This article was downloaded by:[Corcoran, John] On: 6 March 2008 Access Details: [subscription number 791294329] Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK History and Philosophy of Logic Publication details, including instructions for authors and subscription information: http://www.informaworld.com/smpp/title~content=t713812075 Categoricity John Corcoran a a Department of Philosophy, State University of New York at Buffalo, Buffalo, N.Y. 14260, U.S.A Online Publication Date: 01 January 1980 To cite this Article: Corcoran, John (1980) 'Categoricity', History and Philosophy of Logic, 1:1, 187 207 To link to this article: DOI: 10.1080/01445348008837010 URL: http://dx.doi.org/10.1080/01445348008837010 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf This article maybe used for research, teaching and private study purposes. 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Received 14 July 1979 After a short preface, the first of the three sections of this paper is devoted t o historical and philosophic aspects of categoricity. The second section is a self-contained exposition, including detailed definitions, of a proof that every mathematical system whose domain is the closure of its set of distinguished individuals under its distinguished functions is categorically characterized by its induction principle together with its true atoms (atomic sentences and negations of atomic sentences). The third section deals with applications especially those involving the distinction between characterizing a system and axiomatizing the truths of a system. 0. PREFACE Aside from the analysis of the logical structure of mathematical propositions and the formalization of mathematical reasoning, perhaps the most striking achievement of pre-Godelian mathematical logic was the categorical characterization of traditional mathematical systems (Euclidean geometry, the natural numbers, the rational numbers, etc.) viewed as interpretations of formal languages. Section 1 treats historical and philosophical aspects of the notion of categoricity (and, thus, also isomorphism) within the broader context of a discussion of characterization of a mathematical system by means of a set of sentences which hold in it. Section 2 considers mathematical systems which are categorically characterized by means of one (second order) induction principle supplemented only with atomic sentences and negations of atomic sentences (i.e., using no properly first order sentences). In particuIar it is shown that a system can be categorically characterized by such means provided only that it is inductive, i.e. that its domain is the closure of a finite number of its individuals under a finite number of its operations (any number of relations may also be present). This theorem leads immediately to a very weak test of categoricity. Section 3 shows that the test has useful applications in axiomatizing inductive systems. The weakness of the test, especially when viewed in relation to examples given, suggests that the importance of categoricity may have been exaggerated and that the relationship between characterizing a model and axiomatizing its truths, is not as close as had been thought. In particular, the test is used to establish a categorical characterization of the natural number system from which it can not be deduced that zero is not a successor. Other D ow nl oa de d B y: [C or co ra n, J oh n] A t: 18 :4 3 6 M ar ch 2 00 8 188 J . C O R C O R A N examples of deductively very we. 1( theories which are nevertheless categorical are also given. 1. C A T E G O R I C A L C H A R A C T E R I Z A T I O N S O F M A T H E M A T I C A L SYSTEMS By the turn of the century mathematicians had distinguished mathematical systems from axiomatizations. A mathematical system was thought of, in effect, as a class of mathematical objects together with a finite family of distinguished relations, functions and elements.' An axiomatization of a system was often thought of as a set of propositions about the system. Some mathematicians took the notion of a proposition about a system so literally that they could not conceive of a reinterpretation of a set of axioms (Frege lPO6,79). At this time, one must recall, there was no such thing as a formal grammar. Nevertheless, certain mathematicians (e.g. Hilbert 1899; Veblen 1904) conceived of the axioms for a mathematical system as propositional forms interpreted in the given system but admitting of other interpretations as well.2 Today we can speak of mathematical systems without reference to particular formal languages interpreted in them, but we often do not. For example, when we speak of the system of natural numbers we often mean the intended interpretation of one of the formal languages commonly used for number theory. And we have to be reminded of the fact that we can refer to the system of natural numbers in itself, so to speak. We also have to be reminded of the fact that an interpretation of a formal language is not merely a mathematical system but it also involves (among other things) a precise specificatioq of which formal symbols get assigned to which distinguished relations , functions, and elements. In general, a set of formal axioms can be interpreted in a given system in more than one way. Thus, strictly speaking, a I . The term 'mathematical system' was and is widely used in just this sense (compare Huntington (1917, 8) and Birkhoff and MacLane (1944, 1953)). From a philosophical and historical point of view it is unfortunate that the term 'mathematical structure' is coming to be used as a synonym for 'mathematical system'. In the earlier useage, which we follow here, two mathematical system having totally distinct elements can have the same structure. Thus in this sense a structure is not a mathematical system, rather a structure is a 'property' that can be shared by individual mathematical systems. At any rate a structure is a higher order entity. The relation between a given structure and a system having that structure is analogous to the relation between a quality and an object having that quality. For mathematical purposes it would be possible to 'identify' a structure with the class of mathematical systems having that structure, but such 'identification' may tend to distort one's conceptual grasp of the ideas involved. A referee suggested that some readers could confuse 'mathematical system' in the above sense with 'axiom system' in the sense of an axiomatization. This would be analogous to confusing an event with a description of the event or to confusing the set of solutions to an equation with the equation. All three cases confuse subject-matter with discourse 'about' that subject-matter. 