In this paper, by a function of ordinals we understand a function which is defined for all ordinals and each of whose value is an ordinal. In [7] (also cf. [8] or [9]) we defined recursive functions and predicates of ordinals, following Kleene's definition on natural numbers. A predicate will be called arithmetical, if it is obtained from a recursive predicate by prefixing a sequence of alternating quantifiers. A function will be called arithmetical, if its representing predicate is arithmetical.

The cardinals are identified with those ordinals a which have larger power than all smaller ordinals than a. For any given ordinal a, we denote by the cardinal of a and by 2a the cardinal which is of the same power as the power set of a. Let χ be the function such that χ(a) is the least cardinal which is greater than a.

Now there are functions of ordinals such that they are easily defined in set theory, but it seems impossible to define them as arithmetical ones; χ is such a function. If we define χ in making use of only the language on ordinals, it seems necessary to use the notion of all the functions from ordinals, e.g., as in [6].

In the design of multiple input-output logical systems, the authors have previously formulated and proved a computer program for obtaining optimal system representations satisfying three optimality criteria [1]. In that program, the need for obtaining prime implicants occurred. Two algorithms for this purpose are available in the literature; one due to Quine [2] and one due to Nelson [3], [4]. Nelson's, briefly, is to express the function (whose prime implicants are desired) as a product of sums of literals, multiply out, and discard repeated literals save one and proper multiples. (To obtain the function in the initial form, one can start with the complement of the function expressed as a sum of products and complement it using De Morgan's theorem.) In using these algorithms, it became apparent to us that in certain cases one of these two algorithms required less computer time than the other one. In particular, if the given system has a large number of don't cares then the Nelson algorithm seemed to be preferred [5]. Also in the case of a single function, if there are few zeroes in the truth table representation for the function, then Nelson's algorithm is preferable. It also became apparent to us that in certain situations one needs a generalized version of the Nelson algorithm.

In this paper we show that for n-th order languages L′ with p nonlogical constants, 2 ≦ n < ω, 0 ≦ p < ω, a notion of satisfaction can be defined in an n-th order language containing one additional nonlogical constant, say S. By the usual methods we also show that this notion cannot be defined in L′. Hence, in its present formulation, Beth's Theorem in [1] for first order languages has no analogue for L′.

Our defining expression is such that, given any values of the other nonlogical constants and any appropriate m-tuple, it allows us to determine whether or not the m-tuple belongs to the value S of S without considering the totality of objects which are of the same type as S. Whether every definition in an n-th order language is equivalent to one thus “predicative”, and hence whether there is a formulation of Beth's Theorem which generalizes to higher orders, we do not know.

The falsehood for L′ of the analogue of Beth's Theorem implies the falsehood of an analogue of an interpolation theorem for first order languages. The above definability of satisfaction for L′ implies a result on finite axiomatizability in slightly richer languages. Details are given in § 4.

The Stoic “indemonstrables” were inference rules; a rule about rules was the synthetic theorem: if from certain premisses a conclusion follows and from that conclusion and certain further premisses a second conclusion follows, then the second conclusion follows from all the premisses together. Similar things occur as medieval “rules of consequence”, although not usually on a metametalevel; and (with the same proviso) the following might be deemed a contemporary avatar of that Stoic theorem.

If every formula which occurs once or more often in the list A1, A2, …, An, B1, B2, …, Bm occurs also at least once in the list C1, C2, …, Cr then:

This rule [Church: Introduction to Mathematical Logic, 1956, pp. 94, 165], which may be called the rule of modus ponens under hypotheses (MPH), is worthy of attention for the following reasons:

A. MPH and the axioms A ⊃ A yield precisely the positive implicative calculus (and very easily, too).

B. MPH and the axioms A ⊃ f ⊃ f ⊃ A yield a new formulation of the full classical propositional calculus (in terms of f and ⊃).

C. MPH and the axioms ∼A ⊃ A ⊃ A and A ⊃. ∼A ⊃ B yield the classical calculus in terms of ∼ and ⊃.

In the fourth edition of Hilbert-Ackermann's Grundzüge der Theoretischen Logik (Springer-Verlag: Berlin-Göttingen-Heidelberg 1959) the following assertion is “proved” (p. 131):

Ein Ausdruck, der keine freien Individuenvariablen enthält, ist dann und nur dann allgemeingültig, wenn allgemeingültig ist. Die Ausdrücke, die die Identität enthalten, sind dabei eingeschlossen.

As a counter example one can cite: ∃x(¬∃xFx ∨ Fx), which is universally valid yet the generalized form: ∀x(¬∃xFx ∨ Fx) is not (e.g. let the domain of individuals be the natural numbers and let ‘Fx’ be ‘x is factorable’).

It may be of interest to see where the alleged proof went astray. It is argued (correctly) that any expression can only be derived by the given rules from a formula , or from this formula with any other variable in place of y. Moreover (still correctly) any upper formula of such a formula is of the form (“Upper formula” in the transitive sense of an upper formula of an upper formula being an upper formula). But then it is stated (incorrectly) that if is an upper formula of , then is an upper formula of . To show that this is incorrect we here append a proof of the universally valid formula mentioned above:

A class of problems is called decidable if there is an algorithm which will give the answer to any problem of the class after a finite length of time. The purpose of this paper is to discuss the classes of problems that can be solved by infinitely long decision procedures in the following sense: An algorithm is given which, for any problem of the class, generates an infinitely long sequence of guesses. The problem will be said to be solved in the limit if, after some finite point in the sequence, all the guesses are correct and the same (in case there is more than one correct answer). Functions, sets, and functionals which are decidable by such infinite algorithms will be called limiting recursive. These, together with classes of objects which can be identified in the limit, are the subjects of this report.

