N. C. A. DA COSTA AND F. A. DORIA SUPPES PREDICATES AND THECONSTRUCTION OF UNSOLVABLE PROBLEMS INTHE AXIOMATIZED SCIENCES ABSTRACT. We firstreviewour previous workonSuppespredicates andthe axiomati- zationof the empiricalsciences. We then state someundecidability andincompleteness results in classical analysis that lead to the explicit construction of expressions for characteristic functions in all complete arithmetical degrees. Out of those results we show that for any degree there are corresponding 'meaningful' unsolvable problems in any axiomatizedtheory that includes the language of classical analysis. Moreover wealso show that withinour formalizationthere are 'natural' unsolvableproblems and undecidablesentences whichare harder than any arithmeticproblem. As applications wediscuss a 1974 HilbertSymposiumproblemby Arnold on the existence of algo- rithms for the decision of properties of polynomial dynamical systems over Z, prove the incompleteness of the theory of finishNash games, and delve onrelatedquestions. Neither forcing nor diagonalizations are used in those constructions. 1. INTRODUCTION Suppes predicates were the starting point in the recent development of a technique for the construction of both algorithmically undecid- able sets of objects in physics and undecidable 'meaningful' sentences about physical objects. (See for details (da Costa, 1988; Suppes,1967, 1988).) That technique allowed us to prove the undecidability and incompleteness of most of classicaland quantum physics, providedthat they are givena first-order axiomatization (through Suppes predicates) that includes the language of classical elementary analysis (da Costa, 1991 a, 1991b, to appear, 1994 a; Stewart, 1991). Actual examples dealt with the proof of the incompleteness of chaos theory (da Costa, 1991 a) and with the related question of the existence of problems in dynam- ical systems theory whose solution is equivalent to solving very hard Diophantine problems, such as Fermat's Conjecture (da Costa, 1994 a). The same techniquereached beyondphysics and led to the proof of the incompleteness of the theory of Hamiltonian models for the dynamics of economical systems (Lewis, 1991b), while providing a partial result related to the recent proof by Lewis of the noncomputability of Arrow- Debreu equilibria (da Costa, 1992 d; Lewis, 1991 a). We can still list 151 P. Humphreys (cd.), Patrick Suppes: Scientific Philosopher, Vol. 2, 151-193. @ 1994 Kluwer Academic Publishers. Printedin theNetherlands.
ABSTRACT. We firstreviewourpreviousworkonSuppespredicatesandtheaxiomati-zationof theempiricalsciences. We thenstate someundecidability andincompletenessresults in classical analysis that lead to the explicit construction of expressions forcharacteristicfunctions inall complete arithmeticaldegrees. Out of thoseresults weshow that for any degree there are corresponding 'meaningful' unsolvable problemsin any axiomatizedtheory that includes the language of classical analysis. Moreoverwealso show that withinour formalizationthereare 'natural' unsolvableproblemsandundecidablesentences whichare harder than any arithmeticproblem. As applicationswediscuss a 1974HilbertSymposiumproblemby Arnold on the existence of algo-rithmsfor the decisionof properties of polynomialdynamical systems over Z, provethe incompletenessofthe theory of finishNash games,anddelveonrelatedquestions.Neither forcingnor diagonalizations are used in thoseconstructions.
1. INTRODUCTION
Suppespredicates were the starting point in the recent developmentof a technique for the construction of both algorithmically undecid-able sets of objects inphysics and undecidable 'meaningful' sentencesaboutphysicalobjects. (See for details (da Costa,1988; Suppes,1967,1988).) That technique allowed us to prove the undecidability andincompletenessof mostof classicalandquantum physics,providedthatthey are givena first-order axiomatization (through Suppespredicates)that includes the language of classical elementary analysis (da Costa,1991a,1991b, to appear,1994a;Stewart,1991). Actual examplesdealtwith theproofof the incompleteness of chaos theory (daCosta, 1991a)and with the related question of the existence of problems in dynam-ical systems theory whose solution is equivalent to solving very hardDiophantine problems, such as Fermat's Conjecture (daCosta, 1994a).The same techniquereachedbeyondphysics andled to theproofof theincompleteness of the theory of Hamiltonian models for thedynamicsof economical systems (Lewis,1991b), whileproviding a partial resultrelated to the recentproofbyLewis of the noncomputability of Arrow-Debreu equilibria (da Costa, 1992d; Lewis, 1991a). We can still list
among its consequencestwoexamples that are discussedin the presentpaper: a resultonone ofArnolds problemsconcerningalgorithms forpropertiesofpolynomial dynamicalsystems over the integers (Arnold,1976a) and theproofof the incompletenessof the theoryof finite gameswith Nashequilibria (da Costa,1992d).
Our main theorems depend in an essential way on a lemma ofRichardson (Richardson,1968);noncomputabilityinaxiomatizedphys-ical theories was anticipated by Scarpellini (Scarpellini, 1963) andbyKreisel (Kreisel,1976). Also, we wish to emphasize that no forcingisneeded for our independenceresults.
We obtain here a whole plethora of new intractable questions inSuppes-axiomatizedtheories. Thepresent resultsare new in thefollow-ingsense: allourpreviousexamples for undecidabilityand incomplete-ness withinaxiomatizedphysicscanbe formally reduced toelementaryarithmetic problems. However, that reduction cannot always be madein the present case, as some of our new examples are not elementarynumber-theoretic problems in disguise; they stand beyond the pale ofarithmetic.
There are even weirder situations: we obtain formal expressionsthat describe physical systems such that nothing but trivialities can beprovedabout them. For wecanexplicitlyconstructundecidable familiesofobjects within aclassical first-order languageLt such that:
No nontrivial properties of those families can be algorithmicallydecided.No assertion about the system can be reduced to an arithmeticassertion,thatis tosay, the systemlies fully outside thearithmeticalhierarchyand belongs to the nonarithmetical portion of set theory(if weare working,say, withinZermelo-Fraenkel set theory).Those results are consequencesof general incompleteness theo-rems that apply to anynontrivialpropertyPin the theoryT; thosetheorems extend a previous one (Proposition 3.28 in (da Costa,1991a) that originated in a suggestedbySuppes.
Againwehavea correspondingincompleteness theoremas thereareformal expressions for systemsall of whose nontrivial properties mustbe formulated as undecidable sentences. (Again no propertyof thosesystemscanbereduced to anarithmetic property.) Thoseundecidabilityand incompleteness results are to be found below in Propositions andCorollaries 3.28, 3.30,3.37, 3.41,3.47,3.49, 4.1.
153SUPPESPREDICATESANDUNSOLVABLEPROBLEMS
Section 2 of this paperreviews the theory of Suppes predicates forempirical theories in the da Costa-Chuaqui version (da Costa, 1988,1990b, 1992a). Section 3 deals with the undecidability and incom-pleteness of classical analysis in its several aspects; the main pointsare theexplicitconstruction of expressions for characteristic functionsof any complete degree in the arithmetical hierarchy, and the gener-alincompleteness theorems about expressionsofelementary functions.Section 4discusses ourexamples,whileSection5commentson possibleimplications of our results.
Preliminary Concepts andNotation
We are here interested in formal languages strong enough to representall the usual mathematical theories. For a review of the main ideassee (da Costa, 1990b, 1991a). Those formal languages are built outof a finite alphabet, and its sentences- 'meaningful assertions' -arefinite sequencesof letters from thebasic alphabet. We therefore reduceeverything to finite sequencesof letters. (As an example,an intuitivelyinfinite set suchasastraight lineontheCartesianplaneR2isrepresentedby the finite sequenceof symbols {(x,y) € R2 :y =2x 4- I}, abbre-viations such as R for the set of realsbeing allowed as their definitionsare reducible to finite sequencesof letters from the theory's alphabet.)Different formal expressions may represent the same 'intuitive' mathe-matical object;however in most everydaysituations ourformal systemswill not be strong enoughto decide,given twoexpressions,whether ornot they represent the same object, even if they are strong enough toproveallusual mathematical results.
To be more specific, we suppose that our theories are formalizedwithina first-order classical predicatecalculus with equality. (It is alsoconvenient to suppose that our formal theories T include Russell's isymbol; in that case the extended theory is a conservative extensionofthe theory without that particular variable-binding termoperator.)
We follow the notationof (daCosta, 1991a) with afew changes thatare explicitly indicated; in particular v will denote the set of naturalnumbers, Z is the set of integers, and R are the real numbers. LetTbe a first-order axiomatic theory that contains formalized arithmeticN and such that T is strong enough to include the concept of set andclassical elementary analysis. (We can simply take T= ZFC, whereZFC is Zermelo-Fraenkel set theory with the axiom of choice.) IfLt
154 N.C. A. DA COSTA AND F. A. DORIA
is the formal language of T, we suppose that we can form within Ta recursive coding for Lt so that it becomes a setLt G T of formalexpressions in T. Objects in T will be denoted by lower case italicletters, such as x,y, z, f,g, a,...Predicates in T will be noted P,Q,. ..The use of Greek letters and more particular notational features(suchas p, qfor polynomial functional symbols) will be clearfrom thecontext. (Predicates are openone-variable formulae in the language ofT.)
