Not so long ago I had my first exposure to axiomatic set theory, and I've found myself in a rut... How do you justify implementing logical operators in the axioms of set theory when logical operators are binary functions, and functions are reliant on set theory as their basis? Can you meaningfully construct logic without that dependence, or am I missing something ?
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How do you justify implementing logical operators in the axioms of set theory when logical operators are binary functions, and functions are reliant on set theory as their basis? Can you meaningfully construct logic without that dependence, or am I missing something ?
The short answer is, yes, you can construct a formal system without any set theory at all, using a bare minimum meta-system that merely allows you to describe the string manipulation rules for the formal system you wish to construct. See this brief outline of the building blocks you will need. This also explains why we can construct a proof verifier (a program in any typical programming language) for formal systems such as ZFC set theory. We can say that the proof verifier captures the formal system. With a bit more assumptions in our meta-system (see the linked post) we can then validly say of some formal systems (that are either based on first-order logic or can interpret arithmetic) that either the formal system is inconsistent (it proves "$\bot$" or "$0 = 1$" respectively) or it is not. This relies more or less on the assumption that any given program on any given input either halts or it does not halt. Do you believe this? If so, then you get very basic proof theory.
Note that up to this point all we have done is to construct a formal system via some program as a concrete representation of it. Since the program can be literally anything, the question remains of whether the corresponding formal system is meaningful or not. As described above, the question of whether ZFC set theory is inconsistent is meaningful under very weak assumptions, but that is far from saying that ZFC itself is meaningful.
That is precisely where your question of meaningfulness of functions sits. In ZFC we happily define a function to be a set of pairs that satisfy a certain condition. We can hence define things like binary relations and operations so that we can talk about first-order logic within ZFC. But as you noticed, ZFC itself is a first-order theory, and the very language we are using in ZFC already comes with all the first-order logic notation! So what is happening? The answer is that if you use ZFC you assume it has meaning, and then you utilize that to describe other things. This is why you define within ZFC the semantics of first-order logic simply in terms of what ZFC thinks. For instance we define that $M \vDash P \land Q$ iff ( $M \vDash P$ and $M \vDash Q$ ). The first "$\land$" is merely a symbol from the perspective of ZFC, while the second "and" is what you 'think'.
But how do we agree on what "and" means? We could go back to syntax and say that it means that when we say "P and Q" we are simply saying both "P" and "Q" simultaneously, and that is precisely the reason for the rules for "$\land$" in natural deduction. Still, it does not resolve the circularity, because one cannot understand "and" without already knowing it... For the same reason, we cannot grasp the meaning of a rule (syntactic or not) unless we already understand basic logical notions. For instance a general kind of inference rule is of the form "If you have deduced $x$ and ... and $y$ then you can deduce $z$."
That is why ultimately mathematics is circular at the bottom. But let me repeat the distinctions that you should make in foundations of mathematics:
If you accept the validity of specifying a formal system via string manipulation rules, then you can construct any practical formal system including set theories like ZFC,NBG,MK and type theories like MLTT,CoC. Whether what you construct has any meaning is not answerable without more assumptions.
If you further accept that the halting problem is well-defined, then as I described above you can talk about basic things like whether the formal systems you constructed are consistent.
And so on. The more meaning you want to ascribe to a formal system, the more assumptions you need in your meta-system. In particular, if you want to talk about models of a first-order theory, you are going to need the meta-system to already have some notion of collections (whether sets or types).
