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Which one is more fundamental, Set theory or Axiomatic system? Which one can be defined without the other?

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    oh no, its the chicken and the egg. – 2017-01-24
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    [Axiomatic set theory](https://plato.stanford.edu/entries/zermelo-set-theory/) is an axiomatic system for set theory... – 2017-01-24

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When begining any mathematical theory, you need axioms, same is for sets. The very basics of Mathematics is true-false accounts and tautologies, and predicate counting of fist order (forall and exists) and after that you could do something.

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    But isn't predicate counting of first order somehow part of the set theory itself? Doesn't it look like a circular definition? – 2017-01-24
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    No, it does not. Predicate counting are how exists and forall behave without sets let us say $(\forall x) \Longrightarrow (\exists x)$ and this x has some property P, which does not need to be enough to form a set, you can search that in elementar logic, there are counterexamples that not every property is coming to form a set. – 2017-01-24
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    Thanks @nikola for the reply. Would you please elaborate a little bit further on what you mentioned in your last comment? – 2017-01-26
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    Forall and exists address to, well, some property P of given element $x$, if you tell that something holds for all $x$, then you have said that all of those $x$ have that property P. Now, let us have the sentence $(\forall x \in X) x \in x$, that is totally correct if you are relying on the predicate account exactly, but if you form a set with that property, you will end up in contradiction. – 2017-01-26
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    But when we say (∀x∈X) , aren't we talking about a set named X which already exist? I mean as soon as we talk about (∀x∈X) or (∃x∈X) we are already assuming the set X exists. Am I right? – 2017-02-06
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    In regards to above comment, per what is being said in this page [link] (http://philosophy.stackexchange.com/questions/4165/how-do-quantifiers-work-in-predicate-logic) "The ∀ symbol is more powerful in this way — it allows us to express a notion without having to refer to every object meeting some criterion;" .... So basically by using ∀ we are referring to a set and its members implicitly, isn't it? – 2017-02-06