No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
Two Suggestions for Ramsey-Reducts of Infinite Theories
ResearchGate has not been able to resolve any citations for this publication.
Ramsey's device [2], for eliminating theoretical terms from a theory assumes that the theory has only finitely many axioms. Paul Berent [1], has recently suggested a modification of Ramsey's method that is designed to cover the infinite case. But, his modified method is quite unlike Ramsey's in that it involves an ascent from the original theory to a suitable metatheory. Thus, it is perhaps of some interest to note that there is an extension of Ramsey's method that covers the infinite case and at the same time is much closer in spirit to his original suggestion.
By a theory we shall always mean one of first order, having finitely many non-logical constants. Then for theories with identity (as a logical constant, the theory being closed under deduction in first-order logic with identity), and also likewise for theories without identity, one may distinguish the following three notions of axiomatizability. First, a theory may be recursively axiomatizable, or, as we shall say, simply, axiomatizable . Second, a theory may be finitely axiomatizable using additional predicates ( f. a. +), in the syntactical sense introduced by Kleene [9]. Finally, the italicized phrase may also be interpreted semantically. The resulting notion will be called s. f. a. +. It is closely related to the modeltheoretic notion PC introduced by Tarski [16], or rather, more strictly speaking, to PC ∩ AC δ .
For arbitrary theories with or without identity, it is easily seen that s. f. a .+ implies f. a .+ and it is known that f. a. + implies axiomatizability. Thus it is natural to ask under what conditions the converse implications hold, since then the notions concerned coincide and one can pass from one to the other.
Kleene [9] has shown: (1) For arbitrary theories without identity, axiomatizability implies f. a. +. It also follows from his work that : (2) For theories with identity which have only infinite models, axiomatizability implies f. a. +.
Let C be the closure of a recursively enumerable set B under some relation R. Suppose there is a primitive recursive relation Q , such that Q is a symmetric subrelation of R (i.e. if Q ( m, n ), then Q ( n, m ) and R ( m, n )), and such that, for each m ϵ B, Q ( m, n ) for infinitely many n . Then there exists a primitive recursive set A , such that C is the closure under R of A . For proof, note that , where f is a primitive recursive function which enumerates B , has the required properties. For each m ϵ B , there is an n ϵ A , such that Q ( m, n ) and hence Q ( n, m ); therefore the closure of A under Q , and hence that under R , includes B . Conversely, since Q is a subrelation of R, A is included in C . Finally, that A is primitive recursive follows from [2] p. 180.
This observation can be applied to many formal systems S, by letting R correspond to the relation of deducibility in S, so that R ( m, n ) if and only if m is the Gödel number of a formula of S, or of a sequence of formulas, from which, together with axioms of S, a formula with the Gödel number n can be obtained by applications of rules of inference of S.
The first order functional calculus was proved complete by Gödel in 1930. Roughly speaking, this proof demonstrates that each formula of the calculus is a formal theorem which becomes a true sentence under every one of a certain intended class of interpretations of the formal system.
For the functional calculus of second order, in which predicate variables may be bound, a very different kind of result is known: no matter what (recursive) set of axioms are chosen, the system will contain a formula which is valid but not a formal theorem. This follows from results of Gödel concerning systems containing a theory of natural numbers, because a finite categorical set of axioms for the positive integers can be formulated within a second order calculus to which a functional constant has been added.
By a valid formula of the second order calculus is meant one which expresses a true proposition whenever the individual variables are interpreted as ranging over an (arbitrary) domain of elements while the functional variables of degree n range over all sets of ordered n -tuples of individuals. Under this definition of validity, we must conclude from Gödel's results that the calculus is essentially incomplete.
It happens, however, that there is a wider class of models which furnish an interpretation for the symbolism of the calculus consistent with the usual axioms and formal rules of inference. Roughly, these models consist of an arbitrary domain of individuals, as before, but now an arbitrary class of sets of ordered n -tuples of individuals as the range for functional variables of degree n . If we redefine the notion of valid formula to mean one which expresses a true proposition with respect to every one of these models, we can then prove that the usual axiom system for the second order calculus is complete: a formula is valid if and only if it is a formal theorem.