Symmetry motivates a new consistent fragment of NF and an extension of NF with semantic motivation
M. Randall Holmes · 2013
A sequence of theories of sets and classes is presented, indexed by a natural number parameter k. The criterion for determining which classes are sets has to do with symmetry. The theories with k 2 are shown to entail Quine’s New Foundations, which is interesting as the criteria for sethood do not involved typed set theory or syntax of formulas. The theories with k = 0;1 are shown to be consistent. The case k = 0 is related to recent work of Sergei Tupailo. Consideration of the case k = 1 yields a new subtheory of New Foundations extending the theory recently described by Tupailo and the theory NF3 shown to be consistent by Grishin in 1969; the consistency proof for the case k = 1 thus yields a consistency proof for a new fragment of NF.