2. Resnik (1974, esp. pp. 390-392) discusses this particular aspect of Hilbert's thought as reflected in the so-called 'Frege-Hilbert controversy'. The interpretation of Hilbert advanced here is in full agreement with Resnik. D ow nl oa de d B y: [C or co ra n, J oh n] A t: 18 :4 3 6 M ar ch 2 00 8 CATEGORICITY 189 set of formal axioms does not delimit a class of mathematical systems but rather it delimits a class of interpretations of its language. (An exact definition of 'interpretation' is given in sub-section 2.2 below.) In the rest of this paper certain confusions can be avoided by keeping the above distinctions in mind. Especially important is the fact that, strictly speaking , a set of formal sentences true in a given interpretation should be regarded as an axiomatization of the interpretation rather than an axiomatization of the underlying mathematical system. Let K be a set of non-logical constants and let LK be a formal language having K as its set of primitives. Let i be an interpretation of LK and let T(i) be the set of sentences of LK true in i. A complete axiomatization of i requires the choice of a subset A of T(i) which logically implies the rest. An axiomatization of a given interpretation provides a description of the interpretation. Without meaning to suggest that one can uniquely describe an interpretation by means of a set of sentences, Hilbert (e.g. in 1899) permitted himself remarks to the effect that a set of sentences can 'define' an interpretation . Frege's criticism of Hilbert shows that Frege misunderstood Hilbert's remarks as implying the possibility of unique axiomatic characterizations (Frege 1899, 6-10). But Hilbert's reply shows that Hilbert was fully aware of the impossibility of such characterizations. Hilbert (1899L, 13-14) wrote: '. . . each and every [satisfiable] theory can always be applied to infinitely many systems of basic elements'. (See also footnote 2 above.) Even today one occasionally finds a passage which admits of the same misconstrual . For example, Kac and Ulam (1968, 171) write: 'The axioms are meant to describe simple properties of the objects under consideration; one hopes that in these properties the essence of the objects will be captured completely '. Nevertheless, by the turn of the century, at least, it had become clear that truth in a formal language has nothing whatever to do with the 'essence' of the objects in an interpretation, but rather depends solely on the form of the interpretation or, as it is sometimes put, on the formal interrelations among the objects. The notion of isomorphism between two interpretations was adopted as a mathematical formulation of the idea of two interpretations having the same form.3 3. A mathematically precise definition of isomorphism is given in sub-section 2.3 below. It is important to realize that the concept comes into play in a context where the language is fixed and the interpretations are changed (not when the interpretation is f w d and the language is changed): This confusion is rather widespread in the informal parts of the literature of the recent past. For example, in introducing the proof that any two mathematical systems satisfying the integral domain postulates are isomorphic, Birkhoff and MacLane say that the postulates 'are true of the integers not only as expressed in the usual decimal notation; they are also true of the integers expressed in the binary, ternary or any other scale!' (1944, 37). Notice that the Birkhoff and MacLane remark is true and that it would still be true regardless of whether the postulate set in question were categorical. Their remark is totally beside the point and could only be made in this context by persons confusing change of system with change of notation. D ow nl oa de d B y: [C or co ra n, J oh n] A t: 18 :4 3 6 M ar ch 2 00 8 190 J. CORCORAN One occasionally reads that if two interpretations are isomorphic they are 'identical except for the names' of the elements and relations (Fraleigh 1967, 55) or that isomorphic interpretations 'differ only in the notation for their elements ' (Birkhoff and MacLane 1953, 33). Such remarks can be misleading becau'se generally two isomorphic interpretations have dzgerent elements and relations. The whole point is that the two have the same 'form', and what sets of objects the two each involve, be they identical or different, is beside the point. (Compare footnote 3 above.) For example, let LK be the algebraic language based on one binary operation symbol *. One familiar interpretation takes as domain the four so-called complex units, 1, -1, i and -i and as interpretation of * it takes multiplication. Another interpretation takes as its universe the four social classes of the Kariera society and as interpretation of * it takes the function which yields the class of a child when applied to the class of its father and the class of its mother. ~evi - '~ t rauss has discovered that such a function exists and, indeed, that the interpretation just mentioned is isomorphic to the interpretation in the complex units (compare Barbut 1966). Surely one would not want to say in this context that a complex unit and a social class differ only in notation. The form which is common to these two interpretations is, of course, the so-called 'the Klein group'. The insight that truth in a formal language depends solely on the form of the interpretation (and is independent of content or matter) is partly reflected in the fact that isomorphic interpretations have the same set of truths, i.e. if i and j are isomorphic then T(i) = T( j). Moreover, it has been clear at least since the turn of the century (Hilbert 1899L, 14) that given any interpretation i, there are other interpretations isomorphic with i but having no content in common with i. The existence of such isomorphic 'images' implies, of course, the impossibility of uniquely characterizing an interpretation by means of a set of sentences in a formal lang~age.~ Accordingly, it is sometimes said that the best possible characterization of an interpretation would be a 'characterization up to isomorphism', where a set A of sentences is said to characterize i up to isomorphism if every interpretation which satisfies A is isomorphic to i. Thus instead of an ideal of exact characterization, mathematicians adopted the ideal of characterization up to isomorphism, and terminology was introduced to indicate the property of sets of sentences which characterize up to isomorphism the interpretations they characterize (Veblen 1904,346). More precisely, a set of sentences A is said to be categorical if every two interpreta4 . As late as 1944 some writers were still not clear about this point. For example, Birkhoff and MacLane 1944 are clear that, within the class of formal languages that they were using, no postulate set could distinguish between isomorphic systems; but they did not see that this is a feature of all classes of formal languages. They wrote: '. . . no postulate system for the integers (or the type which we have used) could distinguish between two isomorphic systems' (Birkhoff and MacLane 1944,37). D ow nl oa de d B y: [C or co ra n, J oh n] A t: 18 :4 3 6 M ar ch 2 00 8 CATEGORICITY 191 tions which satisfy A are is om or phî.