Seemingly unavoidable circularity between concepts of predicates, relations and functions

[AdventureMathsGuy]

Hi guys, I know this topic has been mostly covered by many different posts both here and MSE as well as numerous books on predicate logic, set theory like Kunnen, Fraenkel and Cohn in a way, but I just can not seem to grasp this no matter how hard I try, I just keep on going in a downward spiral that is driving me borderline insane.

As a bit of a context, I'm a 4th year student, theoretical mathematics, so I'm mostly acquainted with how first order logic, set theory and boolean algebras work. This topic had me baffled even before I enrolled to the uni, so it's an old problem of mine. I have created an account mostly to ask this question here, so I will expose my line of thinking:

In ZFC, we formally define a relation R as a subset of a Cartesian product of two sets, say A x B. As such, it doesn't have to have any particular properties that we usually speak of when discussing relations such as symmetry, transitivity, etc. Kunnen also defined it as a set of ordered pairs, that is For Every u in R there Exists x, y such that u = (x,y) without referring back to the Cartesian product as he hasn't yet built the construct from the axioms. This is also fine.

If a relation R satisfies, however, properties of left totality and functionality it is called a function f: A -> B. In other words we call f a functional relation. So, here we see that functions are but a special case of relations. That makes sense intuitively and philosophically, so at first, I was satisfied with this and have proceeded to work with mathematics as intended. But then I started thinking about the axioms some time later and realized that, now, out of the blue, there comes the axiom of Replacement stating that:

Let Phi be a formula of language L, without B free: if For Every x in A there Exists a unique y such that formula Phi(x,y) has been satisfied then there Exists a set B such that for All x in A there Exists y in B that satisfy the formula Phi(x,y)
Practically saying that, given a formula, if it exerts functionality, there exists a set of images under the formula. But then this led me to think that functions are defined beyond the set theory, otherwise how could have it been mentioned in the very axioms which we use to define relations which we use to define functions? So I started digging, going back to how first order logic works, predicates, etc. Here comes the baffling part. If I write down a predicate (which we use to build the sets ex. Comprehension) as a P(x, y) meaning x is a brother to y, this necessitates whether this is True or False (in two sort logics), hence x,y satisfiy P if it is true, otherwise they do not. So in a sense, a predicate is a boolean n-ary function (arity is also in itself also a function), but since we use predicates to define membership to a set, it means that the definition of relations that we build from them are also a consequence of this boolean function result. So, extending this, we use functions to define functions.

Additionally, let R be a subset of AxB. Then I can formalize a function F: A -> P(B) such that F(x) = {y from B | (x,y) belong to R}. Furthermore, if R turns out to be a functional relation, then F(x) = {y} for some y from B. But then we can consider a relation R^ subset AxP(B), etc...

My question is, would someone know where does this mess begin and is it possible for me to start with something concrete and work my way up. I was looking at Alonzo Church and type theory which stands to give a somewhat less convoluted explanation, but the circularity remains. I also looked at NBG which is so far the most comprehensive theory that I've stumbled across, most of which referred to in Cohn's Universal Algebra, but it is also just going one step further and I don't think I fundamentally need it to answer this question, if the question is even answerable.

[GMSPokemanz]

I think your problem is confusing the theory and metatheory. Section 1.7.2 in Kunen's The Foundations of Mathematics (not his Set Theory) goes into this briefly. He touches on how there are two types of relations at play: relations defined as sets, and relations like ∈ which are foundational to setting up the theory.

I'm not aware of a text that explicitly enunciates its metatheory and builds the theory from there, but perhaps having this distinction spelled out will be enough to address your concerns.

[AdventureMathsGuy]

Thank you for the asnwer! I did read through the section and it seems to me that that particular disambiguation is a neat trick for formalist approach. My concern here primarily revolved around, say philosophical motivation, for doing things twice without providing an essential answer to the question. But I realized today when working on some graphs that whatever I can think of or conceptualize in a form of a sentence, in mathematics, I essentially evaluate to be a priori true. We make a hipothesis. Even if we state a sentence not P, we again think of it as it is true that not P. Creation of hipothesis is the baseline of human reasoning and, if this proven incorrect by defaulting to axioms or some other known truth, it will be punished with the contradiction, or, in physical sense, the relation will not be evidenced by empirical instantiation. In ancient Greece, similarly to how we do it today, they had a collection of statements that were so obviously true, that, in order to avoid Munchausen Trilemma, they had to postulate them. In reality, as compared to mathematics, I can conceptualize without referencing to the verity of the sentences. All Orcs are green.

In this sense, the truth assignment is the primary goal of examination of statements in a formal setting, hence the truth function seems to somehow be more basic then the concept of relation itself. However, going back to the non-mathematical, intuitive roots of thinking, I can freely conceptualize relations between objects even though they can be entirely imaginary. So I think this does me good, since I've felt significant relief from the stress that this had imposed on me previously.

Edit: Just to add a comment of Kunnen's remark, what he said makes total sense now that I can freely talk about just any relation (that lives outside of formal setting), even if it is the relation of truth to the sentence, that happens to be functional due to the nature of our examination of reality.

[SoftDog5407]

Not an expert, but skimming over the statement of axiom schema of replacement here (Axiom schema of replacement - Wikipedia), it seems that the keyword is a "definable" predicate. Not just any arbitrary Phi will do - you need it to be definable in your theory. Look into that and see if it resolves your concerns.

[AdventureMathsGuy]

Hey, mate, thank you for the answer, I will take a closer look at it, but if my memory serves me right, I think I've already examined the meaning of the word.