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The consistency of some 4-stratified subsystem of NF including NF3

Published online by Cambridge University Press:  12 March 2014

Maurice Boffa  and
Paolo Casalegno
Maurice Boffa
Affiliation:
Faculty of Sciences, University of Mons, 7000 Mons, Belgium
Paolo Casalegno
Affiliation:
Scuola Normale Superiore, 56100 Pisa, Italy
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Extract

As is well known, NF is a first-order theory whose language coincides with that of ZF. The nonlogical axioms of the theory are: Extensionality. (x)(y)[(z)(z ∈ x ↔ z ∈ y) → x = y].

Comprehension. (Ex)(y)(y ∈ x ↔ ψ) for every stratified ψ in which x does not occur free (a formula of NF is said to be stratified if it can be turned into a formula of the simple theory of types by adding type indices (natural numbers ≥ 0) to its variables).

Before stating our result, a few preliminaries are in order. Let T be the simple theory of types. If ψ is a formula of T, we denote by ψ+ the formula obtained from ψ by raising all type indices by 1. T* is the result of adding to T every axiom of the form ψ ↔ ψ+. A formula of T is n-stratified (n > 0) if it does not contain any type index ≥ n. A formula of NF is n-stratified if it can be turned into an n-stratified formula of T by adding type indices to its variables. (In practice, we shall allow ourselves to confuse an n-stratified formula of T with the corresponding n-stratified formula of NF). For n > 0, Tn (resp. ) is the subtheory of T (resp. T*) containing only n-stratified formulae. For n > 0, NFn is the subtheory of NF generated by those axioms of NF which are n-stratified. Let = 〈M0, M1,…,=, ∈〉 be a model of T.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 50 , Issue 2 , June 1985 , pp. 407 - 411
Copyright
Copyright © Association for Symbolic Logic 1985

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References

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