^ (Because of the peculiarity of 'every', a contradictory set A is vacuously categorical.) Incidentally, Veblen noted that the term 'categorical' was suggested by the philosopher John Dewey (ibid.). By the middle of the frst quarter of this century categorical characterizations of several important interpretations had been established. Many of these results are reported in Huntington 1905. It became common to 'identify' the intended interpretation of a formal language used to discuss a standard mathematical system with the system itself. For example, the phrase 'the system of natural numbers' sometimes indicates the intended interpretation of a language LK, where K is a set of 'arithmetic primitive symbols'. Using this terminology it can be said that the following systems had been categorically characterized: the natural numbers, the integers, the rationals, the reds, the complex numbers, and Euclidean space.6 Investigation of formal languages led to the further insight that whether a categorical characterization is possible depends not only on the form of the interpretation in question but also on the logical devices (variables and logical constants) available in the language chosen. For example, if L K is a first-order 5. Calvin Jongsma (University of Toronto) pointed out that some early mathematicians and logicians understood Veblen to be applying the term 'categorical' to systems that would now be called 'semantically complete' or, to use Church's phrase, 'complete as to consequences', as opposed to deductively complete (Church 1956, 329; compare Skolem 1928,523). Categoricity, of course, implies semantic completeness but the converse does not in general hold, as can be seen from Skolem's paper (ibid.). Later I noticed that Bukhoff and MacLane treat the categoricity of the axiom set for integral domains in a section called 'Completeness of the postulates for the integers' (1944,36). It is worth noting that, in that section, there is not a word about completeness in either the semantic or deductive senses. Incidentally, the mathematical use of the t e n 'categorical' is certainly due to Veblen (1904,346). However, it is not at all clear that Veblen uses it in its modem sense. In fact, he seems to be using 'categorical' to mean 'semantically complete'. The earliest use of 'categorical' in the modem sense is no later than Young (1911,49). 6. The fact that categorical characterizations of the traditional mathematical systems were see-consciously obtained by early mathematical logic suggests that the discovery of such characterizations may have been a stated goal of the field (see footnote 7 below). Developments leading up to research aimed at categoricity results are not well-known. An idea similar to isomorphism is attributed to Galois (1811-1832) and the tmn is found in an 1870 paper of Carnille Jordan (Kline 1972,765,767). Related ideas are found in Cantor's Grundhgen of 1883 (Jourdain 1915,76, 112) and in Dedekind (1887,93). Cantor and Dedekind each have theorems which can easily be applied to yield categoricity results, but neither seemed to have the idea of characterizing a class of systems by means of sentences (or propositional functions). The earliest genuine categoricity result I know of is due to Huntington (1902). Kline Ands that 'this notion [categoricity] was first clearly stated and used by . . . Huntington in a paper devoted to the real number system' (1972, 1014). Kline is referring to Huntington 1902, which proves the categoricity of s set of 'uxiorns' for 'absolute continuous magnitude'. It was alleged, falsely and without justification, by Young (1911, 154) that Hilbert's Founhtions of geometry (1899) contains a proof that Hilbert's axiomatization of geometry is categorical. Ironically, Hilbert 1899 does not even show awareness of semantic completeness, despite Veblen's apparent comment (I 904,346) to the contrary. D ow nl oa de d B y: [C or co ra n, J oh n] A t: 18 :4 3 6 M ar ch 2 00 8 192 J. CORCORAN language (i.e. one having only individual variables) without identity then no categorical characterizations are possible, and if LK is a frst-order language with identity, then only finite interpretations can be categorically characterized. Kreisel (1965, 148) points out that this limitation of first-order languages came as a surprise to logicians,' and he also makes the interesting observation that all finite interpretations are categorically characterizable in first order languages so that being finite and being first-order categorically characterizable are equivalent properties of interpretations. Many writers, includhg Kreisel (ibid.) and Montague (1965, 136), have noted that the many known categorical characterizations of the familiar classical systems all involve languages of second order, at least. However, if one moves beyond a first-order language with identity by the smallest possible amount, i.e. by allowing one one-place predicate variable, then not only are some infinite interpretations categorically characterizable but many important infinite interpretations are so characterizable. For example, if mathematical induction is written (Po & Vx(Px r> Psx)) 3 v y Py then the natural number system (relative to a primitive 0 for zero and s for successor) is categorically characterizable. Likewise, the integers, the reds and other important systems are also categorically characterizable using these 'slightly augmented first-order languages' CMontague 1965). To be more precise, define the formulas of the slightly-augmented language with non-logicalprimitives K, abbreviated 'SULK', to be exactly the formulas of the first-order language with identity but based on K + {PI, where P is a oneplaced predicate symbol not in K. The sentences of SULK are the formulas which lack free occurrences of the individual variables. The truth conditions for sentences of saLK are exactly these of the first-order sentences involving K + {PI except that a sentence S(P) involving P i s true under an interpretation i iff it is satisfied by every assignment of a subset of the universe of i to P. Thus every sentence S(P) is understood to be universally quantified with respect to P taken as a variable. Henceforth, P is called 'a one-placed predicate variable'. Note that S(P) and --S(P) are contraries, not contradictories. Note that SULK is not equivalent to the language in which there is universal quantification of P, unless one requires (1) that only one occurrence of the universal quantification of P is allowed per sentence, and (2) that the single universal quantification must occur at the front. Thus for example, VPS(P) is 7. How much of a surprise this was is another matter. Ellentuck says: 'One of the earliest goals of modem logic was to characterize familiar mathematical structures up to isomorphism . . . in a first order language' (1976, 639). In the opinion of this author it is doubtful whether any logicians held this as a goal, at least for very long. By the time of Skolem 1920, it was clear that no uncountable systems (e.g., geometry, the reds, or the complex numbers) could be categorically characterized in fust order, and there appears to have been very little interest in first order languages before that. D ow nl oa de d B y: [C or co ra n, J oh n] A t: 18 :4 3 6 M ar ch 2 00 8 CATEGORICITY 193 equivalent to the SULK sentence S(P), but -VPS(P) is not in general equivaleht to a SULK sentence. In particular, the existential predicate-variable quantifier is not definable in SULK. If somehow required to classify a slightly-augmented language as first-order or as second-order, many