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On Languages Which are Based on Non-Standard Arithmetic1)

Published online by Cambridge University Press:  22 January 2016

Abraham Robinson
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The natural numbers play a part in the formulation of logical syntax inasmuch as they are used to count the symbols in a sentence, or the sentences in a proof, etc. In the present paper, we shall study an infinitary logical calculus which is based on replacing the ordinary natural numbers in the capacity just mentioned, by a non-standard model of arithmetic. (Compare refs. 3, 5, 6 for some other logical calculi of infinitary nature). Thus, our calculus will include formulae of length n for any natural number n, finite or infinite, in the chosen non-standard model of arithmetic. Evidently, the study of such formulae can be of interest only if we introduce concepts which are beyond the power of expression of the notions borrowed from the standard case. It turns out that the concept of truth in a model, when defined by means of Skolem functions has this character and involves a curious phenomenon which is analogous to one first pointed out by H. Steinhaus and J. Mycielski for another kind of infinitary language. Thus, while in the standard predicate calculus the negation of a sentence in prenex normal form is reduced to prenex normal form by changing the type of the quantifiers and by shifting the sign of negation, this procedure is not legitimate when truth is defined in this way in our calculus.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 22 , June 1963 , pp. 83 - 117
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1963

Footnotes

1)

The work on the present paper was carried out in part while the author held a grant from the National Science Foundation (No. G 14006) at the University of California in Berkeley.

References

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