Fromtime totime we willplay with thedistinctionbetween anobjectand the expression in Lt that represents it. If x,y are objects in anintended interpretationof thetheoryT,theyareingeneralnotedby term-expressions £, £ thatbelongto the formal languageLt ofT. Ingeneralthere is no 1-1correspondencebetween objects and expressions; thuswemayhavedifferent expressionsfor the same functions: 'cos 7r' and'0' are both expressions for the constant function 0. If £ is a set in anintended interpretationofT, wenoteby \x] a setof expressionsfor theelements in x. We allow the following abuseof language: predicates Psometimes apply to objectsin Tandsometimes apply to expressionsinLt(P(OY> meaning will be clear from context.
Inparticular we notice that since our theory Tincludes formalizedarithmetic N, we will sometimes need the distinction between a partialrecursive function and thealgorithm that computes it. For any compu-tation 4>(x), (f>(x)Imeans that the computation converges (stops andproduces an output), while (f>(x)|means that thecomputation diverges(entersa never-endingloop).
We emphasize that proofs in T are algorithmically defined ways ofhandling the objects ofLt', for the conceptof algorithm see (da Costa,1991a;Rogers,1967).
2. SUPPES PREDICATES
Suppespredicates (or Bourbaki species of structures) were first usedby Suppesin the 'fifties as a way of directly axiomatizing any mathe-matically-based theory. Suppes'smain contention is that"to axiomatizea theory is to define a set-theoretic predicate." A formal treatment wasthengivenbydaCostaandChuaquiin1988,andimmediate applicationsensued. (For details, applications and references see (Bourbaki, 1957,1968; daCosta 1988, 1990b, 1992a;Suppes, 1967, 1988). Everything
SUPPES PREDICATES ANDUNSOLVABLEPROBLEMS 155
here is supposed tohappen within our arithmetically consistent theoryT.
Structures andPredicatesA mathematical structure w is afinite orderedcollection ofsets (whichmay be particularized to relations and functions) of finite rank over theunion of the ranges of two finite sequencesof sets, xj,xi,...,xm andy\,yii ■ " " 2/n» wherem>0andn > 0.If wearedoingourconstructionswithin ZFC, w is thus aZFC set. Thexs and the ys are called thebasesetsofw; the xs are theprincipalbasesets,while the ysare the auxiliarybasesets.
The auxiliarybase sets canbe seen as previously defined structures,while the principal base sets are 'bare' sets; for example, if we aredescribing a real vector space, the set of vectors is the only principalbase set, while the set of scalars,R, is the auxiliary base set.
ASuppespredicateoraspeciesofstructuresin the senseofBourbakiisa formula of set theory whose only free variables are thoseexplicitlyshown:
P defines it; as a mathematical structure on the principal base setsx\,...xm,withtheauxiliarybase sets y\,...,yn,subject to restrictionsimposed on w by the axioms we want our objects to obey. As theprincipal sets x\, ...vary over a class of sets in the set-theoreticaluniverse,we get the structures of speciesP,orP-structures.
The Suppespredicate is a conjunction of two parts: one specifiesthe set-theoretic process of construction of the P-structures,while theother imposesconditions that mustbe satisfiedby the P-structures. Thissecond piece contains the axioms for thespecies of structures P.
We write theSuppespredicate for ageneralw as follows:
The auxiliary sets are seen as parameters in the definition of w. Allofeverydaystandard 'professionals' mathematics canbe formalizedalongthoselines.
DeducedandDerivedStructuresGiven a structure wof species P(w,x\,... ,xm,2/1,... ,yn), let 21,... ,zp,bep(p> 0) sets of finite rank over the union of rangesof thesequences
5" " ",%mi y\i
- - " yn,
also let v\,...,vq (q > 0)be qarbitrary sets. If the Suppespredicate
defines w* as a structure on the principal base sets z\, ... with the v\,...as auxiliary base sets, we say that the structure w* of speciesP* isdeducedfrom the structure w of speciesP.
Wecan obtain new structures out of (sets of) already defined struc-tures by the means of two basicprocedures:
1.With the help of set-theoretic operations, such as Cartesian prod-ucts, passages to the quotient, and the above-described operationof deductionof structures;
2. Through the imposition of new axioms to already existing set-theoretic structures.
Therefore we can introduce the notion of derived structure. Whenwe define anew structure w fromaset s ofother structures with the helpof the twoprocedures described above, we say that w is derivedfromthe structures s. The Suppespredicate of w canbe expressedin termsof the Suppespredicates of theelements of s. The conceptofdeductionof structures is aparticular case ofderivation of structures.
The set s is the set of groundstructures for w.Finally, let w and w1be twostructures of species Pand P', respec-
tively. We suppose that P andP'differ only in connection with theirsets of axioms,but that the conjunction of the axioms of P' implieseach axiom of P, with quantifiers restricted to sets of finite rank overtheunion of the rangesof the base sets for w. If that is the case,we saythat the P'-structure is richer than the P-structure (or thatP' is richerthan P). For instance, the species of commutative groups is richer thanthespeciesof groups.
The Q'-structure g' is then derived from the Q-structure g if Q' isricher thanQ,or Q'canbeobtained from Qin the way we havealreadydescribed above. The above ideas can also be extended to the conceptofpartial structures introduced by (da Costa, 1990a).
P*{w*,zi,...,zp,vi,...,vq)
SUPPES PREDICATES AND UNSOLVABLEPROBLEMS 157
TheAxiomatics ofEmpirical TheoriesAs a first approximation we seeempirical theories as triples
where (i) Mis aSuppes-Bourbakispecies of mathematical structures;(ii) A is the theory's 'domain of definition',and(iii) pgives the 'inter-pretation rules' or 'characteristic examples' that relate Mand A. Wecan be more specific about (ii) and (iii), however,as we did elsewhere(da Costa, 1992c); in any case we consider A to be a set-theoreticconstruct. (In that case p in general contains nonrecursive aspects (daCosta,1991a, toappear, 1992a).)
We follow the usual mathematical notation in this subsection. In par-ticular,Suppespredicates are written in a more familiar but essentiallyequivalent way. Therefore somesymbols willhave a different meaningthanin theremainingportions of thepaper.
The species of structures of essentially all physical theories canbe formulated as particular dynamical systems derived.from the P =(X,G,fi), where X is a topological space, G is a topological group,andp is a measure on a setof finite rank over XUG. Thus we can saythat the mathematical structures of physicsarise outof the geometryofa topological space X: physical objects are those that exhibit invari-ance properties with respect to the action of G and the main speciesof structures in 'classical' theories can be obtained out of two objects,a differentiable finite-dimensional real Hausdorff manifold Mand afinite-dimensional Lie group G.
DEFINITION 2.1. The species of structures of a classical physicaltheory is givenby the9-tuple
1. The GroundStructures. (M,G), where Mis a finite-dimensionalreal differentiablemanifold andGisafinite-dimensional Liegroup.
A=(M,A,p),
E= (M,G,P,T,A,I,G,B,V<p = l)which is thus described:
158 N.C. A.DACOSTA AND F. A. DORIA
2. The Intermediate Sets. A fixedprincipal fiber bundle P(M,G)over M with G as its fiber plus several associated tensor andexteriorbundles.
3. TheDerivedFieldSpaces.Potential spaceA,field spaceTand thecurrentor source spaceX. A,TandTarespaces (in general,man-ifolds) of cross-sectionsof thebundles that appear as intermediatesets in our construction.
4. AxiomaticRestrictions on theFields. Thedynamicalrule Vy? = t,the relation ip = d(a)a between a field <p c T and its potentiala c A, together with the corresponding boundary conditions B.Here d(a) denotes a covariant exterior derivative with respect totheconnection form a,and V a covariantDirac-like operator.
5. The Symmetry Group. Q C Diff(M) ® Q', where Diff(M) isthe group of diffeomorphisms of M and Q' the group of gaugetransformations of the principalbundle P.
6. The Space ofPhysicallyDistinguishableFields. If /C is one of theT, A or Z field manifolds, then the space of physically distinctfields is
K./G.
(In more sophisticated analyses wemust replace our conceptof theoryfor a more refined one. Actually in the theory of science we proceedas in the practice of science itself by the means of better and betterapproximations. However for the goalsof thepresentpaperour conceptofempirical theory isenough.)
We show elsewhere (da Costa, 1992a) that what one understandsas the classical portion ofphysics fits easily into the previous scheme.We discussindetail two examples,Maxwellian theory andHamiltonianmechanics.
Maxwell's Electromagnetic Theory
Let M= R4,with its standard differentiable structure. Let us endowMwith the Cartesian coordination induced from its product structure,and let 7/ = diag(— l,+l,+l,+l) be the symmetric constant metricMinkowskian tensoronM. If the F^v {x) are components of a differ-
SUPPES PREDICATES ANDUNSOLVABLEPROBLEMS 159
entiable covariant2-tensor field onM,p,v = 0, 1,2,3, thenMaxwell'sequations are:
Thecontravariant vector fieldwhosecomponentsaregivenby the setoffour smooth functions j^(x) onMis the current thatserves as sourcefor Maxwell's fieldF^. (We allow piecewise differentiable functionsto account for shock-wave-like solutions.)
Itisknown thatMaxwell's equations areequivalent to the Dirac-likeset
and
(where the {7^ :p — 0,1,2,3} are the Dirac gamma matrices withrespect to 77). Those equation systems are tobe understood togetherwith boundary conditions that specify aparticular field tensorF^ 'outof the source jv (Doria, 1977).