Without qualification, set will mean set of numbers; function will mean number-theoretic function of 1 variable, possibly partial; functionals will take numerical values and have any number of numerical and/or function variables, the latter ranging solely over total functions of 1 variable. Thus a function is a special case of a functional, x will invariably stand for a numerical variable; φ for a function variable; g for a guess (a number); n for the numerical variable which indexes the guesses, referred to as the time.

The purpose of this paper is to present two groups of results which have turned out to have a surprisingly close interconnection. The first two results (Theorems 1 and 2) were inspired by the following question: we know what sets are “decidable” — namely, the recursive sets (according to Church's Thesis). But what happens if we modify the notion of a decision procedure by (1) allowing the procedure to “change its mind” any finite number of times (in terms of Turing Machines: we visualize the machine as being given an integer (or an n-tuple of integers) as input. The machine then “prints out” a finite sequence of “yesses” and “nos”. The last “yes” or “no” is always to be the correct answer.); and (2) we give up the requirement that it be possible to tell (effectively) if the computation has terminated? I.e., if the machine has most recently printed “yes”, then we know that the integer put in as input must be in the set unless the machine is going to change its mind; but we have no procedure for telling whether the machine will change its mind or not.

The sets for which there exist decision procedures in this widened sense are decidable by “empirical” means — for, if we always “posit” that the most recently generated answer is correct, we will make a finite number of mistakes, but we will eventually get the correct answer. (Note, however, that even if we have gotten to the correct answer (the end of the finite sequence) we are never sure that we have the correct answer.)

Attention was directed to modal systems in which ‘necessarily α’ is interpreted as ‘α. is and always will be the case’ by Prior in his John Locke Lectures of 1956. The present paper shows that S4.3, the extension of S4 with

ALCLpLqLCLqLp,

is complete with respect to this interpretation when time is taken to be continuous, and that D, the extension of S4.3 with

ALNLpLCLCLCpLpLpLp,

is complete with respect to this interpretation when time is taken to be discrete. The method employed depends upon the application of an algebraic result of Garrett Birkhoff's to the models for these systems, in the sense of Tarski.

A considerable amount of work on S4.3 and D precedes this paper. The original model with discrete time is given in Prior's [7] (p. 23, but note the correction in [8]); that taking time to be continuous yields a weaker system is pointed out by him in [9]. S4.3 and D are studied in [3] of Dummett and Lemmon, where it is shown that D includes S4.3 and

CLCLCpLpLpCMLpLp.

While in Oxford in 1963, Kripke proved that these were in fact sufficient for D, using semantic tableaux. A decision procedure for S4.3, using Birkhoff's result, is given in my [2]. Dummett conjectured, in a conversation, that taking time to be continuous yielded S4.3. Thus the originality of this paper lies in giving a suitable completeness proof for S4.3, and in the unified algebraic treatment of the systems. It should be emphasised that the credit for first axiomatising D belongs to Kripke.

I have argued in [1] that a concept bearing some resemblance to ‘p is the answer to d’ (p a proposition and d a question) can be defined wherever d has the form,

‘For which a's is it the case that A (a)?’ (Qa)A(a)

where a is a variable and A a wff containing a. To say that p is the true and complete answer to (Qa)A(a) is expressed as saying that p is logically equivalent to the true conjunction of A(a) or ~A(a) for each a. It is defined as;

Such a concept of answer is like Belnap's [2] direct true answer to a complete list question, or like Harrah's use [3] (p. 43) of the notion of a state description. The main difference between my approach and that of Belnap and Harrah is that while they are concerned to develop a formal metalanguage for discussion of questions and answers I am concerned to express, as far as possible in existing systems, certain interrogative statements; in particular statements of the form ‘— is the (an) answer to —’.

While the account in [1] does give a formal analysis of one ‘answer’ concept there are respects in which it is inadequate.

1. Since it uses entailment (or strict implication) to define the relation between p the answer and d the question we can shew that if p is the answer to d and q is logically equivalent to p then q is the answer to d.

From the work of Kleene, Post and Davis it is well-known that the arithmetic sets can be characterized as those sets recursive in ∅(n) for some natural number n, where ∅(0) = ∅ and . Actually the arithmetic sets which can be expressed in prenex form with n alternating quantifiers (applied to recursive predicates) are recursive in ∅(n). Hence, starting with ∅, the “jump operation”, which takes a set A into the set , serves to increase the “complexity” of sets in a uniform way.

As far as extending the ω-sequence of degrees of unsolvability (i.e. the degrees represented by the ∅(n)) into the 2nd number class, there is an immediate problem. One knows from Spector [17], corollary 2, p. 585, that there is no l.u.b. for the ω-sequence. So, at ω one must pick a degree in some other “natural” way. Unfortunately, what has seemed “natural” to some mathematicians has not seemed natural to others. Kleene and Davis (cf. [10] and [3] respectively) extended this arithmetic hierarchy of degrees of unsolvability by making use of natural number notations for a certain segment of ordinals of the 1st and 2nd number classes, the “constructive” ordinals. Using the Church-Kleene system S3, one can “sum up” previously obtained sets at the limit notations in a way that is certainly natural from a notational point of view.