The Wiki mostly consist of definitions or specifications of sets. All entries of the Wiki can be directly accessed via the interactive graph and here is where is all starts. At the root of the graph, there are two more prominent set theory entries worth mentioning:

• Set theory language - introducting the first order formal language for set theories.
• Set theory axioms - discussing possible axioms of the theory.

Notice tha they are both written in a more elaborative style and also don't share the same structure as the entries specifying sets, which is extensively discussed on the entry notation page. The language presented in the first entry is completely standard and in any case necessary for all the set specifications given in this Wiki. Given these motivations, I want to shortly comment on the axioms:

A collections of axioms for set theory determine how to work with the binary set membership predicate “$\in$” and thereby also always implicitly define a notion of “set”. Now say we are working with a set theory $\mathcal T$, given via such a collection of axioms, and consider a predicate $P$. If “$\forall x\ (x\in s \Leftrightarrow P(x))$” is a predicate specifying a term $s$, then the meta-sentence “$\exists s\ P(s)$ is true” can be understood as saying “as far as the theory $\mathcal T$ is concerned, the collection of terms for which $P$ hold represent an instance of a set.” Now two theories $\mathcal T$ and $\mathcal T'$ which are defined via different existence axioms will generally not agree if that meta-sentence is true, i.e. they might not agree if there is a set, which ony might denote by $s$ or some other letter, with the claimed properties. In general, the majority of set theory axioms are about existence of certain sets. One often uses the expression “these axioms are stronger than these” to say that one theory lets one proof more sets to exists than another theory. A more conservative set theory is less likely to contain an inconsistency and will in general also admit less non-constructive proofs, which might be considered a pro or a con.

But all of the above doesn't matter much for this Wiki, as it is more about the specifications of objects in the formal language than theory depended truths about them. I want to stress here, that this Wiki is not about proof or truth - it's a place where one can check definitions and then immediately see their generalizations and special cases. The sets defined on this website don't necessarily have to be understood in the context of any specific set theory. But in any case, as a rule, they certainly all exist according to the pretty generous Tarski-Grothendieck set theory and its axioms are discussed in the entry linked above.

Wikipedia:

Logical basics:

• First order logic, Theory (formal logic), Logic (template),

A list of set theories written down in the language of first order logic as well as

• Set theories in first order logic,

Selected set theories:

• Tarski-Grothendieck set theory, Zermelo–Fraenkel set theory,

Some general remarks on the implementation of common mathematics within these theories:

• Set theory (template), Implementation of mathematics in set theory