The symmetrygroup of theMaxwell field equations is theLorentz-Poincare group that actsonMinkowski spaceMand inan induced wayonobjectsdefined overM.However,since weare interestedin complexsolutions for the Maxwell system, we must find a reasonable way ofintroducingcomplexobjects inour formulation. One may formalize theMaxwellian system as agauge field. We sketch the usual formulation:again we start fromM= (R4,rj), andconstruct the trivial circlebundleP — M x Sl over M, since Maxwell's field is the gauge field of thecirclegroup 51(usually writtenin thatrespectasU(1)). We form the set£ of bundles associated to P whosefibers are finite-dimensional vectorspaces. Theset of physical fields in our theory is obtained out of some
<v^=r,
d^p+ dpFay+dvFpu= 0
V(/? = L,
where
tp = (1/2)^7^,
t-Jul,fi,
V =ypdp,
160 N.C. A.DACOSTA AND F.A.DORIA
of the bundles inS: the set of electromagnetic field tensors is a set ofcross-sectionsof thebundle F =A20s1(M)of all 2-formsonM, where s1 is the group'sLie algebra. To be more precise, the setof all electromagnetic fields is T C Ck(F),if we are dealing with Ck
cross-sections (actually a submanifold in the usual Ck topology due tothe closurecondition dP =0).
Finally we have twogroup actions onT: the first oneis theLorentz-Poincare action L which is part of the action of diffeomorphisms ofM; then we have the (here trivial) action of the group Q1 of gaugetransformations of P when acting on the field manifold T. As it iswellknown,its action is not trivial in thenon-Abelian case. Anyway,it always has a nontrivial actionon the space A ofall gauge potentialsfor the fields in T. Therefore we take as our symmetry group Q theproduct L ® Q' of the (allowed) symmetries ofMand the symmetriesof theprincipal bundle P.Formathematical details see (Doria, 1981).
We must also add the spaces A of potentials and of currents, X,as structures derived from M and Sl. Both spaces have the sameunderlying topologicalstructure; they differ in the way the group Q' ofgauge transformations acts upon them. We obtainI=A1® sl(M) andA=1=Ck(I). Notice thatXjQ' =X while A/Q' A.
Therefore we cansay that the 9-tuple
whereMis aMinkowski space,and B is a set ofboundary conditionsfor our field equations V</? = t, represents the species of mathematicalstructures of aMaxwellian electromagnetic field,where P,TandQarederived fromMand Sl.TheDirac-like equation
should beseen as an axiomatic restriction on our objects;theboundaryconditions B are (i) aset of derived speciesof structures from MandS\ since, as we are dealing with Cauchy conditions,we must specifya local or global spacelike hypersurface C in Mto which (ii) we addsentencesof the form Vx G Cf(x) = /o(x), where /o is a setof (fixed)functions and the / areadequate restrictions of the field functions andequations toC.
{M,SI,P,T,A,g,X,B,V<p = i)
V<£> =L
SUPPES PREDICATESAND UNSOLVABLEPROBLEMS 161
HamiltonianMechanicsHamiltonian mechanics is the dynamics of the 'Hamiltonian fluid'(Arnold, 1976b). Ourgroundspecies ofstructuresarea2n-dimensionalrealsmooth manifold,and the real symplectic group Sp(2n,R). Phasespaces in Hamiltonian mechanics are symplectic manifolds: even-dimensionalmanifolds likeMendowed with asymplectic form, that is,a nondegenerateclosed 2-form Q, on M. The imposition of that formcan be seen as the choice of a reduction of the linear bundle L(M)to a fixed principal bundle P(M,Sp(2n,R));however givenone suchreduction it does not automatically follow that the induced 2-form onMis a closedform.
All other objects are constructed in about the same way as in thepreceding example. However, we must show that we still have here aDirac-like equation as the dynamicalaxiomfor the species ofstructuresof mechanics. Hamilton's equations are
where %x denotes theinterior product with respect to the vector field XoverM, andhis theHamiltonian function. Thatequationis (locally,atleast) equivalent to:
or
whereLxis theLie derivative withrespectto X.Theconditiondip = 0,with (f = ix&>, is the degenerateDirac-like equation for Hamiltonianmechanics. We do not get a full Dirac-like operator V d becauseM, seen as a symplectic manifold,does not havea canonical metricalstructure,so that wecannotdefine (through the Hodgedual) acanonicaldivergence 6 dual to d. The group that acts onM with its symplecticform is the group of canonical transformations; it is a subgroup of thegroup of diffeomorphisms ofM so that symplectic forms are mappedontosymplectic forms underacanonical transformation. We can take as'potential space' thespace of allHamiltonians onM(which is arathersimple function space),andas 'fieldspace' thespaceofall 'Hamiltonianfields' of the form ixsl-
ix& — —dh,
LxSl =0,
d{ixQ) =0,
162 N. C. A. DA COSTA ANDF A. DORIA
The constructionofSuppespredicates for gravitation theory(generalrelativity), classical gauge fields, Kaluza-Klein unified field theories,and Dirac'selectron theory seen asa classical field theory can be foundin (da Costa, 1992a). We notice that Dirac-like dynamical equationshavebeenobtained for all those (Doria, 1975, 1986). Themathematicalbackground is in (Cho,1975;Dell, 1979;Kobayashi, 1963).
3. UNDECIDABILITY ANDINCOMPLETENESS
Wenow review previousmaterial andobtainnew resultson theundecid-ability and incompleteness of classical analysis; the chief new resultsare the construction of several intractable problems and undecidablesentencesin Twhichcannotbe reduced(in T) toarithmetical problems.
DEFINITION3.1. Tis arithmetically consistent if andonly if the stan-dardmodel Nfor N is a model for the arithmetic sentencesofT. ■
Now let \V~\ be the algebra of polynomial expressions on a finitenumber of unknowns over the integers Z; we identify \V\ with theset of expressions for Diophantine polynomials in T. Let \£~\ be theset of expressions for real elementary functions on a finite number ofunknowns,while [JF] is the set of expressions for real-valuedelemen-tary functions on asingle variable (daCosta, 1991a).
Givenapolynomialexpressionp(x\,...,xm,yi,...,2/n),letrm(xi,,xm) be the function that effectively codes m-tuples of natural
numbers (xi, ..., xm) by a single natural number (Rogers, 1967,
Let us inductively construct out of a polynomial qm{x\,...,xn) anRn-defined and R-valued function aqm given through the followingsteps:
InitialStep. Suppose that we are given the expressions qm as below:
"Ifqm(x\,...,xn) =c, where cis a constant, then we put aqm=
|c|+2;"If qm{x\,"" ",xn) = xi, then aqm =x]+2.
p. 63). Letr =rm((...». We abbreviatep(xi,.. .,xm,2/i,...,y„) =Pr(yu---,yn)-
SUPPES PREDICATES AND UNSOLVABLEPROBLEMS 163
Induction Step. We suppose that qm is given as indicated. We thenobtain the corresponding aqm as follows:
Supposenow that weisolate someof the variables in ourpolynomi-als as parameters. Then
Hereg is composedn times.Given apolynomial expressionpm(x\,...,xn) 6 \V~\,we define:
DEFINITION 3.4. The maps i' : \V\ -+ \T] and i" : \V] -+ \T],givenby:
whereais Richardson's FirstMap;and
areRichardson 'sSecondMapof the first (i1)and second {i")kinds. ■
COROLLARY 3.5. Given apolynomial expressionpm G Lt, thereis an algorithm that allows us to obtain expressions t!pm £Lt andi"Pm €Lt for the imagesofpmunder Richardson 'sSecondMap.
Proof. Immediate, from the definition of i'and i".We assert:
PROPOSITION3.6 (Richardson's Functor). Letpm(xux2,...,xn) =0be afamily ofexpressionsforDiophantine equationsparametrizedbythepositive integer m in an arithmetically consistent theory T. Thenthere is an algorithmic procedure a : \V~\ — * \£~\ such that out ofpm €V we canobtain an expression
fm €£, such that fm =0ifandonly iffm <1ifandonlyifthere arepositive integersx\,x 2
,... xnsuch thatpm(x\,...,xn) =0.Moreover, therearealgorithmic proceduresi',i":P — * Tsuch that
we can obtain out ofanexpressionpm two other expressionsfor one-variable functions,gm(x) = i'pm(x\,...)andhm(x) = i"pm(x\,...)such that there arepositive integers x\,...withpm(x\,...) =0ifandonly ifgm{x) =0andhm(x) < 1,forall real-valued x.
(2) Pm(xi,...,Xn) »-> t"[pm(xu... ,X„)](x)= t /[pm(xi,...,X„)](x) - \ ,
Jm\X\,X 2, ..",Xn) —Q!Pm \XliX 2, "" " ,Xn),
SUPPES PREDICATESAND UNSOLVABLEPROBLEMS 165
Proof See (da Costa, 1991a).
PROPOSITION3.7. IfTis arithmetically consistent,and ifweadd theabsolute valuefunction \x\to \T] andcloseit to obtainanextendedsetofexpressions \T*~\, we have:
1. (Undecidability) We canalgorithmically constructin Tadenumer-able family ofexpressionsfor real-valued,positive-definite func-tions km(x) > 0 so that there is no generalalgorithm to decidewhether onehas,forall realx,km(x) =0.
