Talk:New Foundations

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Ordered pair[edit]

The statement "ordered pairs [...] in NF and NFU are defined in the usual way" might be misleading since the usual way is not stratified if X and (X,Y) shall be assigned the same type. —Preceding unsigned comment added by Stephan Spahn (talkcontribs) 14:20, 13 May 2011 (UTC)

This entry is a joy to read[edit]

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. 00:03, 10 July 2006 (UTC)

McLarty's results[edit]

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)

Are the quantifiers reversed?[edit]

The paragraph about comprehension has this formula:

\forall x^n \exists A^{n+1} [x^n \in A^{n+1} \leftrightarrow \phi(x^n)]

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

\exists A^{n+1} \forall x^n [x^n \in A^{n+1} \leftrightarrow \phi(x^n)]

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

 \forall x^n [x^n \in A^{n+1} \leftrightarrow \phi(x^n)]

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)

We do not take a position on this?[edit]

Could someone please clarify the referent of "we" in the sentence starting "We do not take a position on this..."? This impacts its meaning. If it's the editorial "we" then it's a simple statement of fact about the responsible editor, but if it denotes the readers then it would seem to be more of an advisory of the form "one should not take a position on this." --Vaughan Pratt (talk) 12:12, 12 April 2009 (UTC)

Since taking position on anything would be against the NPOV rule of Wikipedia, an explicit statement of this is redundant and should be removed. -- (talk) 20:41, 22 July 2012 (UTC)
Even worse, a bit later there are sentences going "(some people say such-and-such, but) We claim that this-and-that". –Henning Makholm (talk) 19:28, 4 September 2012 (UTC)
Much of the article was written by an expert on NF, many years ago (see the page history). It definitely needs to be copyedited to remove most of the uses of "we" and tighten up the sourcing. But at the same time I think it is important to remember it was written in a different period of Wikipedia, and given the authorship there is little reason to be excessively skeptical of the content. — Carl (CBM · talk) 19:46, 4 September 2012 (UTC)

How NF(U) avoids the set-theoretic paradoxes has randomly scrambled sentences[edit]

The mentioned paragraph seems to have been edited to trash. I do not know the original work, so I can't correct it. If someone with more knowledge would look through this, it would be helpfull. Rubybrian (talk) 20:34, 15 November 2008 (UTC)

U in NFU[edit]

What does the U stand for? --Abdull (talk) 20:30, 6 September 2010 (UTC)

Urelements. --Trovatore (talk) 20:38, 6 September 2010 (UTC)

Summary: what's it for?[edit]

It would be nice if this article included, as the second paragraph, a summary of what NF/NFU is "good for" viz, why its interesting and fruitful to pursue (e.g. by hinting at important results). This should come before the definition of TST, and summarize the rather long article that follows. linas (talk) 15:15, 27 July 2011 (UTC)

Choice and Infinity[edit]

The article makes numerous references to Choice and Infinity, without defining them. Presumably they refer to the Axiom of choice and Axiom of infinity respectively, and all the editing that needs be done is to make links out of the first occurrencies, but I would be more comfortable if some expert on NF could confirm that these are the correct interpretations. For example, might one need some special care when stating them? The statement given in Axiom of infinity is probably not stratifiable at all! (talk) 15:02, 12 September 2012 (UTC)

Article on NF has to say that it is Non well founded![edit]

How can we have an article on Quines New Foundations which doesn't mention anywhere that it is a non well founded set theory!

I.e. a set theory which permits

…xn ∈ xn-1 ∈ …x3 ∈ x2 ∈ x1 .

This should be featured prominently and a whole section devoted to it. I was going to link to this article in something I'm writing (on ET maths) - as the logic sections in wikipedia are generally good - but this is a major omission.

Also it could do with a philosophical section, explanation of the philosophical reasons for focusing on a stratified formula, rather than forcing well foundedness on the sets themselves.

See Quine's New Foundations

I assume this is just an oversight as the article is good otherwise.

I'm not specialist in Quines NF so am a little hesitant about editing the article myself. Especially since it is clearly written by specialists and is otherwise thorough. But you could take the Stanford University account as a starting point. I recommend someone does this.

If nobody else feels up to it, I can do a "stub" type section on this. Robert Walker (talk) 12:57, 15 August 2014 (UTC)

Since this had no reply, I've added an extra para mentioning the well foundedness in the intro. Perhaps someone else would like to write more, as I think it deserves a paragraph on this topic somewhere on the page - on the philosophical motivation of NF as a theory that lets you reason about non well founded sets. Robert Walker (talk) 14:13, 31 August 2014 (UTC)

Consistency of NF[edit]

Randall Holmes has announced a consistency proof for NF on his home page. And Murdoch J. Gabbay has a preprint here announcing the same thing. r.e.b. (talk) 21:48, 30 July 2015 (UTC)