mathematical logicians would probably not hesitate to call it second-order. However, if slightly augmented languages are so classified it must be noticed that they are weaker than t h e w a l second-order languages (Enderton 1972, 268-269) in four ways. In the first place, SULK has no function variables. Second, it has no n-ary predicate variables for n greater than one. Third, instead of infinitely many one-placed predicate variables SULK has but one. Fourth, instead of formulas with arbitrarily many universal predicate-variable quantifications arbitrarily deeply imbedded, SULK contains only formulas with at most one such quantification occumng at the front, i.e. not imbedded at all. Thus SULK would be an extremely weak second-order language. In fact, Church has implicitly classified SULK as an applied first-order language with identity (1956, 548). In the opinion of the author, (1) classification of SULK as a first-order language would introduce confusion because there are many properties usually thought of as intrinsically first-order but which do not hold of SULK, and ( 2 ) classification of it as second-order would tend to mask its expressive weakness and its simplicity. It would seem best simply to refer to SULK by the name given above, or by the name 'slightly augmented first-order language'. The rest of the paper treats categorical characterization in the context of slightly augmented first-order languages." Section 2 establishes an extremely weak sufficient condition for categoricity which is nevertheless useful in constructing categorical sets of sentences. The main theorem is that an interpretation which satisfies an induction principle is categorically characterized by its induction principle together with its true atomic sentences and the negations of its false atomic sentences. In effect, we show that the form of an inductive interpretation is determined by its atoms. The proof of the theorem involves no reference to truth-functional combinations or to quantifications (except, of course, to those involved in induction principles). Section 3 applies the result of section 2. The paper is intended to be largely self-contained. Moreover, since terminology in logic has not yet been 8. The idea of categoricity is attractive even to logicians who want to avoid quantification over 'higher-order objects'. For example, in order to save categoricity in contexts devoid of such quantificiation Ellentuck 1976 goes to infintary languages, and Grzegorczyk 1962 restricts the interpretations to what he calls 'constructive models'. Other writers 'weaken' the idea of being categorical to being 'categorical in a power'. An axiornatization A is said to be categorical in k, where k is a cardinal number or power, if any two models of A whose universes have cardinality k are isomorphic. For further discussion see Enderton (1972, 147). This writer, however, has no interest in avoiding quantification over high-order objects, something he regards as a fundamental aspect of mathematical language. The motivation for considering slightly augmented languages is to isolate an idea which could have served as the core idea in many known categoricity proofs. D ow nl oa de d B y: [C or co ra n, J oh n] A t: 18 :4 3 6 M ar ch 2 00 8 194 J. CORCORAN standardized, it was thought worthwhile to repeat some rather elementary definitions. However, when this is done it is only done to the extent necessary for the immediate purposes at hand. 2. CATEGORICITY IN saLK Sub-sections 2.1 and 2.2 below deal with 'grammatical' and semantic preliminaries . Since categoricity is a purely semantic concept having no intrinsic dependence on object-language deductions, no system of formal proofs is provided. If the reader wishes to have a system of formal proofs for SULK it is sufficient to take any standard system for first-order with identity, e.g. Mendelson (1964, 57, 75) and for the monadic predicate variable take the 'rule of substitution' (which amounts to regarding a sentence involving P as a scheme). Sub-section 2.3 repeats the standard, exact notions of isomorphism and categoricity. Sub-section 2.4 associates with each interpretation a 'bar interpretation'. Sub-section 2.5 proves the main theorem. 2.1. Syntax for atoms and induction formulas Let K be a finite set of non-logical constants containing at least one individual constant and at least one function symbol. Besides these K can contain any number of individual cgnstants and, for each n > 1, any number of n-ary functions symbols and any number of n-ary relation symbols. For each such K, TK is the set of constant terms of K, i.e. TK is the closure of the set of individual constants of K under the operations of attaching an n-ary function symbol f in K to a string t, . . . t,, of n constant terms (formingft, . . . tJ. An atomic sentence of K is an identity 't, = t,' or a string 'RT, . . . t,' where R is an n-ary relation symbol in K and t,, . . . , and t, are all constant terms. The negation of an identity is written 't, # t,', and the negation of 'Rt, . . . t; is written '--Rt, . . . I,'. Atomic sentences add their negations are called atoms. Let P be the monadic predicate variable. If K = {O, s) where 0 is an individual constant and s is a monadic function symbol, then the induction formula for K is the following: I {O, s} (W & V x(Px 3 Psx)) 3 v y Py. If K = {O, 1, s, + I where 0 and S are as above, 1 is an individual constant, and + is a binary function symbol then the induction formula for K is as follows: I {O, 1, s, + I ((Po & PI) & vx,x, ((Px, & Px,) 3 (Psx, & P + x,x,))) 3 Vy Py. In general an induction principle has the form I (B & Vx, . . . x, (IH 2 IC)) 2 Vy Py, where B is the so-called 'basis', IH is the 'induction hypothesis', and IC is the D ow nl oa de d B y: [C or co ra n, J oh n] A t: 18 :4 3 6 M ar ch 2 00 8 CATEGORICITY 195 'induction conclusion'. Thus in order to define the induction formula IK for an arbitrary K it is suficient to define each of the parts, BK, IHK and ICK. The basis BK is the conjunction of all the formulas PC, where c is an individual constant of K. Let m be the maximum of the degrees ('arities') of the function symbols in K. Then the induction hypothesis IHK is the conjunction Px, & . . . & Px,. For each n-ary function symbol f in K, form Pfx, . . . x, The conjunction of all such formulas involves only the m variables x,, . . . , x, and is called the induction conclusion, ICK. Thus IK, the induction formula for K, is the following: IK (BK & Vx, . . . x, (IHK 3 ICK)) 3 Vy Py. The fact that I K is not uniquely determined is not important. 2.2. Semantics for atoms and induction formulas An interpretation i of LK is an ordered pair (u, d) where u is a non-empty set and d is a function defined on K and such that dc is in u if c is an individual constant, df is an n-ary function defined on u and taking values in u iff is an nary function symbol, and dR is set of n-tuples of members of u if R is an nary relation syrnb01.