THE EQUIVALENCE OF NF-STYLE SET THEORIES WITH TANGLED TYPE THEORIES -THE CONSTRUCTION OF OMEGA-MODELS OF PREDICATIVE NF (AND MORE)

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 ome ga-models of NFU (NF with urelements) by R. B. Jensen, and, like the c onstruction of Jensen for NFU, it can be used to construct alpha-model s for any ordinal alpha. The construction proceeds via a model of a ty pe theory of a peculiar kind; we first discuss such ''tangled type the ories'' in general, exhibiting a ''tangled type theory'' (and also an extension of Zermelo set theory with DELTA0 comprehension) which is eq uiconsistent with NF (for which the consistency problem seems no easie r than the corresponding problem for NF (still open)), and pointing ou t that ''tangled type theory with urelements'' has a quite natural int erpretation, which seems to provide an explanation for the more natura l behaviour of NFU relative to the other set theories of this kind, an d can be seen anachronistically as underlying Jensen's consistency pro of for NFU.