New Foundations is consistent

Change Notes: 8/18/2015: imported the revision of the “relevant FM interpretation” found in the latest series, which actually describes what happens in every clan in the main construction without oversimplification. 6/24/2015: 3 am debugging wording in various places. 6/23/2015: 7:46 am MST some typos corrected external permutations seem to be needed to handle the elementarity argument. I have talked through this with Nathan’s group and I believe I know what is needed. corrected action of an injection of P (∅) into P (A), requring the range to be included in clan(P (A1)) and noting the reason. Installed definition of external permutations and proofs of their properties, and corrected the elementarity proof. 6/22/2015: 3:30 pm there is an omission of important detail in the discussion of elementarity, which I have partly made good here. 2:30 pm Some detailed edits and corrections as I read through this. I have not examined this version so carefully in a long time. It does hold up on examination, but there are minor slips here and there, and there are places where clarification of structure is needed. 9 am Corrected one place where the definition of B << A is misstated: in all occurrences, it is supposed to include 0 ∈ B (though this may be misguided or unnecessary as discussion below points out). Corrected the definition of formal element by adding a star (under set codes). Nathan Bowler’s reading group felt that images of argument lists under allowable permutations were not necessarily argument lists because of an issue with representatives. On discussion with Nathan, I think the text as it stands is correct. Some additional exposition of the confusing point might be wanted. adding language about a point noticed by Nathan and his group: In this note I use the abbreviation clan[A] for clan(P (A)) which is used in a later version. Nathan points out that we can prove in fact that P(clan[A]) = τ(A) for any clan index A. The equation to be shown is (if B << A = Bn) |P |B|−|A|+2(clan[B])| = |P(clan[A])|.