I am currently working through Hungerford's Algebra, which in the first few pages claims to rely on the (or perhaps a) Gödel-Bernays axiomatization of set theory. I am looking for a citable reference with an explicit statement of what would then be considered the Gödel-Bernays axioms.
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1$\begingroup$ "Set Theory and the Continuum Problem" by Smullyan and Fitting uses these axioms. But if you ignore results concerning classes, you get exactly the same results about sets if you use the ZFC axioms. When your goal is to go through Hungerford, this is really a side-issue. $\endgroup$– Michael GreineckerCommented Feb 8, 2021 at 10:32
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$\begingroup$ If you just want to know what you can do with the classes that the G–B axioms characterise, then you can do worse than math.stackexchange.com/a/258408/3835, noting that the category of GB classes behaves in a structurally similar way to the category of ZF classes. However, looking at the book, it seems to me that the treatment of classes/sets is self-contained, and gives you all you need. $\endgroup$– David Roberts ♦Commented Feb 8, 2021 at 11:22
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1$\begingroup$ The axioms are listed on page 10 of CLASSICAL SET THEORY:THEORY OF SETS AND CLASSES, by TARAS BANAKH, available via: arxiv.org/pdf/2006.01613.pdf $\endgroup$– Ali EnayatCommented Feb 8, 2021 at 20:44
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