2. (Incompleteness) ForamodelMsuch that Tbecomes arithmeti-cally consistent, there is an expressionfora real-valued functionk(x) such thatM |= Vx <E Rfc(x) =0 while TV- Vx € Rfc(x) =0andTY3x G Rfc(x) 0.
Proof. See (da Costa, 1991a).
If km (as inProposition 3.7)results out ofpm,we writekm= Xpm-
EqualityIs Undecidable inLtCOROLLARY 3.8. IfTisarithmetically consistent then for anarbi-trary real-defined and real-valued function f there is an expressionf G Lr such thatM f= £ = /,whileTY^= fandT V- -.(£ = /).
Now let Mn(q) be the Turingmachine of index n that acts upon thenatural number q (Rogers, 1967). Let 9(n,q) be the halting functionfor Mn(q), that is,0(n,q) =1if and only ifMn{q) stops over q, and0(n,q) =0ifand only ifMn(q) does not stop overq.
We need adefinition and a lemma.
DEFINITION3.9. For the followingreal-defined andreal-valued func-tions:
(i) m={-*:*<2:
Proof. Put £ =/+k(x), for fc(x) as inProposition 3.7.
166 N.C. A. DACOSTA ANDF. A. DORIA
LEMMA 3.10. IfT is (arithmetically) consistent, then each of thefollowing operationsgenerates the others within T:
(1) +,x,|
(2) +, X,7/(...).
Proof. Immediate.
Let pn,q(xi,X2,...,xn) be a universal polynomial (Jones, 1982).Since \T*~\ has an expression for |x| (informally one might have|x| =+Vr),ithasanexpression for thesign functiona(x). Thereforewe can algorithmically build within the languageof analysis(where wecan express quotients and integrations) an expression for the haltingfunction 6{n,q):
PROPOSITION 3.11 (The Halting Function). IfT is arithmeticallyconsistent, then:
/-n / \ fl,a;>0,(2) "M-10,x< 0.
/ox" f x — y, x — y >0,(3) x-y= { r,
y' ~nv J y [0, x—2/ <0.
( +1, x > 0(4) a{x) = < 0, x=0
[ -1, x<o
(3) +,x,(...-...).
(4) +,X,<7(...).
o(n,q) = a(Gnjq),
+00 2
a - f Cn.i(x>ce~xdxG"'""J 1+Cn,,(i)"*'
SUPPESPREDICATES AND UNSOLVABLE PROBLEMS 167
Proof See (da Costa,1991a).
There follows:
PROPOSITION 3.12. IfT is arithmetically consistent then we canexplicitly and algorithmically construct in Lt an expression for thecharacteristic functionofasubsetofujofdegree0".
Remark 3.13. That expression dependsonrecursive functions definedon ujandon elementary real-defined andreal-valued functions plus theabsolute value function,aquotient and an integration, as in the caseofthe9 function givenby Proposition3.11. ■
Proof. Wecouldsimply use Theorem9-IIin (Rogers, 1967,p.132).However,for the sakeofclarity wegiveadetailed,albeit informalproof.Actually the degree of the set described by the characteristic functionwhoseexpressionweare going to obtain willdependonthe fixed oracleset A; soour construction is ageneral one.
LetA C ujbe a fixedinfinite subsetof the integers:
DEFINITION3.14. The jumpofA is written A';A1= {x :<j>£{x) |},where 4>^ is the A-partial recursive algorithm of index x. ■
1. An oracle Turing machine (f>^ with oracle A can be visualizedas a two-tape machine where tape 1is the usual computationaltape, while tape 2 contains a listing of A. When the machineenters the oracle state so, it searches tape 2 for an answer to aquestionof theform 'is w G AT Onlyfinitely manysuch questionsare asked during a converging computation; we can separate thepositive andnegative answers into two disjoint finite sets DU(A)and D*(A) with (respectively) the positive andnegative answersto those questions; notice that Dv C A, while D* Cuj— A. Wecan view these sets asorderedk- and A;*-pies;vand v arerecursivecodings for them (Rogers, 1967). The DU(A) and D*(A) setscan be coded as follows: only finitely many elements of A arequeriedduring an actual convergingcomputation with input y; if
) — ,...,Xr ).
168 N. C. A.DA COSTA AND F. A.DORIA
k' is thehighestinteger queriedduringone suchcomputation, andif dA C ca is an initial segmentof the characteristic function ca,we take as a standby for D and D* the initial segment dA wherethe length l(dA) =k'+1.We can effectively listall oraclemachines with respect to a fixedA, so that,givena particular machine we can computeits index (orGodel number) x,and given x we can recover the correspondingmachine.
2. Given an A-partial recursive function <ft£, we form the oracleTuring machine that computes it. We then do the computation4>x(y) = z tnat outputs z. The initial segment dy tA is obtainedduring the computation.
3. The oracle machine is equivalent to an ordinary two-tape Turingmachine that takes as input (y,dy^)',2/ ls written on tape 1whiledVyA is written on tape 2. When thisnew machine enters state so itproceedsas theoracle machine. (For an ordinary computation, noconvergingcomputation enters so, anddViA is empty.)
4. The two-tapeTuringmachine canbemadeequivalentto aone-tapemachine,where some adequatecodingplaces onthesingle tapeallthe information about {y,dy^)- When this thirdmachine enters soit scansdy>A-
5. We can finally use the standard map r that codesn-ples 1-1 ontoujandadd to theprecedingmachine aTuring machine that decodesthe single natural number r((y,dy,A)) into its components beforeproceedingto thecomputation.
Let w be the index for that lastmachine; we noteit <j>w.If x is the index for 4>£, wenote w =p{x), where pis the effective
1-1procedure described above that maps indices for oracle machinesinto indices for Turingmachines. Therefore,
Now let us note the universal polynomial p(n,q,X\,...,xn). Wecan define the jump of A as follows:
With the helpof the Amap definedfollowing Proposition3.7, wecannow form a function modelled after the 9 function inProposition 3.11;it is the desired characteristic function:
(Actually we haveprovedmore; we have obtained
We recall (Rogers,1967):
DEFINITION3.15. The complete Turingdegrees,0,0',...,o^\p < uj, are Turing equivalence classes generated by the sets 0, o', 0",
Now let 0(n)be thenth complete Turing degrees in the arithmeticalhierarchy. Letr{n,q) =mbe thepairing function in recursive functiontheory (Rogers,1967). For 9(m)= 9(r(n,q)), we have:
COROLLARY3.16 (Complete Degrees).IfTisarithmetically consis-tent, forallp £ uj the expressions9p(m) explicitly constructedbelowrepresentcharacteristic functions in the complete degrees0^p\
for ca as inProposition 3.12.
Incompleteness TheoremsWe now stateand prove several incompleteness results about N anditsextension T;they will be needed when weconsider ourmain examples.Werecall that
'— ' is aprimitive recursive operationon uj.
CO/(x) =0(p(x),(x,4jO/».
ca'(x) = 9{p(x),{x,dX)A )),
with reference to anarbitrary A Cuj.)We write 9^2\x) =cr(x).
0<p>...
Proof. FromProposition 3.12,'0(°) = c0(m) =O,
< 0O)(m) =cy(m) =0(m),0(n)(m),
170 N.C. A. DACOSTA ANDF A. DORIA
The starting point is the following consequence of a well-knownresult:
PROPOSITION 3.17. IfTisarithmetically consistent, then we canalgorithmically constructapolynomial expression g(xi,...,xn) overZ such thatM(=Vxi,...,xn G ujq(x\,...,xn) >0,but
Proof. Let(GItbe an undecidable sentenceobtained for T withthehelp of Godel's diagonalization; letn^ be its Godelnumber andletrriTbe theGodelcodingofproof techniquesinT(of theTuringmachinethat enumerates all the theorems of T). For an universal polynomialp(m,q,x\,...,xn) we have:
COROLLARY 3.18. IfH is arithmetically consistent then we canfindwithin itapolynomialp asinProposition 3.17 ■
Wecan also stateand prove a weaker versionof Proposition 3.17:
PROPOSITION3.19. IfTis arithmetically consistent, there is apoly-nomial expressionover Z.p(x\,...,xn) such thatMf= Vxj,.. .,xn Gujp(xi,...,xn) > 0, while
and
Proof. See (Davis, 1982): if p(m,x\, ..., xn), m = r(q,r),is a universal polynomial (r is Cantor's pairing function (Rogers,
TV- Vxi,...,x„ G wg(xi,... ,xn) >0
and
TV- 3xi,"" ",xn G ujq(x\,...,xn) =0.
g(xi,...,xn) = (p(mT,n^xv... ,xn))2.
TFVxi,...,x„ G wp(xi,...,xn) >0
TP3xi," " ",xn G ujp(x\,... ,xn) =0.
SUPPES PREDICATES ANDUNSOLVABLE PROBLEMS 171
1967), then {m :3xi... G ujp(m,x\,...) =0} is recursively enu-merable but not recursive. Therefore there is an mo such that Vxi...uj(p(m0,xi,...))2 > 0. ■
Predicates orproperties in T are representedby formulae with onefree variable in Lt.
