Talk:Von Neumann–Bernays–Gödel set theory

message: I expanded and rewrote the History section. One goal was to show how von Neumann's work grew out of previous work in set theory, and to indicate the innovations he was making. On Bernays' work, I mention that his working with sets and classes follows existing traditions in set theory and logic. Finally, I added a few more details on Bernays' and Gödel's work.
I also corrected an error in the old History section: Bernays did not discover that the axiom system could be finitely axiomatize—von Neumann's original system only used finitely many axioms. See my discussion in Finite axiomatization.--RJGray (talk) 21:01, 12 April 2013 (UTC)
thread_title:  Rewrote History section 

message: Four and a half years ago, in the last paragraph of #Two sorted theory, I wrote: "We are still left with the question of why this Wikipedia article is giving an exposition of NBG in two-sorted logic. As Carl points out, Mendelson's book, which is referenced by this article, treats NBG with a single sort. Personally, I would like to see an exposition that is closer to Gödel's and Mendelson's. I think it would be a more accurate representation of NBG and would be simpler for readers not familiar with many-sorted logic. Also, we can then remove Rp(A,a) ("set a represents class A"), which is only mentioned once in the Ontology section, and also remove the abuse of notation a=A, which means the same as Rp(A,a) and which is only used three times in the Axiomatizating NBG section." 
Since a rewrite has not appeared, I decided to do one. Before I go into details, I would like to thank Jochen Burghardt for all his help on the rewrite. He read through a couple of editions of this rewrite and made many excellent suggestions (including the extension lemma used in the proof of the class existence theoem). He also pointed out some errors. His work not only led to a more accurate article, but also to a more readable, comprehensive, and visually appealing article. He also suggested and implemented the diagram on the History page that gives the history of the approaches that led to NBG. This diagram is a compact representation of the evolution of NBG. 
My approach to writing the article was to not only look at Gödel's 1940 monograph and the original Wikipedia article, but also the French Wikipédia article (:fr:Théorie des ensembles de von Neumann-Bernays-Gödel) and the German Wikipedia article (:de:Neumann-Bernays-Gödel-Mengenlehre). Both the French and German articles use a single sort. The German article is short and follows Mendelson. The French one is much longer and references both Gödel and Mendelson. Also, it has a proof of what Mendelson calls the class existence theorem, which shows how to build classes for any given formula. The French proof is nearly complete as far as atomic formulas are concerned, but it just mentions the axioms used for the inductive part of the proof. I have given the details. 
What I like best about the original Wikipedia article is that the editor(s) gave a proof of the very important class existence theorem. However, concerning the axiomatization, the article states: "The finite axiomatization presented below does not necessarily resemble exactly any NBG axiomatization in print."
I believe that it's best to follow Gödel's axiomatization and proof closely enough so that readers who get interested in the original proof will find that the proof in the Wikipedia article prepares them to read the original proof. I made three small modifications that simplify Gödel's proof for Wikipedia readers. 
The first modification is a small change to one of Gödel's axioms. (I've converted Gödel's axiom to use the standard definition of n-tuples with new components on the right rather than his definition with new components on left):
:\forall A \,\exists B \,\forall x \,\forall y \,[(x, y) \in B \iff x \in A]
In words: If A is a class, there is a class B whose ordered pairs have the property that their first component is a member of A. Gödel's axiom only specifies the ordered pairs of B. It may contain members that are not ordered pairs. So the first modification is to go back to the axiom of Bernays that Gödel modified:
:\forall A \,\exists B \,\forall u \,[u \in B \iff \exists x \, \exists y \,(u = (x, y) \land x \in A)]
In words: If A is a class, then A \times V is also a class. 
Bernays' axiom is stronger because it specifies all the members of B. It allows the proof of the class existence theorem to produce unique classes at the end of the basis and inductive steps of the class existence theorem's proof. In a footnote, I explain the difference between the proof in the rewritten article and Gödel's proof, and I point out how his approach allows him to use his weaker axiom. 
Using Bernays' axiom along with a second modification—proving the theorem immediately for class variables rather than for "symbols for special classes"—reduces Gödel's four-theorem approach to his theorem M4 (he proves theorems M1, M2, M3, and M4) down to proving two theorems.
