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Arithmetic and the theory of types

Published online by Cambridge University Press:  12 March 2014

M. Boffa
M. Boffa*
Affiliation:
Faculté des Sciences, Avenue Maistriau, 15, 7000 Mons (Belgique)
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Extract

A hundred years ago, Frege proposed a logical definition of the natural numbers based on the following idea:

He replaced this circular definition by the following one:

He tried afterwards to found his theory over a notion of class satisfying a general comprehension principle:

Russell quickly derived a contradiction from this principle (the famous Russell's paradox) but saved Frege's arithmetic with his theory of types based on the following comprehension principle:

In 1979, talking at the Claude Bernard University in Lyon, I remarked that 3 types suffice to provide Frege's arithmetic, showing in fact that PA2 (second order Peano arithmetic) holds in TT3 + AI (theory of types 0, 1, 2 plus a suitable axiom of infinity). I asked whether TT3 + AI was a conservative extension of PA2. Pabion [3] gave a positive answer by a subtle use of the Fraenkel-Moskowski method. This result will be improved in the present paper, with a view to getting models of NF3 + AI in which Frege's arithmetic forms a model isomorphic to a given countable model of PA2.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 49 , Issue 2 , June 1984 , pp. 621 - 624
Copyright
Copyright © Association for Symbolic Logic 1984

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References

REFERENCES

[1]Boffa, M. and Crabbé, M., Les théorèmes 3-stratifiés de NF3, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris, Série A, vol. 280 (1975), pp. 1657–1658.Google Scholar
[2]Grishin, V., Consistency of a fragment of Quine's NF system, Soviet Mathematics–Doklady, vol. 10 (1969), pp. 1387–1390.Google Scholar
[3]Pabion, J. F., TT3 I est équivalent à l'arithmétique du second ordre, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris, Série A, vol. 290 (1980), pp. 1117–1118.Google Scholar
[4]Wȩglorz, B., A model of set theory over a given Boolean algebra, Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 17 (1969), pp. 201–202.Google Scholar