DEFINITION3.20. Apredicate PinLt is nontrivial if there are term-expressions(,CG LT such that ThP(£) andT h -iP(C). ■
Then:
PROPOSITION 3.21. IfH is arithmetically consistent and ifP isnontrivial then there is a term-expression(G Ln such thatN |= P(C)whileNV P(()andNV -.P(C).
Prao/ Put (=£ 4- r(xi,...,xn)v, forr=1-erg2,gas inPropo-sition 3.17 (or asp in 3.19). ■
Remark 3.22. Therefore everynontrivial arithmetical propertyP intheories from arithmetic upwards turns out to be undecidable. We cangeneralizethat result to encompass other theories T that include arith-metic; seebelow. ■
We now give alternative proofs for well-known results about thearithmetical hierarchy that will lead to other incompletenessresults:
DEFINITION3.23. The sentences £, £ G Lt are demonstrably equiv-alent if andonly ifTh £ <-» (. ■
DEFINITION3.24. The sentence£ G Lt is arithmetically expressibleif and only if there is an arithmetic sentence£ such that Th^<->(. ■
Then
172 N. C. A. DACOSTA ANDF. A. DORIA
PROPOSITION3.25. IfT is arithmetically consistent, thenfor everym G uj there is asentence £ G T such thatM |= £ while forno k < nis there aEfc sentence in Ndemonstrablyequivalent to £.
Proof. The usual proof for N is given by Rogers (Rogers, 1967,p. 321). However, we give here a slightly modified argument thatimitates Proposition3.19. First notice that
is recursively enumerable but not recursive in 0(m). Therefore 0(m+Uis not recursively enumerable in 0(m), but contains a proper 0(m)-
recursively enumerable set. Let us take a closer look at those sets.We first need a lemma: form the theory T m̂+l^ whose axioms are
those for T plus a denumerably infinite set of statementsof the form'n0 G 0(m)', 'ni G 0(m)',...,that describe 0<m). Then,
LEMMA 3.26. IfT n̂+l^ is arithmetically consistent, then 4>xm (x) J.ifandonly if
Proof. Similar to theproofin thenonrelativized case; see (Machtey,1979,pp. 126ff). ■
Therefore wehave that the oraclemachines (f>xm (x) [ ifandonly if
However, since 0(m+1) is not recursively enumerable in 0(m) thenthere will be an indexmo(0 (-m^) = (p(z),(z,d 0(m> )) such that
while it cannotbe provednor disproved within J'(m+1) - itis thereforedemonstrably equivalent to anm+i assertion. ■
Now let g(mo(0(m) ),xi..)= p(rao(0(m) ),xv ■ " -))2 be asinPro-position 3.25. Then:
COROLLARY3.28. IfTisarithmeticallyconsistentandifLT containsexpressionsfor the9^ functionsasgiveninProposition3.16, thenforanynontrivialpredicatePinN there isaC, GLt such that theassertionP(C) isT-demonstrably equivalent to andT-arithmetically expressibleas a nm+i assertion, but not equivalent to and expressible as anyassertion with a lower rank in the arithmetic hierarchy.
Proof. As in theproofofProposition 3.21, we write:
wherep{...) is as inProposition 3.25.
Remark 3.29. Rogers discusses the rank within the arithmetical hierar-chyof well-known openmathematical problems (Rogers, 1967,p.322),suchas Fermat's Conjecture-whichin its usual formulation is demon-strablyequivalent toaniproblem,orRiemann'sHypothesis,alsostatedas a111problem. Rogersconjectures thatourmathematical imaginationcannothandle more than four or five alternations of quantifiers. How-ever theprecedingresult shows thatany arithmetical nontrivialpropertywithin Tcan give rise to intractable problemsof arbitrarily high rank.
We obviously need the extensionT D N, since otherwise we wouldnot be able to find an expression for the characteristic function of aset
/?(m+l ) =a(C(mo(0 (n))),
G(mo(0("))) =+[°C(rno(^),x)e-° 2
C(mo(0(n)),x) = Ag(mo(0(n)),xi,...,x r),
M |= /3(m+l) = 0 butfor alln < m+ 1, T^ V flm+^ = 0 andT{n) y. _,^(m+l) _
o). ■
C =S + [1- ff(p(mo(0m),x 1,...,xn))>,
174 N. C.A.DACOSTA AND F. A. DORIA
with a high rank in the arithmetical hierarchy within our formal lan-guage. ■
An extension of theprecedingresult is:COROLLARY 3.30. IfT is arithmetically consistent then, for anynontrivialpropertyP there is a £ G Lt such that the assertion P(()is arithmetically expressible, M |= P(Q but it is only demonstrablyequivalent to a Tln+\ assertion andnot to a lower one in the hierarchy.
Proof. Put
where oneuses Corollary 3.27.
Undecidable Sentences OutsideArithmeticWerecall:
DEFINITION3.31.
for x,y G uj.
Then:
DEFINITION3.32.
where c^{u)(m) is obtainedas inProposition 3.12.
Still,
DEFINITION3.33.
C =
0^= {(x,t/) :xG0(y)},
#H(m) = C0(w)(m),
o(w+i) _ (oHy_
SUPPES PREDICATES AND UNSOLVABLEPROBLEMS 175
COROLLARY3.34. O 1̂ ) is thedegreeo/0(w+l).
COROLLARY 3.35. 9^u+l\m) is the characteristic function ofanonarithmetic subsetofujofdegree0^u+l\ ■
PROPOSITION 3.37 (Nonarithmetic Incompleteness). IfT is arith-metically consistent then givenanynontrivialpropertyP:(1) There is a family ofexpressions (m GLt such that there is no
generalalgorithm to check, for every m G uj, whether or notP((m).
(2) There is an expression( G Lt such thatM |= P(C) while T VP(()andTY-^P(().
(3) Neither £ mnor £ are arithmetically expressible.
Proof. We take:
(3) Neither 9^UJ+l\m) nor /3W+l) are arithmetically expressible. ■Remark 3.38. We have thus produced out of every nontrivial predi-cate in T intractable problems that cannotbe reduced to arithmeticalproblems. Actually there are infinitely many such problems for everyordinal a, as we ascend the set of infinite ordinals in T. Also, thegeneral nonarithmetic undecidable statement P(() has been obtainedwithout the help of anykind of forcing construction. ■
/^+1) =(7(G(mo(0(v;) )),
m («M« Tc(m0(»(")),x)e-'1G(mo(0 ))=il+ C(ra„(oH),z)^
We motivateour nextintractability results with aconcreteexample fromHamiltonian mechanics.
Suppose that we have axiomatized Hamiltonian mechanics over afixedphase spaceM with the help of aSuppes predicate within a first-order theory T (which we can take to be ZFC) as done in Section 2.We then have a predicate H(£) inLt that asserts, 'the expression£ isa Hamiltonian function. We can enumerateall other predicatesP& inLt,k anatural number,and wecan alsoenumerateall the theorems inT. We start such anenumeration,and select theorems of Twhich havethe form:(l)For&eLr,Tr-#(&).(2) For fc,Zj,i/j,G Lr,ThPfc(&) andTh -Pfc fe).
Outof that we list allnontrivial predicatesPfc thatapply to Hamilto-nian functions.
We have proved:
PROPOSITION3.39. IfTis arithmetically consistent, we can obtaina recursive enumerationofpairs ofexpressions £ 2i,&i+h ianaturalnumber, that representdifferentfunctions andsuch that:
where theP{arenontrivialpredicates(relative to theHamiltoniansoverM) in T that rangeoverHamiltonian functions. ■
Remark 3.40. The previous result allows us to obtain, out of an enu-meration of the theorems in Tand of allpredicates Pi in the languageof T, an enumeration of different expressions £2*. £2^+l* ia naturalnumber,for different Hamiltonians thatsatisfy (and donot satisfy)eachpredicatePi. ■
Wecan state:
PROPOSITION3.41. IfTis arithmetically consistent, then there is acountable undecidablefamily Cm ofexpressionsforHamiltonians in the
(1) Both ThHfoi) andTY H(&i+l);(2) T hPifoO o/k/T h -P;(6i+i)/
SUPPES PREDICATESAND UNSOLVABLEPROBLEMS 177
languageofTsuch that there isnogeneralalgorithm to decide,foranynontrivialpredicatePk relative to the Hamiltonians over M, whetherthatexpressionsatisfies (ordoes notsatisfy) Pk in T.
Moreover, there is noproblem in the arithmetic hierarchy that canbe made equivalent to the (algorithmetically unsolvable) decisionpro-ceduresfor Cm-
Sketch of the proof. We proceed in a stepwise manner. Let £1,...,be an infinite countable sequenceof mutually independentGodelsentences in T. Let m\, ...,be the corresponding Godel numbers.Form the0{
= 6(mi). Put: eJ=1-9j,c) =9j,alljG uj.
Let mrn be a variable that ranges over all 2nbinary sequences oflengthn. Code thoseby orderedn-tuples of O's and l'sand establishamap/ between those n-bit binary sequencesandalln-factor products£f e2 ... £"" ', so that in the 7-th position (0)1 ■-» ej(e|). Given m,rn,the associated product is /(rn). Given a specific model for T, for aprescribed length, all such sequencesequal0but for a single one, thatequals 1.
Order thepredicates Pi,P2,...;we write thatTh P»(C2x-i) whileTY^P^).
