDIT.
Finally, the bijective property of x, y, z assures their dynamic character. DIT [x,y,z]
defined in terms of x, y, z is fully dynamic. For this reason we have called our model
DynamicIdentityTriple.
3.3. DynamicIdentityTriplewithSymmetry(DITS). LetusslightlymodifyDIT and
addasymmetryconditiononz[(yz)=(zy)].
Lemma3.2. [(yz)=(zy)]=⇒[(xz)=z]
6 LIDIAOBOJSKA
Proof. (xz)= ((zy)z)= (z(yz))=(z(zy))= (zx)= z (cid:3)
(Ax3.4) (Ax3.1) (Ax3.4) (Lm3.1)
Lemma3.3. [(yz)=(zy)]=⇒[(xy)=(yx)]
Proof. (xy)= ((zy)y)=((yz)y)= (y(zy))= (yx) (cid:3)
(Ax3.4) (Ax3.1) (Ax3.4)
Lemma3.4. [(yz)=(zy)]=⇒[(xx)=x]
Proof. x= (zy)= ((xz)y)= (x(zy))= (xx) (cid:3)
(Ax3.4) (Lm3.2) (Ax3.1) (Ax3.4)
Lemma3.4providesausefulextensiontoAxiom3.3.
Because the symmetry condition [(yz)=(zy)] is not a theorem of DIT, we define a
DynamicIdentityTriplewithSymmetry(DITS)byaddinganotheraxiom:
Axiom3.5. (yz)=(zy)
Definition3.3. ADynamicIdentityTriplewithSymmetryDITSisamodelsatisfyingAx-
ioms3.3,3.4and3.5.
3.4. Dynamic Generative System (DGS). We can “open” DIT and assume that Ax-
ioms3.3and3.4aretrueforanyvariabley. Usingclassicalquantificationrules,weobtain
amodifiedmodel,whichwewillcallDynamicGenerativeSystem–DGS=[x,y,z], with
thefollowingaxioms:
Axiom3.6. ∀y:(xy)=y
Axiom3.7. ∀y∃z:(zy)=x
AnalogouslytoLemma3.1,weobtain:
Lemma3.5. ∀r:(rx)=r
Proof. Assume∀r:(rx)6=r:
∀r,y:(ry)6=((rx)y)= (r(xy))= (ry)—contradiction
(Ax3.1) (Ax3.6)
(cid:3)
NotethatLemma3.5doesnotexcludethepossibilityofr=x.
Hence,weget(xx)=xbyAxioms3.6and3.7orbyAxioms3.3,3.4andthesymmetry
condition(Axiom3.5).
3.5. Dynamic GenerativeSystemwith Symmetry(DGSS). As in the case of DIT, we
canaddasimilarsymmetryconditiontoDGS.
Axiom3.8. ∀y∃z:(zy)=(yz)=x
Definition 3.4. A Dynamic Generative System with Symmetry (DGSS) is a model that
satisfiesAxioms3.6,3.7and3.8.
Lemma3.6. ∀x,y:[x=y]=⇒∃s,t:[(sx)=(ty)=x]∧[s=t]
Proof. Theexistenceofsandt isassuredbyAxiom3.7:
=⇒ ∃s:x=(sx)
(Ax3.7)
=⇒ ∃t:x=(ty)
(Ax3.7)
Furthermore,Axiom3.7guaranteestheequivalenceof(sx)and(ty):
(sy)=(sx)= x= (ty)=(tx)
(Ax3.7) (Ax3.7)
Finally,weprovetheequivalenceofsandt:
s= (sx)=(s(ty))= (s(yt))= ((sy)t)=(xt)= t
(Lm3.5) (Ax3.8) (Ax3.1) (Ax3.6)
(cid:3)
ARELATIONALAXIOMATICFRAMEWORKFORTHEFOUNDATIONSOFMATHEMATICS 7
Lemma3.7. ∀x,y:[x=y]=⇒∃s,t:[(xs)=(yt)=x]∧[s=t]
Proof. Theexistenceofsandt isassuredbyAxiom3.7:
=⇒ ∃s:x=(sx)
(Ax3.7)
=⇒ ∃t:x=(ty)
(Ax3.7)
Furthermore,Axiom3.7guaranteestheequivalenceof(sx)and(ty):
(xs)= (sx)= x= (ty)= (yt)
(Ax3.8) (Ax3.7) (Ax3.7) (Ax3.8)
s= (sx)=(s(yt))= ((sy)t)=(xt)= t
(Lm3.5) (Ax3.1) (Ax3.6)
(cid:3)
Corollary 3.1. The symmetry condition (Axiom 3.8) assures the uniqueness of z in Ax-
iom3.7: ∀y∃˙z:[(zy)=x].
