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The classical and the ω-complete arithmetic1

Published online by Cambridge University Press:  12 March 2014

A. Grzegorczyk ,
A. Mostowski  and
C. Ryll-Nardzewski
A. Grzegorczyk
Affiliation:
Warsaw
A. Mostowski
Affiliation:
Warsaw
C. Ryll-Nardzewski
Affiliation:
Warsaw
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Extract

We consider two formal systems for the theory of (natural) numbers, both of which are applied second-order functional calculi with equality and the description operator. The two systems have the same primitive symbols, rules of formation, and axioms, differing only in the rules of inference.

The primitive logical symbols of the systems are the improper symbols (,), the prepositional connectives ∨, &, ⊃, ≡, ~, the quantifiers ( ), (E), the equality symbol =, the description operator ι,-infinitely many distinct individual (or number) variables, and for each positive integer k infinitely many distinct k-place function variables. Our systems have in addition the following four primitive nonlogical (or arithmetical) constants:0, 1, +, ×.

The classes of “number formulas” (nfs) and “propositional formulas” (pfs) are defined inductively as the least classes of formal expressions (i.e. of concatenations of primitive symbols) satisfying the following conditions:

(1) 0, 1, and the number variables are nfs.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 23 , Issue 2 , June 1958 , pp. 188 - 206
Copyright
Copyright © Association for Symbolic Logic 1958

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Footnotes

1

The present paper is an outgrowth of joint work of the authors carried out in the Seminar on the Foundations of Mathematics in the Mathematical Institute of the Polish Academy of Sciences in the academic year 1955–1956.

The work started with some observations of Mostowski concerning the so-called models absolute for the natural numbers (see [15]). Ryll-Nardzewski then established the connection between the validity of formulas in these models and their provability in formal systems containing the rule ω (his work was done independently of the work of Orey [16], which has meanwhile appeared in print). We then became interested (chiefly on the suggestion of Ryll-Nardzewski) in the problem of carrying over the metamathematical theorems about the system of arithmetic with finitary rules of inference to the system in which the rule ω is also assumed. The way of attacking this problem was suggested by Grzegorczyk and the details were afterwards elaborated and discussed by the three authors jointly.

For the sake of comparison of the finitary system and the system with the rule ω, we thought it useful to present here once again some results concerning the finitary system. In this section our work is for the most part not original and we have suppressed almost all proofs. Also in the final section, in which we are dealing with the system containing the rule ω, we have repeated a number of results due to other authors which we quote in appropriate places.

It will be seen from our paper that the parallelism between the two systems discussed below goes indeed very far. However we leave open the question of the deeper sources of these analogies.

We are much indebted to Dr. J. W. Addison for his helpful suggestions which have enabled us to improve considerably our previous text.

References

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