Now list all finite binary sequences and select from those an infi-nite set of mutually incompatible sequences such as 1, 01,001,0001,...Note the incompatible set {t\,t2,.. .};if Tj is any sequence, noteTj>,Tjii,..., its extensions. (Ageneralextension is notedr^.)
The expression we require is:
(s2* denotes sumoverall extensionsof equallengthin the factors c.) ■
Remark 3.42. Notice that, as each9^k\m) is either 0 or1, C willequalasingle oneamong theQ,despite the fact that itis anexpressionwith acountably infinite recursively defined number of symbols. Also,sincetheactual value of C can only be determined if we solve an intractableproblem in the arithmetic hierarchy, and since it contains (possiblyrepeated)representative expressions ofallproperties inT, we will onlybe able to check for any property if we first solve a complete problemof degree> o'.
Moreover, since every arithmetical statement has a finite, bound-ed degree in the hierarchy, nontrivial properties of C cannot be madeequivalent to anything in the hierarchy.
Finally, when we (informally, butin an actual computation) 'openup' theexpression for C, weobtain an infinite formal sum whose termsare products of iterated integrals (the expressions for the 9^); suchexpressionsare certainly uncommon in classical mechanics,but theyare the standard staple of quantumand particle physicists. So, nothingout of the mainstream here. ■
SCHOLIUM3.43. The lowest degree in(m maybe arbitrarily high.Proof. For anyp>owecan substitute9p+k for 9k in theexpression
for Cm-
COROLLARY 3.44. The decision problem within an arithmeticallyconsistentTforCm cannotbe madeequivalent to adecisionprobleminthe arithmetic hierarchy.
Proof. From Remark 3.42.
Remark 3.45. Rogers gives the proofby a diagonal argument (Rogers,1967, Section 14.8, Theorem XIII) of the following assertion: if theaxiomatization for ZFC is consistent then there is a sentencein ZFCthat cannotbe madeequivalent to an arithmetic sentencewith the toolsof set theory. As in most proofsby diagonalization the counterexampleobtainedisalegitimate assertioninZFCbut itasmeaninglessasGodel'soriginal undecidable sentencein arithmetic. However,out of theprevi-ousresult (Proposition3.41) wecan immediately obtainanonarithmeticsentencein set theory. ■
Sincethe set of theorems of T(supposed [arithmetically] consistent)is a creative set (Rogers, 1967), its complement is productive. There-fore, we can add theaxiom 'There isno solution foxpo{x\,...,xn) — 0over thenaturalnumbers' toanew polynomialp\(x\,...,xr) such thatthe Diophantine equationp\ = 0has no solution in M,but such thatagain wecannotprove that fact from T. Again, due toproductivenesswe obtain athirdpolynomial,p2,whichleads to a thirdundecidable sen-tence and to another extended theory T". We thus generatea sequence
SUPPESPREDICATESAND UNSOLVABLEPROBLEMS 179
of unsolvable Diophantine equations po —o,p\ =0,...,pj =0,...,which lead to undecidable sentences in the theories T(°) =T,rpl rp(i)j. .., j. ...
That construction is equivalent to obtaining a recursively enumer-able set out of a productive set (Rogers, 1967); no such constructionwillexhaust the productive set,and severalalternative recursively enu-merable subsets of theproductive set can be found.
We now use those pj as follows: if A is the map defined followingProposition3.7, we form the sequenceki(x)= Xpi,iG uj,and obtain:
However, T^V Pi =0, allk < i.Proof. See (da Costa, 1991a).
Then,
PROPOSITION3.47. IfT is arithmetically consistent, thenfor everyrecursively enumerable extension of theaxioms ofT there is aHamil-tonian C all whose nontrivialproperties cannot be proved within thatextension.
Proof. The expression we require is:
(J2* denotes sum over all extensions of equal length in the factors c.)Here the #'s in the r'sare substituted for /3's. ■
Theseexampleslead to the immediate proofofageneralundecidabil-ity and incompleteness theorem that deals with those systems for which
no nontrivialpropertycan be algorithmetically decided orproved. LetQ{x, ai,a 2
,.. ",an)be a Suppespredicate on the fixedparameters a\,Suppose given an enumeration of the predicatesPk in T. Again
we suppose that:WFor&GLr.ThQte).(2) For&,&,** j,cLT,TY Pfc(&) andTh -Pfc fe).(3) Out of that we list all nontrivial predicates Pk that apply to Q-
defined objects.Out of that weobtain:
PROPOSITION3.48. IfT is arithmetically consistent, we can obtaina recursive enumeration ofpairs ofexpressions 2i, * anaturalnumber, that representdifferent objectsandsuch that:
where the Pi arenontrivialpredicates (relative to Q) in T that rangeoverQ-objects. ■
Then:
PROPOSITION3.49. IfTisarithmetically consistent then:Undecidability. There isa countablefamily (m ofexpressionsforQ-objects in T such that there is no generalalgorithm to decide,for any nontrivial Q-property Pk in T whether that expressionsatisfies (or doesnotsatisfy) Pk.Incompleteness. There is a Q-objectfor which none of the non-trivial Q-propertiescan beprovedwithin T. ■
So, incompleteness of the nastiestkindis to be expectedeverywherein the axiomatized sciences. For density theorems related to thoseintractable problems see (da Costa, 1993a).
4. TWO APPLICATIONS
We discusshere twoproblems thathavebeen recently handled with thehelp of the present methods. Arnolds 1974 questions on polynomial
(1) Both Th Q(i2i)andTY Q(&i+i).(2) T hP*fe)«^T h -Pifei+i),
SUPPES PREDICATES AND UNSOLVABLEPROBLEMS 181
dynamical systemsand the incompleteness of the theoryof finite Nashgames. Full details will appearin references (da Costa, 1992d,1992e).
PolynomialDynamicalSystems Are Undecidable andIncomplete
In the 1974 AMS Symposium on the Hilbert Problems, V.I.Arnoldsuggested the following questions for the list of 'Problems of PresentDayMathematics' that wascompiled at that symposium:
Is thestability problemforstationarypointsalgorithmically decid-able? The well-known Lyapunov theorem solves the problem inthe absence of eigenvalues with zero real parts. In more compli-cated cases, where the stability depends onhigher order terms inthe Taylor series, there existsno algebraiccriterion.
Let a vectorfieldbe given bypolynomials ofa fixed degree, withrationalcoefficients. Doesanalgorithm existallowing us todecidewhether the stationarypoint is stable?
A similarproblem: Doesthereexistanalgorithm todecide whethera planepolynomial vector field has a limit cycle?
See(Arnold,1976a).We notice that Arnoldsays nothingaboutbound-ary values,despite the fact that the system's behavior may vary wildlyas a function of the initial vales; he also imposes noconditions on thegeometry of the manifolds where those dynamical systems are to befond. Therefore we may offer as a counterexample a specific poly-nomial dynamical system over Z with fixed initial values such that noalgorithm in the sense ofArnold exists for therequiredproperties.
LetPbeanontrivialpredicate (relative to theadequate vector fields)in T. We assert:
PROPOSITION4.1. IfTis arithmetically consistent, then:(1) There is anexpressionfora vectorfield v over Rn in T such that
M\='v is a smooth vectorfieldon Rn,n>2 withpropertyP',andTV -> 'v is a smooth vectorfieldon Rn withpropertyP.
(2) There is an expressionfor a vector field v in T such that thesentence 'v is a smooth vectorfieldon Rn,n> 2 with propertyP'is T-arithmetically expressibleasann+i sentence,m>\,butsuch thatforno m it willbe equivalent toa Tim sentence.
182 N. C. A. DA COSTA AND F. A.DORIA
(3) There is an expressionfor a vector field v in T such that thesentence 'v is asmooth vectorfieldon Rn,n> 2 withpropertyP
'cannotbe taken to be T-demonstrably equivalent to any sentencein the arithmetical hierarchy.
(4) We canexplicitly findanexpressionfor apolynomial vectorfields over7.on Rm,mfixed, such that,for agivenP,M \= P(s),butTV P(s)andTY^P(s).
HereMis amodel whereTis arithmetically consistent.Proof. The first three assertions are immediately proved with the
methods presented in (da Costa, 1991a) and expandedin the previoussection of this paper. Actually they generalizeourprevious results ondynamical systems (daCosta, 1991a).
The lastassertion is the negative solution (in an obvious sense) to aproblem related to Arnolds problem; for details on Arnolds see (daCosta, 1992e). We make two remarks:
" We write x= (xi,...,Xj),where j=dimx." We notice that if the polynomialp(x) is notidentically zero, thengiven the expressionbelow for a smoothelementary function,
if rG R, then for s(x,y) =u(x,y) —r, R2-7 — s_1(0,0) is openin R2-7,where vis taken as areal-valued function onR2j. (See onvaboveDefinition 3.2.)