Moreover,wecanprovethefollowingpropertiesofDGSS:
Lemma3.8. ∀x,y,z:[(zx)=(zy)=x]=⇒[x=y]
Proof. x= (xx)=(x(zy))= ((xz)y)= ((zx)y)=(xy)= y (cid:3)
(Lm3.5) (Ax3.1) (Ax3.8) (Ax3.6)
Analogously:
Lemma3.9. ∀x,y,z:[(xz)=(yz)=x]=⇒[x=y]
Proof. [(xz)=(yz)=x]⇐⇒ [(zx)=(zy)=x]=⇒ [x=y] (cid:3)
(Ax3.8) (Lm3.8)
Finally,weprovethatinDGSSextensionalityandthesubstitutionpropertyofequality
hold:
Proposition3.3. ∀x,y,z:[(zx)=(zy)]⇐⇒[x=y]
Proof.
(1) ∀x,y,z:[(zx)=(zy)]=⇒[x=y]:
[(zx)=(zy)]=⇒ ∃s:[(s(zx))=(s(zy))=x]
(Lm3.6)
[(zx)=(zy)]=⇒[(s(zx))=(s(zy))=x]⇐⇒
(Ax3.1)
[((sz)x)=((sz)y)=x]=⇒ [x=y]
(Lm3.8)
(2) ∀x,y,z:[x=y]=⇒[(zx)=(zy)]:
=⇒ ∀a∃s:(sa)=x
(Ax3.7)
Witha=(zx): x=(sa)=(s(zx))= ((sz)x)=((sz)y)
(Ax3.1)
[x=y]=⇒ [((sz)x)=((sz)y)=x]⇐⇒
(Lm3.6) (Ax3.1)
[(s(zx))=(s(zy))=x]=⇒ [(zx)=(zy)]
(Lm3.8)
(cid:3)
Analogously:
Proposition3.4. ∀x,y,z:[(xz)=(yz)]⇐⇒[x=y]
8 LIDIAOBOJSKA
Proof.
(1) ∀x,y,z:[(xz)=(yz)]=⇒[x=y]:
[(xz)=(yz)]=⇒ ∃s:[((xz)s)=((yz)s)=x]
(Lm3.7)
[(xz)=(yz)]=⇒[((xz)s)=((yz)s)=x]⇐⇒
(Ax3.1)
[(x(zs))=(y(zs))=x]=⇒ [x=y]
(Lm3.9)
(2) ∀x,y,z:[x=y]=⇒[(xz)=(yz)]:
=⇒ ∀a∃s:(sa)=x= (as)
(Ax3.7) (Ax3.8)
Witha=(xz): x=(as)=((xz)s)= (x(zs))=(y(zs))
(Ax3.1)
[x=y]=⇒ [(x(zs))=(y(zs))=x]⇐⇒
(Lm3.7) (Ax3.1)
[((xz)s)=((yz)s)=x]=⇒ [(xz)=(yz)]
(Lm3.9)
(cid:3)
4. APPLICATION IN FOUNDATIONS OF MATHEMATICS
4.1. Relationalsets.
Definition4.1. QisarelationalsetofobjectsxandxisanelementofQifforaspecific
qualityqholds: (qx)=x[x⊑Q,Q= (qx)].
df
Example4.1. Letn bethe qualityofbeingnaturalnumber. Thus, (nx)definestherela-
tionalsetofnaturalnumbersN=(nx). Ifxisanaturalnumber,(nx)=xholds.
If x = y = q then Q = (qq) = q=⇒q⊑q, which can be interpreted as a
df (Axs3.6,3.7)
singleton, a set composedof only one element: q is a relationalset and it is an element
ofitself.
HavinginmindRussell’sparadox[12],onecanaskwhathappenswhenpisconsidered
tobethequalityofnotbeinganelementofitself¬[x⊑x].
LetP= (px),[x⊑P]⇐⇒[x=(px)].
df
¬[x⊑x]⇐⇒¬[x⊑P]⇐⇒[x6=(px)].
Asaresultthereexistobjectswhicharenotrelationalsets.