~ As usual, the denotation dit of a term t under an interpretation i is defined on TK as follows: di[c] = dc and diVt, . . . t,] = (df)di[t,l . . . di[tnl, i.e. the denotation of an individual constant is its interpretation and the denotation of a function symbol attached to terms is the interpretation of the function symbol applied to the denotations of the terms. As a result of the way that TK, the set of terms, is defined it is obvious that di is defined on all of TK. The range of diis then the set of objects in u which are denoted by constant terms. For example, if K = { 1, s}, u is taken to be the set of natural numbers including zero, dl is taken to be the number one and ds is taken to be the successor function, the range of d i is only the set of positive numbers; the range of d i need not be, all of u. 9. It is true that, in the strict sense of the term 'set', each element "occurs" only once. Thus when one speaks of 'the set of solutions to an algebraic equation' the word 'set' is not being used in the strict sense. Sometimes the word "family" is used to indicate a 'set' wherein an element can have multiple occurrences. If repetitions of the same element are distinguished by indices, one speaks of an 'indexed family' (Halmos 1960, 34). Thus a system, in the sense of section 1, is a class of mathematical objects together with a finite family of distinguished elements, functions and relations. An interpretation can then be seen as a certain kind of system, namely as a system wherein the family is indexed by a set of symbolic characters (namely, by 9. It is important, however, that the indexing respect semantico/syntactic distinctions, i.e. that elements are indexed by individual constants, that n-ary functions are indexed by n-ary function symbols and that nary relations are indexed by n-ary symbols. Bridge (1977, 6, 16) treats the topic explicitly in this way. For a treatment of semantic categories, see Tarski 1934,2 15-236. D ow nl oa de d B y: [C or co ra n, J oh n] A t: 18 :4 3 6 M ar ch 2 00 8 196 J. CORCORAN Let diTK be the range of di. Notice that because of the way TK is defined, diTK is the closure of the set of objects in u denoted by individual constants under the functions denoted by function symbols, i.e. diTK is a subset of any subset of u containing the said objects and closed under the said functions. Where no confusion results 'di' is sometimes written 'd'. As usual an identity 't, = t,' is true under i if dt, is the same object as dt, (i.e. dt, = dt,) and false under i if dt, and dt, are different (i.e. dt, f dt,). 'Rt, . . . t,,' is true under i if the n-tuple of objects denoted by the terms is in the relation denoted by R((dt,, dt,, . . . dtJ E dR), and 'Rt,t,. . . t,' is false under i, otherwise. Negation, of course, reverses truth-values. For a given interpretation i = (u, d) an assignment UP to the monadic predicate variable P is simply a subset of u. When UP is assigned to P, (1) BK is true under i if UP contains the objects denoted by individual constants; (2) Vx, . . . x,,,(IHK 3 ICK) is true under i if a P is closed under the functions denoted by function symbols; and (3) VyPy is true if UP = u. More particularly, to say that the induction formula IK is true of a P under i is to say that if UP both contains the objects denoted by individual constants and is closed under the functions denoted by function symbols, then aP is u. And IK is true under i if IK is true of each UP under i. More particularly, I K is true under i iff every subset of u which contains the objects named by individual constants and which is closed under the functions denoted by function symbols is u. Since the range of d \ diTK, is such.a subset, if IK is true under i then diTK = u. Conversely, if diTK = u then I K is true under i. Thus, to say that IK is true under i is to say nothing but that diTK = u. In other words, the induction formula 'says' that every object is denoted by some constant term. When induction holds in i, we say that i is inductive. 2.3. Isomorphism and categoricity Let i = (u, d) and j = (v, e) be two interpretations of LK. Then i is said to be isomorphic to j if there is a one-one function h from u onto v which 'preserves the structure' in the sense that: if c is an individual constant then h[dc] = ec; iff is an n-ary function symbol then for every b,, . . . , b, in u, and if R is an n-ary predicte symbol then for all b,, . . . , b, in u, if and only if Let K = {0, s} and take i and j as follows. The universes u and v y e both the set I of integers, ds and es are both the successor function, do is zero and eO is one hundred. The function hx = x + 100 is an isomorphism between i and j. D ow nl oa de d B y: [C or co ra n, J oh n] A t: 18 :4 3 6 M ar ch 2 00 8 CATEGORICITY 197 Here (and below) we exploit the set theoretic notion of a function being a set of ordered pairs by writing i = (I, (0, zero), (s, successor)) and j = (I, (0,100), (s, successor)). If i is isomorphic to j then any sentence of SULK true in one is true in the other, i.e. T(i) = T(j). The proof of this is straight-forward but it requires a set of definitions rather more complete than is otherwise required in this paper. Let A and B be two sets of sentences. Then i is a model (or true interpretation) of A if A c T(i) and A logically implies B if every model of A is a model of B. If A logically implies B then, for purposes of smooth expression, B is said to be a logical consequence of A. If A has no models, A is said to be contradictory. Because of the peculiarity of 'every', a contradictory set implies every set. IfA implies B and B = {p} then A is said to imply p. A set A of sentences is categorical if all models of A are isomorphic to each other. Because of the peculiarity of 'all', contradictory sets are vacuously categorical. Notice that if A is categorical then for every sentencep not involving P, A implies p or A implies -p. It fails in the case of sentences involving P only because for them negation does not reverse truth-values: S(P) and -S(P) are contraries, not contradictories. Examples. Let K = (0, s). Let A0 be the set of sentences true in iN = (N, d ) where N is the natural numbers, do is zero, and ds is the successor function. Let SnO indicate n occurrences of S followed by 0. Notice that the true atoms are simply the logical identities (SnO = SnO) and the negations of the other identities ( P O f SmO, for n f m). Let A be the Peano Postulates for zero and successor, i.e. A = {IK, Vxy(sx = sy 2 x = y), Vx(sx # 0)). Reasoning which shows that A is a categorical characterization of iN is familiar (compare Birkhoff and MacLane 1953,54-56). Let A 1 = {IK}. It is obvious that A1 is not categorical because it is satisfied by the unit model i l = ({O}, d ) where do is zero and ds is the identity function. More generally it is clear that for any K, IK and any set of positive atoms is satisfied by the unit model, and thus, if IK and a set of atoms is categorical, a negative atom must be present. Let A2 = {IK, S O f 0,. . . , SnO f 0,. . .