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. Glossary leads to a list of properties and information where they are defined.

Given the motivations stated above, 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. Also all presented relations among the sets in the Wiki are true in that framework.

With the foundational approach to represent concepts as in terms of sets comes some issues which might be considered not only unaesthetic but also unpractical. Two examples to bring the point home:

1. We will adopt the standard definitions for the ordered pair and the natural numbers: The ordered pair $\langle a,b\rangle$ of two sets $a$ and $b$ is identified with the set $\{\{a\},\{a,b\}\}$, because in this way one can distinguish between the first and second element. The number zero $0$ is identified with the empty set $\emptyset$ and the number $1$ is the set which contains the number zero, i.e. $\{\emptyset\}$), because then there is a way to represent addition via set union. Now our theory says $1\in\langle 0, 7\rangle$ is true, while obviously merely being an artifact of our modeling.

2. You can go and define the real numbers $\{1,2.3,2\pi,\dots\}$ as some set $\mathbb R$ with the structure of addition $+$ and multiplication defined on them. Now if you model the set of complex numbers $\mathbb C$, you must make a decision: Do you want the real numbers above to be direct elements of it, or do you rather want to model all your complex number in the same fashion? For example as $a+ib\equiv\langle a,b\rangle$, with real and imaginary part stored in an ordered pair? In principle, the truth of $2\pi\in \mathbb C$ depends on it. Of course, the structure including $2\pi$ and the one including $\langle 2\pi,0\rangle$ are perfectly isomorphic.

Usually, one doesn't ever see people care about these issues. The point is that in most cases, the actual set representations of the concepts in questions are only of foundational concern, while everyday mathematics is only interested in their properties, which often mirror the more practical problem at hand. A more category theoretical approach takes this into account tries to capture mathematical objects more directly via their universal properties. Everything is defined only up to isomorphisms. The cognitive drawback is that always just talking about things which behave like this and might be considered more abstract.

Now what I'm getting at here is that I will try to be formal to an appropriate extend, but I will not deny myself a “structural point of view”. E.g. I will not bother pointing out if the plus sign I'm using is that for real or that for complex numbers. And I will care if I formally defined the triple $\langle a,b,c\rangle$ as $\langle \langle a,b\rangle,c\rangle$ or $\langle a,\langle b,c\rangle\rangle$, because what counts is the defining property: I know how to project out the first, second or third component.

What follows are some Wikipedia articles relating to foundations of mathematics.

• Logical basics:

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

• A list of set theories written down in the language of first order logic, and two examples:

Set theories in first order logic, 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

Set theory language, Set theory axioms, Glossary