From: Mike Oliver Subject: Re: axioms of modern mathematics Date: Fri, 05 Jan 2001 01:35:49 -0800 Newsgroups: sci.math Summary: von Neumann-Goedel-Bernays (NBG) set theory axioms "Jesse F. Hughes" wrote: > > hrubin@odds.stat.purdue.edu (Herman Rubin) writes: > > > > > Von Neumann's thesis gives the first axiomatization of set > > theory with a finite number of axioms. > > How does he deal with comprehension, say? > > If the answer is complicated, a reply of "RTFVNT"[1] is acceptable. > But, if the answer is easy to sketch, I'd be interested. Actually it's not hard at all. What von Neumann does is add a second kind of variable, which ranges over *classes* (that is, arbitrary collection of sets, including those that are "too large" to be sets themselves). Now the comprehension axiom schema we extend to the new language by saying that for any formula phi(x), all of whose *bound* variables range over *sets* only, has an extension (which is a class). (phi is allowed to have other *free* variables besides x, which may be either set or class variables). Replacement is similarly extended, I think. The resulting theory is variously called Goedel-Bernays, Bernays-Goedel, or von Neumann-Goedel-Bernays (NBG) -- I usually use the last name. It turns out that NBG is a *conservative* extension of ZFC -- that is, if you look at all theorems of NBG and throw away all the ones with any class variables, you have exactly the theorems of ZFC. Now you might ask how this gets us any closer to a finite axiomatization, given that we've still specified infinite axiom schemata of comprehension and replacement. Well, in ordinary ZFC, why is it that we can't replace the axiom schema of comprehension which says something like (all phi)(all x)(exists y)(all z) (z \in y <--> z \in x and phi(z) )? The reason is that we don't have a uniform formula Theta(phi,z) that is true just in case phi(z) is true, so we can't write the above axiom in the language of set theory. But we can *almost* do it. After all, given a structure M, we can write a formula Theta(M,phi,z) that is true if and only if M |= phi(z) , using the Tarski definition of truth. Trouble is, the structure M we want here is the universe V of all sets, but V is not a term of the (first-order) language of set theory. But V *is* a term of our new language, the second-order language of set theory in which NBG is written! So now we can express comprehension (and I think also replacement) in a single axiom. OK, I've left out a lot of details here, and I've never personally checked them all, so I'm not certain I haven't left out some other important idea, but I think these are the basics. To check your understanding, figure out whether you can prove in NBG that there is some ordinal alpha such that V_alpha is an elementary submodel of V, and if not, what goes wrong? Posted and mailed. ============================================================================== From: ndm@shore.net (Norman D. Megill) Subject: Re: axioms of modern mathematics Date: 10 Jan 2001 02:52:50 -0500 Newsgroups: sci.math In article <87d7e3usj2.fsf@phiwumbda.dyndns.org>, Jesse F. Hughes wrote: >hrubin@odds.stat.purdue.edu (Herman Rubin) writes: > >> >> Von Neumann's thesis gives the first axiomatization of set >> theory with a finite number of axioms. > >How does he deal with comprehension, say? See p. 230 of Mendelson, Elliott, "Introduction to Mathematical Logic", 4th ed., 1997. There are 7 "axioms of class existence" from which the comprehension schema can be proved as a metatheorem by induction on formula length. --Norm