Abstract
Although this problem is still open, there are a few related results concerning (i) theories equiconsistent with NF and (ii) consistent fragments of NF. Concerning (i) we will discuss the following results(1) Specker's result about the equiconsistency of NF with an extension of the theory of types [16],(2) the equiconsistency of NF with fragments of NF containing NFU [2], [10],and a new one about(3) the equiconsistency of NF with an extension of Zermelo's set theory with the axiom of comprehension restricted to bounded formulas.Concerning (ii) we will give a simplified proof and a generalization of (4) Jensen's result about the consistency of NFU [13].Only a few words will be devoted to(5) Grishin's results and refinements [4], [5], [8], [9], treated with more details in [4].The reader who is unfamiliar with NF or who has a natural repulsion for this system is first invited to read the excellent account contained in [7, Chapter III, §3].
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