The third modification concerns bound variables. In the case of the existential quantifier, I found Gödel's inductive proof somewhat hard to explain because he introduces a non-subscripted variable in the formula for the inductive hypothesis. This formula does not have the form expected by the induction hypothesis since the class existence theorem is stated for subscripted variables. To get formulas with subscripted variables, just replace the bound variables by subscripted variables that continue the numbering of the free set variables through the nested quantifiers. Since bound variables are free for part of the induction, this guarantees that, when they are free, they are treated the same as the original free variables. 
These modifications lead to a short computer program that succinctly captures what is happening for readers who know some elementary programming. Output from this program illustrates how the construction of a class mirrors the construction of the formula defining the class. I included this output because the original article states: "An appealing but somewhat mysterious feature of NBG is that its axiom schema of Class Comprehension is equivalent to the conjunction of a finite number of its instances." So I'm attempting to lift some of this mystery; readers can tell me whether I succeeded.
The subsections Axiom schema versus class existence theorem and Classes and sets contain material borrowed from the French article.  The History section comes from the original article (I added some material here and also Jochen's diagram has been added). The ZFC, NBG, and MK section contains material from the original article's Discussion section together with material from the French article (I also added to this material). The Models subsection comes from the original article's Model theory section with a couple of additions. The Category theory section comes from the original article's Category theory section with a little rewording.
I used Gödel's definition of the formulas used in the class existence theorem—namely, those formulas that quantify only over sets. I avoid the term "predicative" for two reasons: (1) Gödel's definition is simpler and (2) the formulas are not completely predicative but only predicative for proper classes, so the reference to Wikipedia's Impredicativity article is confusing. The Wikipedia reference says: "Roughly speaking, a definition is said to be impredicative if it invokes (mentions or quantifies over) the set being defined." When a formula that quantifies only over sets is used by class comprehension to produce a class that is a set, this set is impredicatively defined because the formula quantifies over all the sets including the set being defined. NBG has to allow this since ZFC allows impredicative definition of sets. So quantifying only over sets only means that proper classes are defined predicatively. By the way, the Impredicativity article states "There is no generally accepted precise definition of what it means to be predicative or impredicative: many different authors have given different but related definitions of what the words mean." 
In the Models subsection, I use the term "model" rather than "intended model." For Peano arithmetic, an intended model exists, but it's hard to justify this term for models of set theory. When Zermelo and von Neumann wrote down their axioms, they intended that their axioms would characterize the universe of sets. Of course, their axiom systems only approximately characterize the universe of sets since, for example, they don't decide the continuum hypothesis.
In 1930, Zermelo was the first to build the models V_\kappa for \kappa strongly inaccessible (see Zermelo's models and the axiom of limitation of size). He didn't consider these models as "intended models" of set theory. Instead, he used this sequence of models to explain that the paradoxes "depend solely on confusing set theory itself … with individual models representing it. What appears as an 'ultrafinite non- or super-set' in one model is, in the succeeding model, a perfectly good, valid set with both a cardinal number and an ordinal type, and is itself a foundation stone for the construction of a new domain [model]." 
I hope that readers will find the rewrite informative and interesting. --RJGray (talk) 16:40, 13 December 2017 (UTC)
replies: text: The first version you gave of the axiom is equivalent to \forall A \exists B \,( \, B \cap (V \times V) \, = \, A \times V \, ) . JRSpriggs (talk) 06:32, 14 December 2017 (UTC)
thread_title:  Rewrite of NBG article 

message: Let's review the facts:
* Every axiom in ZFC is a theorem or axiom of NBG
* Every proof in ZFC is a proof in NBG
* Some theorems of ZFC have drastically shorter proofs in NBG (per Pudlak 1998)
The only way a ZFC theorem can have a longer proof in NBG vs ZFC is if you use some sort of "cumulative proof" metric that penalizes using ZFC axioms in NBG because they are theorems. But in practical mathematics nobody cares whether the result they are using is a theorem or an axiom. "Shorter" refers to the textual length of the proof and it can be quite short if a powerful theorem is used. Since every theorem of ZFC is a theorem of NBG there is no way the shortest proof in NBG can be shorter than that in ZFC.
So I think the current statement
:Even though NBG is a conservative extension of ZFC, a theorem may have a shorter and more elegant proof in NBG than in ZFC (or vice versa).
should be
:Even though NBG is a conservative extension of ZFC, a theorem may have a shorter and more elegant proof in NBG than in ZFC.
Mathnerd314159 (talk) 23:10, 8 October 2022 (UTC)
thread_title:  Shorter and more elegant proofs in NBG vs ZFC