4

During this summer, I am taking an introductory course on "von Neumann-Bernays-Gödel set theory." My professor is really good in this subject and he doesn't use any reference book except his notes. Since I have a hard time to understand some axioms, their need in this theory, and consequences of some of those axioms; I am looking for a good book for self-study.
Free online book or a PDF would be much better.
I really appreciate any help you can provide.

Andrés E. Caicedo
  • 79,201
Bumblebee
  • 18,220
  • 5
  • 47
  • 87
  • 3
    See von Neumann-Bernays-Gödel set theory for references: Bernays, Gödel, Cohen are available. For a textbook see Mendelson. – Mauro ALLEGRANZA Jun 11 '17 at 17:55
  • 1
    Elliott Mendelson, Introduction to Mathematical Logic, CRC Press (6th ed 2015), page 231. – Mauro ALLEGRANZA Jun 11 '17 at 18:09
  • 1
    @MauroALLEGRANZA Aha, that's what I get for trying to search a non-searchable pdf! Derp. You're quite right, it's been a long time since I looked at that book. – Noah Schweber Jun 11 '17 at 18:11
  • 2
    @MauroALLEGRANZA Why don't you add that as an answer? – Noah Schweber Jun 11 '17 at 18:54
  • Maybe you have trouble understanding it because it's a very dubious model of set theory. In my opinion, it might be possible to learn how to write a proof in NBG but that doesn't mean everything it can prove is actually true. I feel like it was created to satisfy some of the intuitive properties of Naive set theory and be a stronger consistent subtheory of Naive set theory than ZF. NBG appears to break the obviously true assumption that a proper class is something and each of them has a set that contains only that class and in fact refutes it. There may be a model of NBG in NF. That is, it – Timothy Jan 05 '19 at 22:07
  • might be possible to find an intuitive injection from strings of characters that represent a statement in NBG to strings of characters that represent a statement in NF such that when one string can be decided from another 2 strings in NBG, its image can also be deduced from the image of the other 2 strings in NF. – Timothy Jan 05 '19 at 22:10
  • 2
    @Timothy Is the implication being made that ZFC is a dubious model of set theory? Because NBG is a conservative extension of ZFC https://math.stackexchange.com/questions/136215/difference-between-zfc-nbg and you mention ZF instead of ZFC so it's not clear to me either way. – hasManyStupidQuestions Jan 08 '23 at 05:14
  • 1
    @Timothy I also don't understand the relevance of your comment about NF. Is the mere fact that there might be an interpretation of NBG in NF supposed to make us skeptical about NBG? Or if that's not what you're saying, what are you saying? – Alex Kruckman Nov 11 '23 at 13:45

2 Answers2

7

For a textbook dealing with NBG set theory, see:

Elliott Mendelson, Introduction to Mathematical Logic, CRC Press (6th ed 2015), page 231-on.

Mauro ALLEGRANZA
  • 94,169
0

Сlassical Set Theory: Theory of Sets and Classes by Taras Banakh

Artem Hak
  • 9