}; It is worth noting that A2 implies Vx(Sx f 0). (IK 'says' that every object is named by a term S"0. The joint effect of the rest of the sentences in A2 is to say that if an object is named by a term its successor is not zero.) Nevertheless, it is clear that A2 is not categorical because it has as a model i2 = ({O, 11, d) where d o is zero and ds is the identity function. Let A 3 be the result of D ow nl oa de d B y: [C or co ra n, J oh n] A t: 18 :4 3 6 M ar ch 2 00 8 198 J. CORCORAN adding to A 2 the rest of the negative atoms, S'O f SW, for n f m and m f 0: Consider a sentence SS"O = SSmO 2 S"O = SmO. If m = n then the sentence is logically true. If m f n then, since the negations of the antecedents are atoms in A 3, the sentence itself is implied by the atoms in A 3 . By the reasoning of the previous paragraph, then, A 3 implies V,'xy(sx = sy I> x = y). Thus A 3 implies all three of the Peano postulates and is thus categorical. From the perspective of the next sub-section, the main feature of A3 is that it is atom-complete in the sense that for every atomic sentence p, either A3 implies p or A3 implies -p. The main theorem is that every atom-complete set containing induction is categorical. 2.4. Bar cointerpretations The denotation function di maps TK into u. Thus d i can be used to define an equivalence relation Ei on TK in the usual way, i.e. let tlEit, iff ditl = d't,. Let 2' be the equivalence class oft, and let TK' be the set of equivalence classes. Let 2' be the function from mi into u such that Z l i = d't. Notice that 2' is oneone from TK~ onto d i ~ K . Let i = (u, d) be inductive. The mapping %'is therefore a one-one onto function from mi, the set of equivalence classes of terms, to the universe of i. Now we define a denotation function 'b on K in such a way that ai is an isomorphism from { r n Y i b } to i = {u, dj. The interpretation ( m i , 'b} thus induced by ii' is called the bar cointerpretation. One could, of course, define (TK', 'b) as 'the isomorphic image of i under the inverse of 2''. However, the information that we need to highlight is brought out better by defining ( m i , 'b) explicitly and then verifying that i is its isomorphic image under ai. (1) The universe of (m ib) is, of course, TK' The interpreting function 8 is defined as follows. (2) 'bc=ti. Now notice that ft, . . . t;=ff,+,.. . ti,, is implied by the following taken together I f = . . . ,I; = I&,. This means that the equivalence class of a term ft, . . . t, is a "function" of the equivalence classes of its components t,, . . . , t, (3) Thus we can define 'bf as follows: . Finally notice that taken together, imply that 'Rt, . . . t,' and 'Rtn,l.. . t2,' have the same truthvalue in i. (4) Thus we can define the relation 'bR to hold of {i.,', . . . , I2 iff 'Rt, . . . tnY is true in i. D ow nl oa de d B y: [C or co ra n, J oh n] A t: 18 :4 3 6 M ar ch 2 00 8 CATEGORICITY To check that ai is an isomorphism, one need only check: (1) that 2' is one-one and onto from TK' to u, (2) that Ji{'bc) = dc, (3) that ;i'{('bf)(i/, . . . , Ii)} = (df)($$f, . . . , diT$, and (4) that (I,, . . . , I , ) E 'bR iff (&:, . . . , air/) e dR. The above reasoning establishes the following: Lemma. Each inductive interpretation is isomorphic to its bar cointerpretation. Now we establish the main lemma of the paper Lemma. Two inductive interpretations which satisfy the same atoms have the same bar cointerpretation. Proof: Let ( TK' '6) and ( m i , jb) be the two bar cointerpretations. To show: ( I ) TK' = TKI, (2) 'bc = jbc, (3) ibf = jbf a d (4) %R = j b ~ . To see (1) notice that t,Eit, holds iff 't, = t,' is true in i. By hypothesis the latter holds iff '1, = 2,' is true in j. Again the latter part holds iff t,Ejt,. Therefore mi = mj. It follows then that 3 = 2j. Thus 'bc = jbc. (2) is established, It also follows that for all t,. ?=?, and, in particular, that -. -fr, . . . t; =frl . . . t,/. Thus 'bf = jbfi (3) is established. To see (4) notice first that (Ti , . . . , i/) E 'bR iff 'Rt, . . . t i is true in i. By hypothesis the latter holds iff 'Rt, . . . t,' is true in j. Again using the definition of bar interpretation, now for j, it follows that 'Rt, . . . t,: is true in j iff SO 'bR = jbR. Q.E.D. 2.5. Main theorem Any atom-complete set of sentences which includes induction is categorical. Proof: Let S be such a set of sentences. If S is not satisfiable then S is vacuously categorical. Assume that S is satisfiable. Let i and j be models of S. Since S is atomcomplete , i and j satisfy the same atoms. Since S includes induction, i and j are inductive. By the lemmas, i and j are isomorphic. Q.E.D. In order to put this theorem in perspective one can note that its import is D ow nl oa de d B y: [C or co ra n, J oh n] A t: 18 :4 3 6 M ar ch 2 00 8 200 J. CORCORAN simply the following. The set of true atoms of an inductive interpretation, taken together with induction, characterize the interpretation up to isomorphism. Thus ifone is characterizing an inductive interpretation, and ifone can tell that the axioms already set down are sufficient to imply induction all of the true atomic sentences and the negations of the false ones, then the goal of categoricity is achieved. It is not hard to see that the same theorem holds in all stronger languages, so it holds for second order languages and for higher order languages. Thus one might wonder whether saLK can be weakened without losing the theorem. The immediate answer is affirmative because in the entireproof not a word was said about any sentences besides (constant) atomic sentences, their negations, and induction. Thus the result holds for inductive atomic languages which are deJned as follows. Let K be a finite set of non-logical constants as above. The logical constants of the inductive atomic language based on K, iaLK are =, and a new symbol 'I' to be explained presently. The sentences are all of the atomic sentences of saLK, their negations, and I. The interpretations are the same as for saLK and the truth conditions for the atoms are the same. But I is true in i iff i is inductive. The question also arises whether the condition of atom-completeness plus induction can be weakened without losing categoricity. It is clear that one cannot 'throw out' the negations of the atomic sentences because the example A2 in sub-section 2.3 above shows that the true atorrlic sentences of an interpretation (plus induction) do not characterize the interpretation up to isomorphism. One can also easily show that if S contains induction and is categorical then S is equivalent to an atom-complete set. Thus the above theorem is the strongest possible in the sense that atom completeness is the weakest possible condition sufficient to guarantee that a set of sentences containing induction is categorical. 