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    Decidability and Essential UndecidabilityAuthor(s): Hilary PutnamReviewed work(s):Source: The Journal of Symbolic Logic, Vol. 22, No. 1 (Mar., 1957), pp. 39-54Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2964057 .

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    THE JOURNALOF SYMBOLICLOGICVolume 22, Number 1, March 1957

    DECIDABILITY AND ESSENTIAL UNDECIDABILITYHILARY PUTNAM

    1. There are a number of open problems involving the concepts ofdecidability and essential undecidability.' This paper will present solutionsto some of these problems. Specifically:(1) Can a decidable theory have an essentially undecidable, axiomatiz-,able extension (with the same constants) ?2(2) Are all the complete extensions of an undecidable theory ever decid-able ?We shall show that the answer to both questions is in the affirmative.In answering question (1), the decidable theory for which an essentiallyundecidable axiomatizable extension will be constructed is the theory ofthe successor function and a single one-place predicate. It will also be shownthat the decidability of this theory is a "best possible" result in the followingdirection : the theory of either of the common diadic arithmetic functionsand a one-place predicate; i.e., of addition and a one-place predicate, or ofmultiplication and a one-place predicate, is undecidable.2. Before establishing the main result, it is convenient to give h simpleproof that a decidable theory can have an axiomatizable (simply) un-decidable extension. This is, of course, an immediate consequence of themain result; but the proof is simple and illustrates the methods that weare going to use in this paper.For this purpose, we select the theory of the successor function (call

    Received October 4, 1956.1 The terminology of this paper is that of Undecidable theories, Tarski, Mostowski,and Robinson, North Holland Publishing Co., Amsterdam (1953). This work will becited as "U.T.".A solution to problem (1) has erroneously been announced by Myhill (Solution of aproblem of Tarski's, this JOURNAL, vol. 21 (1956), pp. 49-51). Actually Myhill hassolved a related problem; but an answer to (1) does not follow from what he proves,as he asserts. (Cf. section 7., below).(Except for section 7) attention in the present paper is confined to theories with afinite number of constants. Kreisel has previously solved problem (1) for theories which

    use an infinite number of constants. A statement of Kreisel's proof appears in hisreview of the article by Myhill just cited; see Mathematical reviews, vol. 17, no. 8(Sept. 1956), p. 815.2 This is easily seen to be a reformulation of a problem in U.T., p. 18. (Cf. thediscussion below). From now on, when the term 'extension' is used, 'with the sameconstants' will be understood.3 This problem was suggested by G. Kreisel whose interest and infectious enthus-iasm have led me to work on these questions.

    39

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    40 HILARY PUTNAMit 'R'). This theory is decidable.4If to R we adjoin two new individualconstants, a and b,5 and no new axioms, we obtain another theory R'.A sentence of R' is valid only if its universal generalizationwith respectto a and b is valid in R; so R' is likewise decidable. In the terminologyof U.T., R' is an inessential extensionof R.

    Let us define 0 as a, 1 as S(a) ("the successor of a"), 2 as S(S(a)), etc.And finally, let us construct a theory L by adding to R' the followingaxioms:(3) -_(b mJ (i = 1,2, . ..)where mi, M2, M3, ... run through the membership of some set of positiveintegers which is recursively enumerable but not recursive (for the sake ofdefiniteness, let us take the set of Gbdel numbersof theorems of quantifi-cation theory).We now claim:

    THEOREM 1. L is recursivelyaxiomatizable,and (simply) undecidable.Proof:i) L is axiomatizable: we have given a recursively enumerable set ofaxioms for L. But, according to Craig's theorem6every theory that canbe axiomatized with a recursively enumerable set of axioms can be axiom-atized with a recursive set of axioms. So L is axiomatizablein the sense ofU.T. (recursively axiomatizable).ii) L is undecidable: it suffices to show that b#n is not a theorem of Lunless it is an axiom; in other words, if n does not belongto the set {ml, M2,

    in3, ... }, then b=n is consistent with all the axioms bomi. For if b-nwere inconsistent with the axioms of L, b=n would have to be inconsistentby itself (sinceit entails all of those axioms), so (b n) would be a theoremof R', and (x)'(x n) would be a theorem of R', which is absurd.iii) L is not essentially undecidable: for if n is any integer not in {ml, m2,

