User talk:Pqnelson

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Class Theory and the Universe

There are currently a few lurkers on ProofWiki who have a finger on class theory, NBG and the concept of the "universe", who have been contributing a certain amount of philosophical discussions around these subjects - but what we don't really have are solid definitions, axioms and proofs that establish the mathematical basis behind it. We have an entry for the Definition:Grothendieck Universe, which I know little about apart from what's here and the cursory coverage of the subject on Wikipedia.

What we really need is someone to put this stuff up on ProofWiki in our time-honoured format, so as to have a definitive series of articles from which the discussions can properly take root. I understand that you have other work you're doing, in particularly your own website, but if you are prepared to start running with this, then your help would be appreciated.

Note we have already started with a basic page on Definition:Class (Class Theory), but, as I say, it contains discussion topics instead of the definition / axiom / proof format that is really required. In particular, the statements concerning Bernays-Godel and Grothendieck approaches need to be turned into links to expositions on these subjects. I also understand that this then overlaps into category theory, which again we have little support for on this site. This also needs to be rectified. --prime mover 07:07, 16 October 2011 (CDT)

What has appeared to happen is that some lurkers are using NBG set theory, others are using Quine's "New Foundations" for classes. For example Universal Class occurs in the "New Foundations" but not in NBG. This is a bit problematical in many regards, but that's the foundations of math: everyone dogmatically believes their own is the best, and all the others can go to hell.

But I think that it should be cleared up, somehow, that the New Foundations is different than the ZFC set theory, and Quine's extension of it in Set Theory and Its Logic (1963) or in Randall Holmes' Elementary Set Theory with a Universal Set (freely and legally available online!). The serious problem is that the quantifiers range over different things: in NBG it's over sets only, whereas the New Foundations (with classes) it's over anything.

How exactly should one handle this situation? I'm not familiar enough with the new foundations to explain the differences with NBG adequately.

Also, you will have to forgive me, since I am not familiar with the house style. For example, citing SGA4 in the page on the Grothendieck Universe may need to be cleaned up...

Pqnelson 10:39, 16 October 2011 (CDT)

What I want to aim for in this site is to make sure that both / all approaches are covered, with appropriate explanatory words: "This is the construction as defined by Grothendieck, compare (link) construction as defined by Quine ..." In that way we can refer to all definitions and approaches, and from there it may be possible to define a synthesis of them all.

As for books ... The "Books" section is intended to hold a reference to all the books that have been used as source works for this site. See the structure of this, check out the BookReference template for how to reference the book. Oh you have done. Better it's not referred to as SGA because we're not all clever fellows who know all the TLAs, use the name in full (unless "SGA" is its name in full, of course). As for links to on-line versions, include them on the page on which the book itself is documented in the Books namespace. --prime mover 11:18, 16 October 2011 (CDT)