At the beginningwe said thatour theorycalls to mind mereology,a sortof collective
settheory,firstformulatedbyS.Les´niewski[8]. Mereologyiscollectiveinthesensethat
amereological“set”isawhole(acollectiveaggregateorclass)composedof“parts”and
thefundamentalrelationisthatofbeinga“part”ofthewhole,anelementofaclass. Thus
beinganelementofaclassisequivalenttobeingasubset(properorimproper)ofaclass.
Inthissenseitisclearthateveryclassisanelementofitself.
Letusnowdefinetheconceptofsubset:
Definition 4.2. A relationalset B [B=(bx)] is a relationalsubset of a relationalset A
[A=(ax)],writtenas(cid:2)(bx)⊆(ax)(cid:3),iff∀x:{(cid:2)(bx)=x(cid:3)=⇒[(ax)=x]}.
Letusverifythecases[x=a],(cid:2)x=b(cid:3)and(cid:2)x=a=b(cid:3):
(1) Ifx=athen(ba)=a= (aa)=⇒{[a⊑B]=⇒[a⊑A]}.
(Lm3.5)
(2) Ifx=bthen(bb)=b= (ab)=⇒{(cid:2)b⊑B(cid:3)=⇒(cid:2)b⊑A(cid:3)}.
(Ax3.6)
(3) Ifx=a=bthen[(aa)=a=⇒(aa)=a]=⇒[a⊑A],andanalogously(cid:2)b⊑B(cid:3).
Nowwecanintroducetheconceptoffunction.
Definition4.3. Abinaryrelation f :A7−→Bisafunctioniff
∀x,y,z:{[x⊑A]∧[y,z⊑B]}=⇒{[(fxy)=(fxz)]=⇒[y=z]},
where[x⊑A]≡[(ax)=x]and[y,z⊑B]≡{(cid:2)(by)=y(cid:3)∧(cid:2)(bz)=z(cid:3)}.
ARELATIONALAXIOMATICFRAMEWORKFORTHEFOUNDATIONSOFMATHEMATICS 9
4.2. Peano’sAxiomsasTheoremsofDGSS. Thenextstepwouldbetoseewhetheritis
possibletoredefinePeano’sAxioms[10]intermsofrelationalsets.
Let ustake n instead of x in Axiom3.6, which standsfor the qualityof beingnatural
number.
(nx) definesa relationalset of naturalnumbers[N=(nx),∀x6=n]. We explicitly ex-
clude n=x, because “n”, the quality of being natural number should not be a number
itself.Furthermore,wedefineasmallestnaturalnumberandcallit“1”,inaccordancewith
theoriginalformulationofPeano[10].
Since1isanaturalnumber,(n1)= 1holds.
(Ax3.6)
Thenwedefineasuccessorfunction(inMathematicsconsideredasaunaryoperation)
(1x): (1x)iscalledasuccessorofx.
Lemma4.1. (1x)isafunctionoftype f :N7−→N.
Proof. Wehavetoprovethat(1x)fulfillstherequirementsofDefinition4.3,i.e.
∀a,b,c:[(fab)=(fac)]=⇒[b=c]with f =1,a=x,b=(1x),c=y:
Atfirst,weshow∀x:[(nx)=x]=⇒[(n(1x))=(1x)]:
(n(1x))= ((n1)x)= (1x)
(Ax3.1) (Ax3.6)
Weshownowthat∀x,y:[(1x(1x))=(1x)y]=⇒[(1x)=y]:
∀x,y:[(1x(1x))=(1xy)]⇐⇒ [((1x)(1x))=((1x)y)]⇐⇒
(Ax3.1) (Pr3.3)
[(1x)=y].
(cid:3)
Notethattermslike(1111)areirreducibleandcannaturallybeinterpretedasnumbers.
Weclaim:
Proposition4.1. [(nx),(1x),1]isamodelofnaturalnumbers.
Proof. WeprovethatallPeanoAxiomsaretheoremsofDGSS:
(1) (n1)= 1=⇒ 1⊑N
(Ax3.6) (Df4.1)
(2) ∀x:[(nx)=x]=⇒[(n(1x))=(1x)]:
(n(1x))= ((n1)x)= (1x)
(Ax3.1) (1)
(3) {∀x,y:[(nx)=x]∧[(ny)=y]∧[(1x)=(1y)]}=⇒[x=y]:
∀x,y:[(1x)=(1y)]=⇒ [x=y]
(Pr3.3)
(4) ∀x:[(nx)=x]=⇒(1x)6=1:
Assume∃x:(1x)=1forx6=n:
(n1)= 1= (1n)
(1) (Lm3.5)
[(1x)=(1n)]=⇒ [x=n]—contradiction.