The first lemma we requireis:
LEMMA 4.2. IfTis arithmetically consistent, then:
together with
(1) «(x, y) =(j+1)V(x)+ £ 0&4(*)1.i=l
(2) tt(x,y) =(j+ l)4b2(x)+J2yM(*)l
(3) yi—
sin7TXi, wi = cos7TXj,
SUPPESPREDICATES AND UNSOLVABLE PROBLEMS 183
is the unique solution of the followingpolynomial dynamical systemwith coefficients in Z over R3j+l endowed with the usual Cartesiancoordinates andthe correspondingEuclideanmetric tensor:
dxi dvdt dxi
for i=l,2, 3,...,j,plus the boundary condition u(0,0) =p(0) (withan obvious abuseoflanguage);
Proof. Equations (4) are immediately integrated (since they aregradientequations)toEquation (2). Equation(4.2) trivially implies thatz = 7r, and therefore (also from Equations (4)) wehave:
(no sumoni). Thatsystemhas the solutions yi = sin7TXi,Wi =cos7TXf.Since the du/dxi arepolynomials overZ whicharenot identically zero,the lemma is proved. ■
We need a secondlemma: here we allow v torange over R — {I}.Therefore for eachxiandfor each yi atmostacountably infinitenumberofpointsare tobe deletedfrom thecorrespondingdomains,thosepointsthat are solutions of the (fixed) equation v — 1= 0 over the reals. We
,« dyi idu(5) Id= W Zdx~- '
dwi i dvdt dxi
" t=°
dyi— = irwi,dxi
dwi-T-
= -*yi,dxi
184 N.C. A. DA COSTA AND F. A. DORIA
then state:
LEMMA 4.3. IfTisarithmetically consistent, then
is the uniquesolution to thepolynomial equations on R — {I},
wherec=—l, v G (— oo, 1) andc=o,v G (1,+oo);and
together with theconditions v(0) =1— ti(0), v G (— oo,1), andv=0/oru G (I,+co).
Proof. Immediate.
We conclude the proof with still another lemma: let S, Tbe poly-nomial vector fields over Z on a A;-dimensional real smooth manifoldMsuch thatT h P(S) andT h + vT). Let us form the vectorfields S = (x, 0)and T =(0,y). Then:
LEMMA 4.4. IfTis arithmetically consistent, then thereis an expres-sion for thepolynomial vectorfieldover Z of theform:
such thatM \= $ =E,butTV $ /E. (AgainMisamodel where Tis arithmetically consistent.)
Proof. It suffices to take thepolynomialp(x)asinPropositions 3.17or 3.19,and getv as inEquations (2) and (3), and finally obtain v outofit. ■
This lemma concludes ourproof.
Let M'be themanifold where the expression for $ is defined. Then
(8) v =l-v
dc(9) Tt
=°'dTj dv(10) Tt= c*
(11) $= £ +7jT
SUPPES PREDICATESAND UNSOLVABLEPROBLEMS 185
SCHOLIUM 4.5. //Tis arithmetically consistent, then T h 'Thedimensions ofMl,m<2k +3(j'+ 1)'. ■
SCHOLIUM 4.6. IfT is arithmetically consistent, then M |= Thedimension ofMl,m= k', while
TV 'The dimension ofM',m =k\and
TV -> 'Thedimension ofM',m =k '.
So we might end up dealing with a planar vector field despite thefact that T will notrecognize it!
COROLLARY 4.7. IfT is arithmetically consistent, then it has anexpression(inM)foraplanarpolynomial autonomousvectorfieldwithcoefficients in Z such that we cannotprove (from the axioms ofT) thatthatfield isplanar. ■
With thehelp of Proposition 3.37 we can state:
COROLLARY 4.8. IfT is arithmetically consistent, then it has inMan expressionforaplanar polynomial autonomousvector field xwith coefficients in Z such that the assertion 'x is planar' cannot bemadedemonstrablyequivalent withinTtoanyarithmeticalassertion. ■
Other consequencesof theprevious resultsareeasily stated andproved.
Remark 4.9. Our exampleofapolynomial vector field withundecidableproperties has either ahighapparent dimension or an apparently veryhigh degree. If we take k = 2 and the original Diophantine universalpolynomialp with 11variables,m=46 and the degree is ridiculouslyhigh. For degree 16 we will have that m = 250. A discussion onthepossibility of reducing those values can be found in (da Costa,1992e). ■
186 N. C. A. DA COSTA ANDF. A. DORIA
IncompletenessofFiniteNash GamesThe results givenhere are presentedin full detail in (daCosta, 1992d).It is intuitively obvious that, for a finite Nash game, we can algorith-mically check for equilibria in it. However, things turnout to be muchmore delicate when we work within aformalization for the theory offinite Nash games. We start from:
DEFINITION4.10. A noncooperativegame is givenby the yonNeu-mann triple T = (N,Si,Ui), with i= 1,2,...,N, where N is thenumber of players, Si is the strategy set of player % and Ui is the real-valued utility functionUi :Yli Si -+ R. ■
Then:
PROPOSITION4.11. IfTisarithmetically consistent, then there is anoncooperativegameTwhereeachstrategysetSi isfinitebut such thatwe cannotcompute its Nashequilibria.
Proof. Let V andT" be two different games with the same numberof players but with different strategy sets and different equilibria. IfP = /3(w+l) in Corollary 3.36, wecan form theutility functions:
Therefore the game T = (N,Si,Ui) does not have a decidable set ofequilibria. ■
SCHOLIUM 4.12. Determining the equilibrium setofT is anonarith-meticproblem. ■
We cantake Tbarely beyondformalized arithmetic N: let N* be thetheory whoseaxioms are N plus acompatible version of the separationaxiom. Therefore we can give an 'implicit' definition for sets of num-bers. We put Ui : Y\iSi —+ uj. We now use q as in Proposition 3.21and form the utility functions Ui = u\ +qu". The remainder of theargumentgoes as usual. Thus:
'I Q II
Ui =Ui+ fJUi .
SUPPESPREDICATESAND UNSOLVABLEPROBLEMS 187
PROPOSITION4.13. 7/N* is arithmetically consistent, then there isagame f inN* such thatthe assertion 'u{ =u\ 'is undecidable m N*.■
5. CONCLUSION
In 1987 the authors started a research program whose goals weretwofold: we wished to axiomatize as much as possible of physics inorder to searchfor physically (i.e. empirically)meaningful undecidablesentencesand physically meaningful unsolvable problems within ourformalizations. The axiomatization program was fulfilled in part withour construction of Suppes predicates for classical physics out of anunified framework that reminds one of ideas and tools from categorytheory (da Costa, 1992a). The incompleteness portion of the programwas certainly much more difficult to pursue: we had two candidatesfor undecidable problemsinphysics,and we hopedthatthose problemsmight lead us to thedesired incompletenessproof. Thecandidates wereclassification schemes in generalrelativity andHirsch'sproblem ontheexistence of algorithms tocheck for chaos in dynamical systems. (Seethe references in (daCosta,1990b, 1991a).
At first webelieved that forcingmodels might provide the indepen-denceproofs we were looking for (daCosta, 1990b,1992b), since forc-ing is an obvious source of so many mathematically relevant undecid-able statements in ZFC.Therefore weexploredour Suppes-formalizedversion of generalrelativity and the corresponding sets of noncompactspace-times, but the results on incompleteness were meager at best.Then,due to a suggestionby Suppes, we turned to Richardson's 1968undecidability results in analysis. It was immediately apparent to usthat those resultsentaileda full-fledged incompleteness theorem, whichweextended,after some failedefforts, to the explicitconstructionof thehalting function for Turing machines within classical elementary realanalysis.
Wethenstarted to turnoutseveralundecidability andincompletenessresults indynamical systemsandrelatedquestions,as wellas counterex-amples to theexistenceofalgorithms of the kind thatHirsch wasaskingfor in chaos theory. Again Suppespointed out to us that there wassome sort of a very generalundecidability and incompleteness theoremat work here; the first version of that theoremis Proposition 3.29 in (daCosta, 1991a).
188 N. C. A. DA COSTA ANDF A. DORIA
That theorem is so general (it reminds one of Rice's theorem incomputer science (Machtey, 1979; Rogers, 1967), from which it canbe derived) that for a moment we felt it might announce some kindof imminent disaster, as every nontrivial property in the language ofclassicalanalysiscanbemadeintoanundecidability andincompletenesstheorem. So, in generalnothing interesting can be decided and verylittle can be proved. (This point wasalso emphasizedby Stewart inhiscommenton our results (Stewart, 1991).) At the same time we receivednotice thatour results were at first met with disbelief by researchers indynamical systems, precisely due to that very general incompletenessstatement;it was thenarguedthatthoseresults werecorrectbut 'strange'due to some undetected conceptual flaw in the current view on thefoundations ofmathematics.
Yet our assertions have a very clear, letus say,practical meaning:if what we mean by 'proof in mathematics is algorithmic proof, thatis to say, something that can be simulated by a Turing machine, thenvery difficult problemsare tobe expectedeverywhereat the very heartof everyday mathematical activity; more and more innocent-lookingquestions are tobe found intractable (as we now see in chaos theory)and(to repeat an example that we haveoffered before) systems willbeformulated that havea tangled,chaotic appearance when approximatedona computer screen,but such thatno proofof their chaotic propertieswill beoffered within reasonable axiomatic systems such as ZFC.
Godel incompleteness isnooutlandishphenomenon;itisanessentialpart of the way we conceive mathematics.
What can we make out of that? We do not think that there is anyessential flaw inthepresent-dayview aboutfoundational concepts;how-ever we think that our incompleteness theorems point out very clearlywhere the problem lies. Nothing will be gained by adding 'stronger'and 'stronger' axioms to our current axiomatizations. But wecertainlymust strengthen our current concept of mathematical proof. Turing-computable proofs are not enough, for Church-Turing computation isnot enough. We must lookbeyond.
