# What set theory is Pinter's Axioms?

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?

• How about "NBG without global choice"? Dec 14 '19 at 8:50
• What are the last two? I have never heard about them. Dec 14 '19 at 9:07
• @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. Dec 14 '19 at 9:14