Talk:New Foundations

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[edit] This entry is a joy to read

It is evident from the writing style alone that the primary author of this entry is Randall Holmes. Thank you very much, Randall, for sharing your knowledge and enthusiasm with the rest of the world. And I can see that your thinking has continued to evolve since you completed your 1998 monograph. You continue to strike me as one of the most philosophically aware mathematicians currently teaching in the USA and Canada. It was by reading you some years ago that I became aware of the extraordinary beauty and power of NFU, which vindicates, I think, Frege's Grundgesetze and Quine's original intuition. I am dismayed at the lack of interest in NFU; in my view, even Tom Forster's monograph does not do it justice. And if it weren't for you, it could truly be said of Quine that, as a mathematician, he would be a prophet without honor in his own country. Nearly all other NFistes are, for some reason, European.

A question. Your 1998 monograph emphasizes a finite axiomatization of NFU, but your entry barely mentions it. Why so reticent? That finite axiomatization banishes once and for all the notion that doing set theory a la Quine style requires a prior commitment to stratification or to some disguised variant of the theory of types. Stratification is, satisfyingly, just an economical way of laying out much of set theory, and requires no ontological commitment of any kind.

Also please discuss briefly McLarty's(1992) negative results on NF and category theory. I am not qualified to say whether McLarty's results are correct, but regardless of their truth status, they deserve mention. The entry should also mention that Saunders MacLane was wrong when he conjectured that Quinean set theory was more hospitable to category theory than ZFC.132.181.160.42 00:03, 10 July 2006 (UTC)

[edit] McLarty's results

McLarty's results are correct. The best way to briefly summarize their import is that the set category of all sets and functions in NF or NFU is not really the correct analogue of the category of sets and functions in ZFC: the correct analogue of the category of all sets and functions over ZFC is the category of all strongly cantorian sets and functions in NF(U), which is a proper class category, and which is cartesian closed. McLarty does not say this (or at least I don't think so); he just briefly proves that the set category is not cartesian closed. Randall Holmes 01:24, 3 July 2006 (UTC)

[edit] Are the quantifiers reversed?

The paragraph about comprehension has this formula:

I read this so that the A^{n + 1} can vary for different choices of xⁿ, which does not look much like a comprehension. There should be different A^{n + 1} for different φ, but for a given φ and a given n, the formula should say that there exists (at least one set) A^{n + 1} which, for each xⁿ contains it if and only if the predicate φ applies:

Am I missing something?

Now the text leading up to this formula already contains the phrase "the set A^{n + 1} exists such that", so perhaps the formula should be only

PerezTerron 16:12, 1 January 2007 (UTC)

I tried to fix it. Does it look OK to you now? JRSpriggs 07:29, 2 January 2007 (UTC)