Permutation methods applied to Quine's “New foundations”

Extract

Permutation methods for proving relative consistency results were first applied to Quine's set theory NF by Dana Scott [5]. He showed that the sentence (∃x)(x = {x}), which asserts the existence of individuals in Quine's sense, and its negation are each consistent relative to NF. In this paper Scott's method is shown to apply to any extension T of NF whose axioms are invariant in a certain sense. (Every stratified sentence is invariant, as are many interesting unstratified sentences, such as Rosser's Counting Axiom and the sentence which asserts that every Cantorian set is strongly Cantorian.) We also use this method to prove a number of interesting relative consistency results for such extensions T of NF.

Our main result (Theorem 2.4) asserts that it is consistent relative to T to assume that every well ordering of a strongly Cantorian set is order isomorphic to the well ordering by ∈ of a von Neumann ordinal. (The restriction to strongly Cantorian sets is necessary.) Therefore, although one cannot develop within NF a satisfactory theory of von Neumann ordinals, due to the many unstratified formulas which arise, our result shows that one can consistently assume as rich a theory as is allowed by the extent to which T provides for strongly Cantorian sets. This seems to be the first such “positive” consistency result for NF, and we think that the permutation method will yield others.