3. APPLICATIONS 3.1. Repetition theory Imagine that one is dealing with 'sets' of 'repeatable' objects where 'multiplicities' are counted. For example 13,31 is the 'set of roots of x2 6x + 9 = 0' but 131 is 'the set of roots of x 3 = 0'. Such 'sets' are called iterates (or heaps or multiplicities: Hailperin 1976, 88). At any rate if I i3 a set of objects being repeated then with each iterate, r, one can associate a unique function f from I into N such that for each repeatable object x, fx is the number of times x is repeated in r. The functions f are called repetitions of I . ' O We consider the case where I = {a, 6). Let u be the set of repetitions of I, i.e. the set of functions from {a, b} into N. Let 0, the null repetition, be the function: 10. Compare footnote 9 above. D ow nl oa de d B y: [C or co ra n, J oh n] A t: 18 :4 3 6 M ar ch 2 00 8 CATEGORICITY 20 1 fa = 0,fb = 0. The a-successor function s, is the function which 'jacks up' the a-component off by one, i.e. sa f (a) = f (a) + 1 and sJ(b) = f (b). Likewise s b is the b-successor function. Thus we are considering an interpretation i = (u, d ) of SULK with K = (0, s,, sb}. A little thought suffices to see that the following axioms are true: A1 VX(S,X f 0 & SbX f O), A2 Vxy((s2 = say 2 x = y) & (sbx = sby 2 x =>, A3 Vx(sasbx = s~s,x), A4 Vx(sax 4 sbx), A5 IK. By looking at what the constant terms denote one can see that an identity 't, = t,' is true in i iff the repetition of {s,, sb} in t , is the same as the repetition of {so, sb} in t,. For example the repetition of ha, sb/ in the terms 4, A51, implies each true identity and the'negation of each false identity. Thus it implies an atom-complete set including induction, which is categorical by the theorem, and so it is categorical itself. 3.2. Other possibilities It is already clear that various versions of Peano arithmetic (or number theory) admit of this treatment. Kleene (1952, 246) has discussed an infinite class of interpretations which he calls 'generalized arithmetics'. It is certainly possible to categorically characterize any one of them and perhaps to give a general formula for treating all of them of course, using the above method. There is an infinite class of theories of strings based on Tarski 1934, 172 and categorically axiomatized in Corcoran, Frank and Maloney 1974. In the same work one finds another infinite class of theories dealing differently with strings based on an idea of Hermes. Both of these classes admit of treatment by this method. In addition it is possible to deal with finitely branching trees and hereditarily finite sets in this way. For purposes of discussion assume that K has no relation symbols. This restriction does not matter in principle but it holds in all the familiar examples which will come to mind. Define an equation in K to be any constant identity (as above) or any sentence of the form Vx, . . . x,t, = t, where t , and t, are terms in K and all variables occurring in t, or t, or both are among x, . . . x,. Let A be a set of equations. A model of A is called an A-algebra. Let I(A) be D ow nl oa de d B y: [C or co ra n, J oh n] A t: 18 :4 3 6 M ar ch 2 00 8 202 J . CORCORAN the set of constant identities implied by A. Let NI(A) be the set of negations of the constant identities not implied by A. If i is a model of A + IK + NI (A) then i is what has been called a 'free' A-algebra. For example if A = { ~ x y z ( x + 0 , + z ) = ( x + y ) + z ) ) and K = i a , ...., a, ,+} then the models of A + IK + NI(A) are the free semigroups on n generators. It is clear that the above methods constitute one approach to getting categorical axiomatizations for the theories of free A-algebras. If A or NI(A) is not recursively enumerable then the above approach may not work at all, even given a maximum of ingenuity. 3.3. Strong induction The induction principle IK treated above is the weakest possible induction principle for LK, a language whose set of non-logical constants is K. We saw that I K is true in i iff every object of u is denoted by a term in TK. The weak induction principles IK are essentially unique, but for each K there are many stronger induction principles and, in fact, there are generally several which are maximally strong. To consider the first class of stronger induction principles for LK consider a proper subset K1 of K which still contains at least one individual constant and one function symbol. The induction principle IK1 used in LK is stronger than IK because it holds.in i iff every object in u is denoted by a term in TK 1 (a proper subset of TK). Clearly, IK 1 implies IK but not vice versa. Thus IK 1 is 'stronger' than IK. For the general definition of strong induction principle, let T be a proper subset bf TK. Let S(K) be a sentence possibly involving the monadic predicate variable. If S (K) has the truth condition that it is true in i iffevery object in u is denoted by a term in T, then S(K) is a strong induction principle for LK. For example take K = {I , +}. Then IK is (PI & Vxy(Px & Py 3 P(x + y)) 3 VzPz. The following, however, is a strong induction principle: (PI & Vx(Px 3 P(x + 1)) 3 VzPz. The truth condition for this sentence is that every object in u is denoted by a term of the following class: 1, (1 + I), ((1 + 1) + I), . . . , which leaves out (1 + (1 + I)), ((1 + 1) + (1 + I)), etc. It is obvious that IK + associativity implies this strong induction principle.ll At any rate, since every strong induc11 . One of the referees wanted to know whether there are 'maximally strong' induction principles, i.e. whether there are strong induction principles which are not weaker than any others. One strengthens a strong induction principle by restricting the class T of terms which it 'forces to cover the universe'. For example, the principle cited above can be strengthened by replacing the (non-constant) term 'x + 1' by ' (x + 1) + 1'. The most restrictive class of terms is, of course, the null set which corresponds to a contradictory 'induction principle', e.g. VyPy. Short of this the strongest induction principles would 'force a unit set to cover the universe', e.g. PC 2 V.vPy. D ow nl oa de d B y: [C or co ra n, J oh n] A t: 18 :4 3 6 M ar ch 2 00 8 CATEGORICITY 203 tion principle implies IK, the above results hold when a strong induction principle is substituted for the induction principle. 3.4. 'Negative' applications In this sub-section let LK be any second-order language with identity and let DK be any sound and recursive system of deductions for LK. For example, let DK be the system of Church I956 suitably extended with rules and axioms dealing with function symbols. In this sub-section it is important to recall the distinction between (1) characterizing an interpretation i by means of a subset of T(i) (the truths of 9 and (2) axiomatizing the truths of i. The point of characterizing i is descriptive and criterial; one aims at distinguishing i from other interpretations. The point of axiomatizing is to form the basis for a deductive development of the truths of i. From the standpoint of characterization the best that can be done (when it can be done) is a categorical characterization. With a categorical characterization an interpretation is distinguished from every other interpretation from which it can be distinguished by formal means. From the standpoint of axiomatization it is clear that the best that can be done (when it can be done) is a deductively complete axiomatization, i.e. a recursive subset of T(i) from which every member of T(i) is deducible by a (finite) deduction. It is obvious that the set of theorems deducible from a set of axioms is necessarily a recursively enumerable subset of the truths no matter which sound, recursive system of deductions is used but that, in general, the set of theorems is sensitive to choice of deductive system. Below we assume a fmed deductive system DK. Some early postulate