The Ricean Objection: An Analogue of Rice's Theorem for First-order Theories

We propose here an extension of Rice’s Theorem to first-order logic, proven by totally elementary means. If P is any property defined over the collection of all first-order theories and P is non-trivial over the set of finitely axiomatizable theories (i.e., P holds for some, but not all theories), then P is undecidable. This not only means that the problem of deciding properties of first-order theories is as hard as the problem of deciding properties about languages accepted by Turing machines, but also offers a general setting for proving several undecidability results in first-order theories. 1 Undecidability in logic and computer science The undecidability of computational problems by Turing machines (TM) is a formidable theoretical result that affects the practice of computer programming. Rather than being isolated, undecidability is an ubiquitous phenomenon, in the sense that any relevant (non-trivial, in the sense of holding for some pograms but not all of them) property about computer programs is undecidable: indeed, H. G. Rice proved more than half a century ago (cf. [Ri]) that any property defined over the set of languages accepted by Turing machines is either trivial or undecidable. Rice’s theorem has an equivalent (perhaps better known) statement in the language of recursive function theory, and it is reasonable to ask for the effects of such ‘‘Ricean’’ objection to logic in general. Some previous work already point to this direction: in [GMS], a book devoted to studying computer models in philosophical research, the first chapter contains two theorems on undefinability of chaos which bear some similarity with Rice’s (sometimes called Rice-Shapiro) theorem in its recursion-theoretical version. A topological version of Rice-Shapiro Theorem appears in [HM]. The uncomputability barriers of Ricean theorems poses hard challenges for the understanding and characterization of the distinction between what is known in programming jargon as ‘‘extensional’’ properties of programs, (i.e. those properties of the class of inputoutput pairs computed by a program), and the ‘‘intentional’’ properties of programs, (i.e. properties of the program code): indeed, the first will suffer from undecidability, while the latter are generally decidable. In logic, analogous to this difference would be the distinction between expressability (properties of theories) versus syntactic-linguistic features of logic theories. Our main result shows that universal undecidability properties granted in computation are also shared by the first-order logical theories. The conclusion is that the problem of Vol. 16 No. 6, © The Author 2008. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org doi:10.1093/jigpal/jzn023 at U niersity of Piburgh on Jauary 2, 2015 http://jigpfordjournals.org/ D ow nladed from 586 The Ricean Objection: An Analogue of Rice’s Theorem for First-order Theories deciding properties of mathematical theories is as hard as that of deciding properties about the language accepted by Turing machines, and this may have interesting consequences. We prove in this note, by an elementary argument, that these ideas can be extended to formal systems not directly related with the notion of computation. By defining for first-order theories analogous concepts as the involved in Rice’s original result, we can show a general form of this theorem to be valid for first-order theories as well. In the next section we discuss the ideas and concepts involved. The formal statement of the problem and its proof is given in the subsequent section. 2 Statement of the problem Given a Turing machineM , let Lang(M ) be the set of words accepted byM . Some immediate questions about the language represented by the TM M can be posed: • Does the empty word belongs to Lang(M )? • Is Lang(M )= ∗, i.e., is it the case that every word over the input alphabet of the TM M is accepted by M? It is possible to show that these and several other problems of the same kind are algorithmically unsolvable. A starting technique for such proofs is to show, by diagonalization, that a first problem is undecidable and then prove, using reduction, that so are the other problems. It is usual to use, as first step, the problem of deciding whether a TM M accepts a word w of its input alphabet: ATM ={〈M ,w〉 | M is a TM and M accepts the word w}. In the previous notation, 〈M ,w〉 is a reasonable codification of an instance of the problem. A. Turing [Tu] proved that ATM , known as the halting problem, is undecidable by Turing machines. The undecidability of the halting problem and some other problems are special cases of the next result. Rice’s Theorem. Let P be a property of languages that is satisfied by some, but not by all, recursively enumerable languages. Then the following problem is undecidable: DP : Given a TM M , does Lang(M ) satisfy P? The proof of this remarkable result can be found in Rice [Ri]. We can interpret a TMM as a finite representation of the possibly infinite set Lang(M ). In exactly the same way, if we interpret a logical theory T as a set of formulas from a specific first-order language, we can try to represent it by a finite set A of axioms (formulas). If Th(A)=T , i.e., the set of formulas provable from A is the set T , then we have found a suitable representation for the theory T . It is possible to proceed in a similar way and formulate some questions about the theory T , finitely represented by A. • Is T a complete theory? • If ψ is a formula in the language of T , does ψ ∈T? • Is T a decidable theory? at U niersity of Piburgh on Jauary 2, 2015 http://jigpfordjournals.org/ D ow nladed from The Ricean Objection: An Analogue of Rice’s Theorem for First-order Theories 587 To answer this and other questions, we chose a sufficiently expressive first-order language. We assume that the language of all the first-order theories refereed will be that generated by the signature ={0,s,+,∗}, where 0 is a constant, s is an unary function and + and ∗ are binary functions. Denote by L the set of valid formulas generated by this signature. This is not a great restriction since the result we present can be easily translated to richer first-order languages. If a theory T is finitely axiomatizable, its codification will be denoted by 〈T 〉. In this way, we are only interested in properties of theories that are finitely axiomatizable, and hence finitely representable. The next results show that some properties of first-order theories are undecidable. Let Q be the finite set of axioms for weak Arithmetic as presented in [EC]. Theorem 1. Th(Q) is an essentially undecidable theory.