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A partial model for Quine's “New foundations”

Published online by Cambridge University Press:  12 March 2014

Václav Edvard Beneš
Václav Edvard Beneš*
Affiliation:
Millington, New Jersey
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Extract

1. In this paper we construct a model for part of the system NF of [4]. Specifically, we define a relation R of natural numbers such that the R-relativiseds of all the axioms except P9 of Hailperin's finitization [2] of NF become theorems of say Zermelo set theory. We start with an informal explanation of the model.

2. Scrutiny of P1-P8 of [2] suggests that a model for these axioms might be constructed by so to speak starting with a universe that contained a “universe set” and a “cardinal 1”, and passing to its closure under the operations implicit in P1-P7, viz., the Boolean, the domain, the direct product, the converse, and the mixtures of product and inverse operations represented by P3 and P4. To obtain such closure we must find a way of representing the operations that involve ordered pairs and triples.

We take as universe of the model the set of natural numbers ω; we let 0 represent the “universe set” and 1 represent “cardinal 1”. Then, in order to be able to refer in the model to the unordered pair of two sets, we determine all representatives of unordered pairs in advance by assigning them the even numbers in unique fashion (see d3 and d25); we can now define the operations that involve ordered pairs and triples, and obtain closure under them using the odd numbers. It remains to weed out, as in d26, the unnecessary sets so as to satisfy the axiom of extensionality.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 19 , Issue 3 , September 1954 , pp. 197 - 200
Copyright
Copyright © Association for Symbolic Logic 1954

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References

REFERENCES

[1]Gödel, Kurt, The consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory, Annals of Mathematics studies, no. 3, Princeton (Princeton University Press) 1940, 66 pp.Google Scholar
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[3]Rosser, J. Barkley, On the consistency of Quine's New Foundations for Mathematical Logic, this Journal, vol. 4 (1939), pp. 15–24.Google Scholar
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[5]Specker, Ernst P., The axiom of choice in Quine's New Foundations for Mathematical Logic, Proceedings of the National Academy of Sciences of the United States of America, vol. 39 (1953), pp. 972–975.CrossRefGoogle Scholar