(Pr3.3)
(5) {[M⊆N]∧[1⊑M]∧{∀k:[k⊑M]=⇒[(1k)⊑M]}}=⇒[M=N]:
With[x⊑M]≡[(mx)=x]:
(m1)=1= (1m)
(Lm3.5)
∀k:[(mk)=k]⇐⇒ [(1(mk))=(1k)]⇐⇒ [((1m)k)=(1k)]⇐⇒
(Pr3.3) (Ax3.1)
[((m1)k)=(1k)]⇐⇒ [(m(1k))=(1k)]
(Ax3.1)
=⇒ (n1)=1
(1)
=⇒ ∀k:[(nk)=k]=⇒[(n(1k))=(1k)]
(2)
∀k:{[(mk)=k]∧[(nk)=k]}=⇒ ∀k:[(mk)=(nk)]≡ [M=N]
(Ax3.2) (Df4.1)
(cid:3)
Naturalnumberscanbedefinedasfollows:
10 LIDIAOBOJSKA
(1) 1isanaturalnumber.
(2) Thesuccessorof1is(11)= 2
df
(3) (12)=(1(11))=((11)1)=(21)= 3
df
(4) (13)=(1(12))=((11)2)=(22)=(2(11))=((21)1)=(31)= 4etc.
df
4.3. The Number Zero. To reconstructnaturalnumberswe have been consideringx in
Axiom 3.6 as a kind of quality and we assumed that n=x6=x∀x. Thus n should be
differentfromallnaturalnumbers.
We now define another fundamental number “0” and put n=0. We can verify that
Proposition4.1holdsforeveryx6=0:
(1) (01)=1
(2) ∀x6=0:[(0x)=x]=⇒[(0(1x))=(1x)]
(3) {∀x,y6=0:[(0x)=x]∧[(0y)=y]∧[(1x)=(1y)]}=⇒[x=y]
(4) ∀x6=0:[(0x)=x]=⇒(1x)6=1
(5) {[(mk)⊆(0k)]∧[(m1)=1]∧
{∀k:[(mk)=k]=⇒[(m(1k))=(1k)]}}=⇒[m=0]
If we would allow x=0 in Proposition 4.1, we would producea contradictionin the
fourthPeanoAxiom:[(10)6=1]and[(10)=1]byLemma3.5wouldimply[16=1].
ThankstoAxiom3.6,whichstates∀x:(0x)=xwecaneasilydefineallnaturalnum-
bersbeginningfrom0:
Example4.2. 2isanaturalnumber,hence(02)=2.
(02)= (0(11))= ((01)1)= (11)= 2.
df (Ax3.1) (1) df
Many mathematicians introduce 0 as a first natural number, but in RC 0 cannot be
consideredanaturalnumber.ThankstoLemma3.5,i.e.(10)=1,read“1isthesuccessor
of 0”, the intended meaning of 0 and 1 is fully preserved, and no additional axioms are
needed.
5. CONCLUSIONS
WefoundtheconceptofFundamentalRelation,asdefinedbyE.DeGiorgi,necessary
butnotsufficienttoexpressadynamicbehaviorofrelations.Todothis,wehaveintroduced
threedistinctunaryrelationswhichformaDynamicIdentityTriple(DIT).Thethreebasic
relationsobeytwoverysimpleaxiomsandifdesiredathirdaxiomofsymmetry(DITS)can
beadded.Therelationsinamodelinteractinawaythatanytwoofthemcannot“operate”
atthesametime:aphenomenonwhichcanbecharacterizedbytheworddynamic.
Usingclassicalrulesofquantification,wemodifythebasicmodelsandobtaintheDy-
namic Generative System (DGS), which can also be enhanced by a symmetry condition
(DGSS).AfterdefiningrulesforaRelationalCalculus,wedefinesomebasicsettheoretic
notions,theconceptoffunctionandfindthatPeano’sAxiomsandextensionalityandthe
substitutionpropertyofequalityaretheoremsofDGSS.Inaddition,itbecomesclearthat
thenumberzerocanbeaddedtothesystemofPeano’sAxioms,butcannotbeconsidered
anaturalnumber.
REFERENCES
[1] Curry,H.B.:FoundationsofMathematicalLogic.McGraw-Hill,NewYork(1963)