6. ACKNOWLEDGMENTS
It certainly is a pleasure to dedicate this work to Pat Suppes on theoccasionof his 70th anniversary. We deem that dedication tobe espe-cially adequatesinceour workon theundecidabilityand incompleteness
SUPPES PREDICATES AND UNSOLVABLEPROBLEMS 189
of axiomatized theories has been inspired from its very beginnings bymany fruitful intuitions and suggestionsof his.
We also wish to acknowledge the constant cooperation andinterestof our coworkers: J.A. deBarros,A. F.Furtado do Amaral,D.Krause,andM.Tsuji. Thepresentideas werealsodiscussed withM.Corrada,R.Chuaqui,and D.Mundici,to whom we owe friendly remarks andcriti-cisms. A first presentationof our incompleteness results was carefullyreadbyM.Hirsch,to whom weare indebted for severalcorrections andimprovements.
The present paper was completed while the second author held avisiting professorship at the University of Sao Paulo's Institute forAdvanced Studies under aFAPESP scholarship program. The authorsalsoacknowledge support from the CNPq(Brazil) scholarship programinphilosophy and wish to thank the ResearchCenter on MathematicalTheories of Communication inRiode Janeiro (CETMC-UFRJ) for theuseof its computer facilities.
N.C. A. da Costa,ResearchGroup onLogic andFoundations,InstituteforAdvancedStudies,UniversityofSao Paulo,05655-010 SaoPauloSP,BrazilF.A. Doria,Research Center onMathematical Theories ofCommunication,SchoolofCommunications,Federal UniversityatRio deJaneiro,22295-900RioRJ, Brazil
REFERENCES
Arnold, V I.: 1976a, 'Problems of Present Day Mathematics', XVII (DynamicalSystems andDifferentialEquations),Proc. Symp. PureMath.,28, 59.
Arnold, V I.: 1976b, Les Methodes Mathematiques de la Mecanique Classique,Moscow: Mir.
Atiyah,M.F: 1979, Geometryof Yang-MillsFields,Pisa: LezioneFermiane.Bourbaki,N: 1957, Theoriedcs Ensembles,Paris:Hermann.Bourbaki,N.: 1968, TheoryofSets,Boston: Hermann andAddison-Wesley.Dalla Chiara,M.L.andToraldodi Francia, G.: 1981,Le TeorieFisiche, Boringhieri.Cho, Y.M.: 1975, J.Math. Phys.,18, 2029-2035.da Costa, N. C. A. andChuaqui, R.: 1988,Erkenntnis, 29,95-112.
190 N. C. A. DA COSTA AND F. A. DORIA
da Costa, N. C. A. and French, S.: 1990a, 'The Model-Theoretic Approach in thePhilosophy of Science',Philosophy ofScience, 57,248-265.
daCosta, N. C. A.,Doria,F. A., andde Barros, J.A.: 1990b, Int. J. Theor. Phys., 29,935-961.
daCosta,N. C.A. andDoria,F. A.: 1991a, Int.J. Theor. Phys., 30, 1041-1073.da Costa,N. C.A. and Doria,F. A.: 1991b, Found. Phys. Utters,4,363-373.da Costa,N. C. A. andDoria,F A.: 1992a, 'SuppesPredicatesforClassicalPhysics',
to appearin: A. Ibarraet al. (Eds.), The Space ofMathematics, De Gruyter.da Costa, N. C. A. and Doria, F. A.: 1992b, 'Structures, Suppes Predicates and
Boolean-ValuedModels in Physics', to appear in: P. Bystrov and J. Hintikka(Eds.),Festschrift inHonor ofV. I.SmirnovonHis 60thBirthday.
da Costa,N. C. A. andDoria,F. A.: 1992c, 'Jaskowski'sLogic', preprintCETMC-10.da Costa,N. C. A.,Doria,F. A., andTsuji, M.: 1992d, 'The Incompleteness ofFinite
NoncooperativeGames withNash Equilibria',preprint CETMC-17.da Costa,N. C. A., Doria,F A., Baeta Segundo, J. A., and Krause, D.: 1992e, 'On
Arnolds1974Hilbert Symposium Problems',inpreparation.da Costa,N. C. A.,Doria, F. A., and Furtadodo Amaral, A. F: 1993a, Int. J. Theor.
'Two Questionson the Geometry ofGauge Fields',Found. Phys., toappear.da Costa,N. C. A. andDoria,F A.: toappear, MetamathematicsofPhysics.Davis, M.: 1982, 'Hilbert's Tenth Problem Is Unsolvable', in: Computability and
StanfordUniversity.Taubes,C.H.: 1987, J. Diff. Geometry,25, 363^112.
COMMENTSBYPATRICK SUPPES
Foroverhalfacenturysince Godel's famousresults, the incompletenessof any axiomatization of classical analysis has beenknown, but in thepast several decades other results have stressed how far we are fromhaving anything like a real algorithmic approach to any significantclass of mathematical problems. One example is the realization thatTarski's decision procedure for elementary algebra is not feasible, inthe technical sense that the computations grow exponentially in thelengthnof any formula whose truth is to be decided. So evenpositiveresults on decision procedures themselves do not guaranteepracticalapplicability. Itused tobe thoughtofaspartof the folklore,butofcoursenot in any sense proved, that 'most' problems in mechanics that werenot toocomplicated to formulate wouldhave relativelystraightforwardsolutions. Therecentdiscoveryofchaotic systemsinallsortsofdomainshas shownhow the problems that fill the textbooks of mechanics are acarefully selected group. That such difficulties were lurking about hasreallybeenknown since theintensiveworkonthe three-bodyproblem inthe nineteenth centuryand theculminating negativeresults ofPoincare.One way to put it is that any undergraduate in physics can derive thedifferential equations governing familiar cases of the restricted three-bodyproblem-therestrictedproblemis when the massof the third bodyis negligibleand therefore does not influence the regular motionof theother two bodies. But the problemof findingmathematical solutions ofthe differential equationsis whollyunmanageable for most cases.
Da Costa and Doria have embarked on a program, as indicated bythe many additional references in theirpaper, toshow how widespreadthepresenceof unsolvableproblems is inphysics andother sciences.
192 N. C. A. DACOSTA AND F A. DORIA
Thatdifficulties inalldirections can be found with alittle effort wasanticipatedin some senseby Richardson's 1968 results onundecidableproblems involving elementary functions of a real variable, but whatNewtonandChico have doneis extend thiskindofresult much further.It has probably been the feelingof most nonlogicians working in theseareas that results on incompleteness or undecidability were really notgoing to stand in the way of standard mathematical progress. Thesefoundational results were quirky oneson the edgesand couldnot haveanything to do with anything lying in the heartland of analysis andmathematical physics. Newton and Chico are certainly in the processof showinghow this is not thecase.Ihave learned a lot from both Newton and Chico over the years,
especially through theirlivelyparticipation inmy seminars atStanford. Iespeciallybenefittedfrom longconversations withChicoDoriaacoupleofyearsago whenhe spentayear atStanford. Infact thoseconversationsabout thefoundations ofphysics,and recentdiscussions with Acacio deBarros,his former student from Brazil, havecompletely rekindled myinterest in working onthe foundations of quantummechanics.
The kind of results that Newtonand Chico have obtained havealsobeen a motivation for the work thatRolando ChuaquiandIhave beendoing on constructive foundations of infinitesimal analysis, especiallyaimed atphysics. Ourobjective hasbeen to giveafree-variablepositivelogic formulation of infinitesimal analysis which is finitarily consistentand which is still strong enough to prove,in somewhat modified form,many of the standard theorems that underlie constructive methods intheoreticalphysics. Itis afeature of theoretical physics that it ismainlyelaborate computations, either of a symbolic or numerical kind from amathematical standpoint. Little is ever done in theoretical physics assuchaboutprovingexistence theorems. Here,of course,Iamdrawingadistinctionbetween theoretical andmathematical physics,a distinctionthatis nowrelatively wellestablished. Ofcourse,necessarily such weaksystems as Rolando andIhave been workingon cannot do everythingthatone wants. We do think thatmuchcanbedoneanditis important tounderstand where the boundaryexists. The important methodologicalpoint is thatby usinginfinitesimals,wecan give afree-variable formu-lation of such standard theorems ofcalculus as themean value theoremor Green's theorem. Ishould mention that these theorems are provedin approximate form, that is, equality is replaced by an equivalencerelation that means there is adifference that is infinitesimal.
SUPPESPREDICATESAND UNSOLVABLEPROBLEMS 193
Ido not mean to suggest that the system that Rolando andIhavebeen workingongetsaroundall theproblemsuncoveredbyNewtonandChico. What it does show is that weak constructive systems, demon-strablyconsistent,are sufficient for a greatdeal of the work.
REFERENCES
Chuaqui,Rolando and Suppes,Patrick: toappear, 'Free-Variable AxiomaticFounda-tionsofInfinitesimalAnalysis: AFragment withFinitaryConsistencyProof.
Suppes,PatrickandChuaqui,Rolando: inpress, 'AFinitarilyConsistentFree-VariablePositiveFragmentof InfinitesimalAnalysis',Proceedingsofthe9thLatinAmericanSymposiumonMathematicalLogic,heldatBahiaBlanca,Argentina, August 1992.
