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I've gotten some contradictory answers on where the definition of the element within a set comes from. From a quick Google search, I got this strange idea that the definition of the element comes from the definition of a set. I reached this conclusion because some websites state simply that the element is

A member of a set

But I feel this results in some type of circular logic (At least I think that's what this logical fallacy is called). Since the definition of a set is

A set is a collection of objects whose contents can be clearly determined. The objects in a set are called the elements of the set. (Robert blitzer college algebra)

I am almost certain that the issue here lies in the definition of an element I found, But I can't seem to find a definition of the element that is independent of the set so to speak.

I hope you guys can help clear this up for me thanks

Official Pyramid
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In the most formal context, namely axiomatic set theory, both "set" and "element" are undefined (aka primitive) terms. That is, there is an (undefined) relation $\in$. The everyday language version of $x\in A$ is "$x$ is an element of $A$". The universe, aka domain of discourse, consists of sets.

Analogy: in modern axiomatic treatments of geometry, "point" and "line" are undefined terms. Euclid's famous "definition", "a point is that which has no part", doesn't really work formally. But it may help convey the idea of "zero size".

The dictionary offers definitions of both "set" and "element", but then the dictionary attempts to define everything. It doesn't worry about circularity. In any axiomatic system, to avoid circularity in our definitions, we have to let some terms be undefined (aka primitive). People sometimes say the undefined terms are "implicitly defined by the axioms"; this is just a more pleasant way of saying they are undefined.

Now for some fine points. In the popular axiom system ZF, everything is a set. The elements of sets are all sets. Everyday mathematical objects, like natural numbers, are "built-up" from sets. For example, 0 is defined as the empty set, 1 is defined as the set containing 0, 2 as the set containing 0 and 1, etc.

A variation called ZFA, "ZF with atoms", permits us to have objects that are not sets. Atoms can be elements of sets. In this system, we could let 0, 1, 2,... be atoms, instead of sets. In ZFA we have an additional undefined term, "atom". We then define "set" as "not an atom". (Or alternately we can make "set" undefined, and define "atom" as "not a set".)

In ZF in completely formal dress, we don't even need a term "set". We use the notation of first-order logic. In that paradigm, we'll have variables like $x$ that implicitly range over the universe of sets. For example, here's the "axiom of the empty set" written out formally: $$\exists x\forall y(\neg (y\in x))$$ We don't have to say explicitly in this notation that $x$ and $y$ are sets. But to discuss this in everyday language, we do use the term. However, we don't need a formal definition of the term "set".

Of course, people rarely learn set theory by starting with a formal axiomatic treatment (nor would I recommend this). So textbooks and web pages and so-forth usually offer informal dictionary-style definitions. But these are meant just as motivation, just to provide intuition.

In a comment, AdamLee123 asks if the "undefined relation $\in$" should be thought of as a set of ordered pairs, since many sources define "relation" this way.

ZF is an example of a first-order theory. The syntax of a first-order theory includes relation symbols, aka predicate symbols. First-order logic includes one "built-in" relation symbol, '='. A particular first-order theory may include other relation symbols according to its needs. For ZF, the only relation symbol needed is '$\in$'.

So much for the syntax. To give meaning (i.e., semantics) to a first-order theory, we usually specify a structure. That consists (in part) of a set called the domain (or universe), and a relation (in the usual mathematical sense) to interpret each relation symbol.

However, this doesn't quite work as usual for ZF. To avoid the paradoxes, ZF doesn't regard certain collections as sets. Loosely speaking, these are collections that are "too big", like the collection of all sets. So in ZF, there is no "set of all ordered pairs of sets", and hence the expression $\{(x,y):x\in y\}$ is meaningless.

But we don't need that to give meaning to the sentences of ZF. All we require is that the expression $x\in y$, for any sets $x$ and $y$, has a definite truth-value. In other words, we can imagine that there is a class of all sets (which is not itself a set), and given two sets $x$ and $y$, either $x$ is an element of $y$ or it isn't. That's enough to ascribe semantics to an assertion like the Pair Axiom, to cite a simple example: $$\forall x\forall y\exists z\forall u(u\in z\leftrightarrow u=x\vee u=y)$$ Depending on your philosophical predilections, you might feel that the semantics of ZF are more informal or problematic or less solid than the syntax, but it's what we have. In any case, you can see how $\in$ is not defined in terms of anything else.

Side note: a so-called standard set-based model of ZF would have a domain that was a set; the element-of relation (for such a model) would be a set of ordered pairs. But such a model would not be the "real universe of all sets", at least for people who believe the phrase has a clear meaning.