    M3, ... }, the theory obtained by adding b=n as sole new axiom to R'4U.T., p. 64. Also, a decision method for the elementary theory of the successor

    function and an arbitrary one-place predicate is given in section 3. below. This can,of course, be used to decide sentences of R.I Logical signs are used as names of themselves throughout the present paper,and not in their object-language use.

    6 Craig's method is as follows (On axiomatizability within a system, this JOURNAL,vol. 18 (1953) pp. 30-32): let {A,, A2, . ...} be a recursively enumerable but not re-cursive set of axioms for a theory T. Replace each axiom Ai by the conjunctionswhose members are Ai repeated n times, for each n such that n is the number of aproof that Ai belongs to the set {Al, A2, ... } (or rather that the Godel number of Aibelongs to the corresponding recursively enumerable set of integers). The recursivenessof the new axiom set follows from the fact that the set of pairs (n, A) such that Abelongs to {Al, A2, .. . } and n is the number of a proof that this is the case (in a suit-able formalism) is recursive.

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    DECIDABILITY AND ESSENTIAL UNDECIDABILITY 41

    is a consistent extension of L, by the remark just made; and this theoryis decidable because it is a finite extension of R' and R' is decidable. q.e.d.

    Thus we have that a decidable theory may have an undecidable andaxiomatizable extension: for R' and L are two theories that stand in justthis relation to one another.

    3. In this section we shall establish the decidability of a certain simpletheory G: this will serve as a basis for showing, in the next section, that adecidable theory (an inessential extension of G) may have an axiomatizableand essentially undecidable extension.

    G is the theory of the successor function and a single one-place predicate;i.e., G has a single monadic operator S, and a single monadic predicate P.The variables of G have arbitrary integers as values; S(x) means "thesuccessor of x"; and P designates an arbitrary class of integers. A sentenceof G is valid if it is true under this unterpretation (for all values of P).For convenience, let us also introduce into G the symbol Pr for "thepredecessor of x." Evidently Pr is definable in terms of S as follows:

    y=Pr(x) =df x=S(y).Instead of S(x) we shall henceforth write x+ 1; and likewise x+2 for

    S(S(x)), x I for Pr(x), etc. Since Pr(S(x)) =x=S(Pr(x)), we can write x,and not (x+l)-1; x+5, and not (x+7)-2; etc.Also, we can "transpose," i.e., rewrite x+3-y-2 as x+5=y or x=y-5.,etc.The decision method for G will be given in four lemmas. We choose the

    decision problem for satistiability as being a convenient form for our pur-poses.

    To introduce a terminology and notation we shall employ from now on:P(x)P(x+ 1)P(x+2) will be written simply PPP, and similarly in similarcases. A formula like PPP (with n terms) will be called a 3-series (n-series).Our procedure in Lemma 1 is in part a variant of a decision procedurefor monadic quantification theory invented by Behmann: following himwe introduce the special symbols (3x)n(...) for "there are at least n x'ssuch that (. . )," and (3]x)n'( .. ) for "thereareexactlyn x's such that ( . )."

    LEMMA1: Every closed /ormula of G can be effectively reduced to-T or 77or to a disjunction o/ formulas o/ the torms:a) (3x)C (where C is an n-series)7 T and 1 ("tee" and "eet") are Quine's symbols for truth and falsity (Methodsof logic, Harvard (1950)). They are always eliminated by elementary reduction

    techniques; or else the whole formula reduces to T or 1. The logical notation in thepresent paper follows Quine (e.g., in using juxtaposition for conjunction; in using both- and - for negation), except that more dots than are strictly necessary for punctuationare sometimes used to facilitate reading. Following U.T., identity is included in eachelementary theory as a logical constant.

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