theories (e.g., Veblen 1904, 346) were clear about the conceptual distinction between characterization and axiomatization and about the possibility of an axiomatically inadequate categorical characterization at least to the extent of explicitly mentioning the possibility that a categorical characterization need not be a (deductively) complete axiomatization. This possibility, of course, entails the possibility of 'logically' incomplete underlying logics (wherein semantic consequences of a given set of axioms are not deducible as theorems). At that time, however, there was no suspicion of the idea of recursiveness, nor, afortiori, of the relevance of recursiveness and recursive enumerability to problems of axiomatizability. Now we can see that if the set of truths of an interpretation is not recursively enumerable then there is no way to give a complete axiomatization even if the logic is complete. It follows immediately from the Godel incompleteness result that a (recursive) set of sentences which provides a categorical characterization need not provide a complete axiomatization. Moreover, in such cases, it follows that there are infinitely many other categorical characterizations each of which provides a better axiomatization in the sense of providing the basis for the deduction of additional theorems not deducible from the fust characterization. Separate from the recursiveness considerations which lead to mismatches D ow nl oa de d B y: [C or co ra n, J oh n] A t: 18 :4 3 6 M ar ch 2 00 8 204 J. CORCORAN between characterization and axiomatization are the so-called 'compactness' considerations which lead to additional mismatches (compare Corcoran 1972, 378) . Since every deduction is finite and therefore involves only finitely many axioms, no consequence of an infinite axiom set which depends on infinitely many of the axioms can be deducible from those axioms. It is compactness considerations rather than recursiveness considerations which are operative in the rest of this discussion. It might be thought that any categorical characterization of an interpretation provides the basis (given a suitable deductive system) for the deduction of the 'obvious' truths of the interpretation. That is, one might expect that any truth not deducible from a categorical characterization must be a 'pathological' or 'complicated' proposition such as a so-called Godel sentence or a statement of consistency. Admittedly this point has not been discussed much in the literature (see Paris 1978). Nevertheless, the test of categoricity given above permits the establishment of categorical characterizations from which the most elementary general truths are not deducible, no matter what sound deductive system is used (regardless of the criterion of recursiveness). Take K = (0, st and take S as the set of all true arithmetic identities (snO = snO) and the negations of all of the false ones. By the above theorem S + IK is a categorical characterization of i = (N, (0, zero), ( s, successor)). Thus S + IK implies Vx(sxf 0). However, it is impossible to deduce p = Vx(sx $0 ) from S + IK using DK or any other sound deductive system DK1 because if, say, p, . . .p ,p is a deduction of p from S + IK and DKl is sound then p is implied by the finite number of premises in p, . . .pn. But it is easy to see that no finite number of sentences in S + IK implies Vx(sx 4 0). Examples of this sort can be multiplied. Take K = {0, s, +). Take i = (N, (0, zero), (s, successor), (+, addition)). For S take Vx(sx # 0), Vxy(sx = sy 3 x = y), and the true identities (snO + smO = sn + "0) and the negations of the false ones. S + IK is categorical but it is impossible to deduce Vxy((x + y ) = 0.I + x)) from S + IK using a sound deductive system. These examples are but other illustrations of the vast difference between characterizing an interpretation and axiomatizing its set of truths. The examples point to the conclusion that the connection between the two is weak. In particular, it is now clear that a 'best possible' characterization can be a very poor axi~matization.'~ The class of categorical characterizations of a given inductive system includes many which are virtually useless as axiomatizations. 12. Since semantic completeness is implied by but does not imply categoricity, it follows that semantic completeness is not sufficient for an axiomatization to be 'good'. In fact, as far as axiomatization is concerned, semantic completeness seems to be beside the point unless supplemented by other conditions formulated in accordance with goals arising in particular cases. These observations are due to George Weaver. D ow nl oa de d B y: [C or co ra n, J oh n] A t: 18 :4 3 6 M ar ch 2 00 8 CATEGORICITY 205 It is clear from a survey of the relevant literature not only that the early postulate theorists were unaware of the recursiveness considerations (as mentioned above) but also that they were unaware of compactness considerations as well. One can not automatically conclude, however, that they were misguided in using categoricity as an index of worth of an axiomatization. One must realize that the above counterexamples all involve infinite sets of axioms whereas the earlier logicians, occasionally explicitly (Veblen 1904, 343), conceived of an axiomatization as inherently finite. And in the opinion of this writer, the philosophical wisdom of abandoning the finiteness condition should be questioned despite the undeniable advances that came as a result of considering the mathematical consequences of relaxing that condition. 3.5. Heuristics The above test of categoricity requires, for its application to a given set of axioms for an inductive system, that one first establish that the axiom set implies each of the true atoms of the system. It is entirely possible that this preliminary step is more demanding in a given case than a straight-forward categoricity proof. However, if one is given the system (interpretation) alone and the problem is to find a manageable categorical set of axioms then the goal of deducibility of the true atoms is often an effective heuristic which leads to the discovery of the required axiom set. ACKNOWLEDGEMENTS Sincere thanks to Nicolas Goodman (Mathematics, SUNYIBuffalo), William Frank (Philosophy, Oregon State University), George Weaver (Philosophy, Bryn Mawr College) and Michael Scanlan (Philosophy, SUNYIBuffalo) for useful criticisms and for pleasant discussions. This paper was presented to the Buffalo Logic Colloquium in November 1977 and to the Association for Symbolic Logic in December 1977. Several improvements in the final draft were made on the advice of Michael Resnik (Philosophy, University of North Carolina) of the Editorial Board of this publication. I would like to emphasize the fact that each of the persons here acknowledged made extensive contributions to this research. BIBLIOGRAPHY Barbut, Marc 1966 'On the meaning of the word "structure" in mathematics', orig. publ. Les temps modernes, no. 246; translated in Lane 1970 by Susan Gray. Birkhoff, G., and S. MacLane 1944 A survey of modern algebra (revised edition 1953), Macmillan, New York. 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