I'm about to study Pinter's "A book of Set Theory", and I just want to know if the Axioms from the book forms ZFC or NBG or etc. There are 13 axioms in the book, which are

A1. Axiom of Extent

A2. Axiom of Class Construction

A3. Every subclass of a set is a set

A4. $\emptyset$ is a set

A5. $a,b$ are sets then $\{a,b\}$ is a set.

A6. $\mathscr{A}$ is set of sets then $\bigcup_{A\in\mathscr{A}} A$ is set

A7. The power set of a set is a set

A8. Axiom of foundation

A9. Axiom of replacement

A10. Axiom of Choice

A11. Axiom of Infinity

A12. Axiom of Cardinality

A13. Axiom of Ordinality

I'm new to set theory, and after some internet surfing, I concluded that these don't form ZFC since the book speaks about proper classes, and not NBG either since it talks only about Axiom of choice, not Axiom of global choice. So what is this system?

  • $\begingroup$ How about "NBG without global choice"? $\endgroup$ Dec 14 '19 at 8:50
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    $\begingroup$ What are the last two? I have never heard about them. $\endgroup$
    – Asaf Karagila
    Dec 14 '19 at 9:07
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    $\begingroup$ @AsafKaragila I got curious and dug it up. They are just provisional axioms to assume a class of ordinals and cardinals exist with the required properties before constructing the classes. $\endgroup$ Dec 14 '19 at 9:14

In my experience, "NBG" doesn't necessarily mean global choice... oftentimes it does, but conventions vary. So it's good form to be explicit. So you could just say "NBG with local choice".

However, that's assuming that, choice aside, this is NBG. The last two axioms I've never heard of, but I'll just assume they are innocuous quirks of the presentation.

(edit: I looked it up and they are unnecessary axioms the author introduces on a provisional basis before proving.)

The main issue has to do with axiom A2. Are the formulas for class construction restricted so that you can't quantify over proper classes? If not, then you aren't dealing with an NBG-like theory, but rather a Morse-Kelley-like one.

(edit: I could be mistaken cause the presentation of logic is too long for me to read right now, but it doesn't look like the author is making this restriction. So seems like Morse-Kelley. Although it’s possible the author never uses the stronger comprehension instances so is effectively working in NBG.)

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    $\begingroup$ I know some people use GB to denote choice-free, GBc to denote axiom of choice for sets, and GBC to denote global choice, as variants of the Gödel–Bernays set theory. $\endgroup$
    – Asaf Karagila
    Dec 14 '19 at 9:08

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