## Journal of Symbolic Logic

### The Equivalence of NF-Style Set Theories with "Tangled" Theories; The Construction of $\omega$-Models of Predicative NF (and more)

M. Randall Holmes

#### Abstract

An $\omega$-model (a model in which all natural numbers are standard) of the predicative fragment of Quine's set theory "New Foundations" (NF) is constructed. Marcel Crabbe has shown that a theory NFI extending predicative NF is consistent, and the model constructed is actually a model of NFI as well. The construction follows the construction of $\omega$-models of NFU (NF with urelements) by R. B. Jensen, and, like the construction of Jensen for NFU, it can be used to construct $\alpha$-models for any ordinal $\alpha$. The construction proceeds via a model of a type theory of a peculiar kind; we first discuss such "tangled type theories" in general, exhibiting a "tangled type theory" (and also an extension of Zermelo set theory with $\Delta_0$ comprehension) which is equiconsistent with NF (for which the consistency problem seems no easier than the corresponding problem for NF (still open)), and pointing out that "tangled type theory with urelements" has a quite natural interpretation, which seems to provide an explanation for the more natural behaviour of NFU relative to the other set theories of this kind, and can be seen anachronistically as underlying Jensen's consistency proof for NFU.

#### Article information

Source
J. Symbolic Logic, Volume 60, Issue 1 (1995), 178-190.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183744684

Mathematical Reviews number (MathSciNet)
MR1324507

Zentralblatt MATH identifier
0819.03044

JSTOR
Holmes, M. Randall. The Equivalence of NF-Style Set Theories with "Tangled" Theories; The Construction of $\omega$-Models of Predicative NF (and more). J. Symbolic Logic 60 (1995), no. 1, 178--190. https://projecteuclid.org/euclid.jsl/1183744684