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I am reading the following quotes. Quote 1 ( source ):

In order to prove some of the fundamental results of set theory, and to begin to define other branches of mathematics based on it, we need to start with some axioms that we can assume to be true. There are many possibilities for choices of axioms, but the most popular set of axioms is the Zermelo-Fraenkel system, or, more generally, Zermelo-Fraenkel with the Axiom of Choice.
We start by constructing the axiom system known as $Z_0$.

Quote 2 ( source ):

The system of axioms $1$–$8$ is called Zermelo-Fraenkel set theory, denoted "ZF." The system of axioms $1$–$8$ minus the axiom of replacement (i.e., axioms $1$–$6$ plus $8$) is called Zermelo set theory, denoted "Z." The set of axioms $1$–$9$ with the axiom of choice is usually denoted "ZFC."

Quote 3 ( source ):

There are many equivalent formulations of the ZFC axioms; for a discussion of this see Fraenkel, Bar-Hillel & Lévy 1973. The following particular axiom set is from Kunen (1980).

When I read this, I have a question: is there any special mathematical term to refer to these "basic" axioms? And if set theories may have infinitely or finitely many other axioms, then what is the mathematical term to refer to all other axioms that are derived from these "basic" axioms? And what is the property that allows to separate these types of axioms?

lyrically wicked
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    Which "basic" axioms do you want to separate from which other "nonbasic" axioms? – Misha Lavrov Sep 14 '18 at 06:11
  • @MishaLavrov: for example, nine axioms of ZFC which are given in all the sources in the question. As far as I understand, these are the "core" of the set theory. Is there some mathematical term dedicated to referring to this "core"? – lyrically wicked Sep 14 '18 at 06:24
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    You seem to have some misunderstanding. The nine axioms of ZFC (two of which are actualy axiom schemas) are all the axioms, not just the "basic" axioms. There aren't any other axioms besides them (in the context of ZFC, at least). – Eric Wofsey Sep 14 '18 at 06:30
  • @EricWofsey: But Wikipedia says that "Given that the number of possible subformulas or terms that can be inserted in place of a schematic variable is countably infinite, an axiom schema stands for a countably infinite set of axioms." Then what does "a countably infinite set of axioms" mean? What axioms are supposed to be in this set? – lyrically wicked Sep 14 '18 at 06:45
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    I'm not sure what you don't understand. The axiom schema consists of all the possible axioms you can construct by substituting some particular formula for the schematic variable. There is one such axiom for each formula. – Eric Wofsey Sep 14 '18 at 06:52
  • @EricWofsey: is there a mathematical term to refer to these possible axioms constructed by substituting? – lyrically wicked Sep 14 '18 at 07:00
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    Uh, they're usually just called something like "instances of the axiom schema" (or "instances of Separation/Replacement" depending on which schema you are using). – Eric Wofsey Sep 14 '18 at 15:08
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    @lyricallywicked Also, they're not possible axioms, they are axioms: ZFC includes all sentences of that form as axioms. – Noah Schweber Sep 19 '18 at 21:38

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