Michael Weiss
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    "People sometimes say the undefined terms are "implicitly defined by the axioms"; this is just a more pleasant way of saying they are undefined." Not exactly. In category theory terms, the axioms do in fact define the primitives, up to a functor. For instance, if we had a new mathematical system where, for every object in Euclidean geometry, we had an analogous object in our new system, and for every axiom we had an analogous axiom, then the new system would in some sense be the "same" as Euclidean geometry. – Acccumulation Jun 18 '23 at 05:02
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    @Acccumulation I don’t think that equivalence of two categories (or even isomorphism) constitutes a definition of the objects and morphisms. But YMMV—it all depends on how you define “definition”. Category theory itself doesn’t start by defining the terms “object” and “morphism”, but just gives the axioms for them. – Michael Weiss Jun 18 '23 at 05:12
  • @Acccumulation In a way, it’s just an updated version of Bertrand Russell’s old witticism, “In mathematics we never know what we are talking about, or whether what we are saying is true.” – Michael Weiss Jun 18 '23 at 05:18
  • The term "undefined" sometimes suggests that something is invalid; for example, we sometimes say that the result of division by zero is "undefined". Is the use in this answer -- where "undefined" just means "we haven't explicitly stated the definition, but we do take this to be a valid object and have axioms about it" -- a standard one? – ruakh Jun 18 '23 at 05:28
  • Should it be $\exists x\forall y \neg(y\in x)$? Otherwise it looks like $\exists x\forall y((\neg y)\in x)$ – Stef Jun 18 '23 at 09:02
  • @ruakh I'd say that division by zero is an invalid step, because the division operation is not defined for a zero second input, and the "result" of division by zero is a nonexistent object. While it's fine to refer to the intended operation, as well as the nonexistent resulting object, by the catch-all adjective "undefined", this is less helpful, methinks, particularly with respect to which axiom/definition needs to be modified or introduced. Related: "indeterminate". – ryang Jun 18 '23 at 10:01
  • @ruakh "division by zero is undefined" is a standard phrase, but it's understood to mean "the result of division by zero is undefined". However "undefined" in the sense in my answer is also standard: see https://en.wikipedia.org/wiki/Undefined_(mathematics). And if you google "undefined terms in geometry", you will get literally millions of hits. – Michael Weiss Jun 18 '23 at 11:25
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    @Stef "$\neg y$" would be meaningless in this context, since there is no $\neg y$ operator in ZF. The standard "order of operations" in predicate calculus says that at the bottom of the parse tree are relations like $y\in x$, then come quantifiers and negation, and then the binary connectives like $\wedge$ and $\vee$. So strictly speaking all parens are superfluous in the example. But extra parens never hurt, so I've added two more for additional clarity (at the expense of a bit of clutter). – Michael Weiss Jun 18 '23 at 11:32
  • @MichaelWeiss: Thanks! – ruakh Jun 18 '23 at 16:23
  • "In any axiomatic system, to avoid circularity in our definitions, we have to let some terms be undefined (aka primitive)" - so, this is where axiomatic systems and natural languages differ, where the natural language has a greater expressive power, as the terms can be "defined" just by examples of valid uses with each others. – rus9384 Jun 19 '23 at 07:36
  • @rus9384 The culprit is not the formal language. Axiomatic developments can be (and usually are) done in natural language. Famous examples: Hilbert's Foundations of Geometry, Landau's Foundations of Analysis. Or look at Coxeter's The Real Projective Plane. The issue is simply circularity. Euclid recognized that you can't prove everything without falling into circular reasoning (or infinite regress). But he failed to note that the same is true for definitions: if you try to define everything, you must fall into the trap of circularity. – Michael Weiss Jun 19 '23 at 12:54
  • @MichaelWeiss Exactly, and the natural language works despite the imminent circularity. Which is only possible because some terms are mutually definitive, i.e. using $A$ and $B$ as $A\circ B$ for the first time would introduce usages to both $A$ and $B$. – rus9384 Jun 19 '23 at 12:58
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    @rus9384 The "trade off" is precision vs. intuition. But it's not really a trade off. Good authors will provide a skeleton of rigorous, precise axiomatic development (if that's the kind of book they're writing), but flesh it out with motivation and intuition. But this won't be labeled "Definition", rather "Example" or just paragraphs of exposition besides the axiomatic core. – Michael Weiss Jun 19 '23 at 13:00
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    @rus9384 Well, introducing $A$ and $B$ in $A\circ B$ before saying what $A$ and $B$ are could be either excellent or terrible pedagogy, depending on how it's done. But I wouldn't say term are "mutually definitive". It's not definitive in the sense of providing precise definitions, nor in the other sense of "definitive". Maybe "mutually supporting". – Michael Weiss Jun 19 '23 at 13:05
  • Sorry for this rather obtuse question (I'm pretty new to Axiomatic Set Theory), but when you say "there is an (undefined) $\mathbf{relation}$ ∈" , is the word "relation" that you are referring to here a subset of some Cartesian Product? (i.e. the usual mathematical definition of a Relation) If so, wouldn't sets have to exist in the first place to in order to talk about ∈? If otherwise, what does the word relation mean in this context? – AdamLee123 Sep 01 '23 at 04:44
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    @AdamLee123 Not obtuse at all. I'll do an addendum to my answer. – Michael Weiss Sep 01 '23 at 13:31
  • @MichaelWeiss Thank you very much for your answer! I would have given 2 upvotes if I could. – AdamLee123 Sep 02 '23 at 07:02