In this paper, we rigorously prove the existence of type-level ordered pairs in Quine’s New Foundations with atoms, augmented by the axiom of infinity and the axiom of choice (NFU + Inf + AC). The proof uses the cardinal squaring principle; more precisely, its instance for the (infinite) universe (VCSP), which is a theorem of NFU + Inf + AC. Therefore, we have a justification for proposing a new axiomatic extension of NFU, in order to obtain type-level ordered pairs almost (...) from the beginning. This axiomatic extension is NFU + Inf + AC + VCSP, which is equivalent to NFU + Inf + AC, but easier to reason about. (shrink)

The main feature of NF/NFU is the notion of stratification, which sets it apart from other set theories. We define stratification and prove constructively that every stratified formula has the (unique) least assignment of types. The basic notion of stratification is concerned only with variables, but we extend it to abstraction terms in order to simplify further development. We reflect on nested abstraction terms, proving that they get the expected types. These extensions enable us to check whether some complex formula (...) is stratified without rewriting it in the basic language. We also introduce natural numbers and a variant of the axiom of infinity, in order to precisely introduce type level ordered pairs, which are crucial in simplifying the definitions in the last part of the article. Using these notions we can easily define the sets of ordinal and cardinal numbers, which we show at the end of the article. The same approach can be readily applied to NF. (shrink)

New Foundations with Urelements (NFU) is a theory that extends Quine’s original theory (New Foundations) by adding “urelements” (atoms). It was discovered by Jensen in 1969, who proved that NFU is relatively consistent with Peano arithmetic and consequently with Zermelo–Fraenkel set theory (ZF). Jensen’s proof is rather hard to follow, so Boffa introduced a more straightforward method of constructing models for NFU from a model of ZF. However, Boffa’s presentation of his construction is extremely terse with many essential details omitted, (...) since he probably only wanted to emphasize the differences from Jensen’s construction. Our main goal is to elaborate on this construction in great detail, and also to prove that the said construction yields some results that are either not present in the contemporary literature, or are accepted as true at face value. Namely, besides proving that this construction gives a model for NFU, we will also use it to define models for NFU with the Axiom of Infinity, NFU with the Axiom of Choice, and NFU with both of those axioms. (shrink)