New Foundations
Consistency

Some mathematicians have questioned the consistency of NF, partly because it is not clear why it avoids the known paradoxes. A key issue was that Specker proved NF combined with the Axiom of Choice is inconsistent. The proof is complex and involves T-operations. However, since 2010, Holmes has claimed to have shown that NF is consistent relative to the consistency of standard set theory (ZFC). In 2024, Sky Wilshaw confirmed Holmes' proof using the Lean proof assistant.

Although NFU resolves the paradoxes similarly to NF, it has a much simpler consistency proof. The proof can be formalized within Peano Arithmetic (PA), a theory weaker than ZF that most mathematicians accept without question. This does not conflict with Gödel's second incompleteness theorem because NFU does not include the Axiom of Infinity and therefore PA cannot be modeled in NFU, avoiding a contradiction. PA also proves that NFU with Infinity and NFU with both Infinity and Choice are equiconsistent with TST with Infinity and TST with both Infinity and Choice, respectively. Therefore, a stronger theory like ZFC, which proves the consistency of TST, will also prove the consistency of NFU with these additions. In simpler terms, NFU is generally seen as weaker than NF because, in NFU, the collection of all sets (the power set of the universe) can be smaller than the universe itself, especially when urelements are included, as required by NFU with Choice.
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