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A finite arithmetic1

Published online by Cambridge University Press:  12 March 2014

Philip T. Shepard
Philip T. Shepard*
Affiliation:
Michigan State University, East Lansing, Michigan 48823
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Extract

In this paper I shall argue that the presumption of infinitude may be excised from the area of mathematics known as natural number theory with no substantial loss. Except for a few concluding remarks, I shall restrict my concern in here arguing the thesis to the business of constructing and developing a first-order axiomatic system for arithmetic (called ‘FA’ for finite arithmetic) that contains no theorem to the effect that there are infinitely many numbers.

The paper will consist of five parts. Part I characterizes the underlying logic of FA. In part II ordering of natural numbers is developed from a restricted set of axioms, induction schemata are proved and a way of expressing finitude presented. A full set of axioms are used in part III to prove working theorems on comparison of size. In part IV an ordinal expression is defined and characteristic theorems proved. Theorems for addition and multiplication are derived in part V from definitions in terms of the ordinal expression of part IV. The crucial final constructions of part V present a new method of replacing recursive characterizations by strict definitions.

In view of our resolution not to assume that there are infinitely many numbers, we shall have to deal with the situation where singular arithmetic terms of FA may fail to refer. For I know of no acceptable and systematic way of avoiding such situations. As a further result, singular-term instances of universal generalizations of FA are not to be inferred directly from the generalizations themselves. Nevertheless, (i) (x)(y)(x + y = y + x), for example, and all its instances are provable in FA.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 38 , Issue 2 , June 1973 , pp. 232 - 248
Copyright
Copyright © Association for Symbolic Logic 1973

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Footnotes

1

Revised from [5]. I am grateful to Professor W. V. Quine for the suggestion of a motif for a reasonably concise presentation of some central formal results of [5]. The present treatment in parts IV and V is new. Also an axiom of the system of [5] has been eliminated in part III of this paper (T29). Other changes from [5] are mainly stylistic.

References

REFERENCES

[1]Goodman, Nelson, A world of individuals, The problem of universals, Notre Dame University Press, 1959. Reprinted in Benacerraf, P. and Putnam, H., editors, Philosophy of mathematics, Prentice-Hall, Engelwood Cliffs, N.J., 1964, pp. 197–209.Google Scholar
[2]Goodman, Nelson, On relations that generate, Philosophical studies, vol. 9 (1958), pp. 65, 66. Reprinted in Benacerraf, and Putnam, , see [1, pp. 209, 210].CrossRefGoogle Scholar
[3]Goodman, Nelson, The structure of appearance, second edition, Bobbs-Merrill, Columbus, Ohio, 1966.Google Scholar
[4]Quine, W. V., Set theory and its logic, second edition, Harvard,University Press, Cambridge, Mass., 1969.Google Scholar
[5]Shepard, Philip T., A nominalistic construal of arithmetic, Doctoral Dissertation, Washington University, St. Louis, Mo., 1969. (Chairman, Robert B. Barrett, Jr.)Google Scholar
[6]Suppes, Patrick, Axiomatic set theory, Van Nostrand, Princeton, N.J., 1960.Google Scholar