In this post, we discuss some questions concerning the philosophical foundations of ZFC.
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Note from the future: as I thought more about some of these questions, this post ended up becoming sprawling, with many “EDIT” tags to reflect new thoughts every few months. I include all my back-and-forth discussion below, but here I’ll include a brief TLDR:
- Is ZFC “valid”? What would that mean? Well, I would define validity to comprise consistency and intuitive validity. I would conclude then that ZFC is valid.
- Is ZFC the most “general” valid axiomatic set theory? I don’t think this question is very formalizable or meaningful to ask mathematically, but intuitively I would say that ZFC is sufficient when it comes to maximum generality.
- Can an axiomatic set theory serve as a formalization of the entire mathematical universe? No, it can’t, since the universe includes objects like numbers and shapes that are informally defined, not just models of specific sets of axioms. Thus, no axiomatic theory, like an axiomatic set theory, can be proven to sufficiently formalize all of the theory of these objects.
- Do results about numbers, shapes, and other concrete objects depend on choice of axiomatic set theory? Yes, they do, since sets are used as intermediaries in proofs even if they are not mentioned in theorem statements, thus the ability or inability to construct various sets or subsets affects the truth of these statements. Furthermore, such usage as intermediaries in proofs is always required, whether it is noted explicitly or done implicitly, as sets are fundamental objects that correspond to any notion of “grouping” or “collection,” which is implicit in math just as functions and numbers are. As a concrete example, the existence of non-measurable subsets for real numbers is dependent on choice of axiomatic set theory.
- Is there a contradiction with mathematical logic’s definitions using sets, but then with logic being applied to axiomatic set theory? No, there isn’t, anymore than there is a contradiction with numbers being used to ultimately develop mathematical logic and study models of properties of numbers. Numbers are “inherently”/informally valid, and they can thus be used to study specific mathematical concepts even if those concepts are satisfied by numbers; the same is true of sets.
To see the full discussion with all the twists and turns, back-and-forth, etc., read on.
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01/23/2023: Let’s backtrack a bit. As a consequence of my post on the definition of mathematical rigor, math must rest on a foundation of axioms which are “physically” or “geometrically” proven. For example, we may produce a “proof without words” via rectangular dots of the commutativity of multiplication (for positive integers.) This has implications for education: the corollary is that we should be free to justify statements (especially early on) via “physical” and/or “geometric” proofs in addition to formal logic (whose logical rules are themselves intuitively understood.) We can build math in this way starting from primitive concepts such as numbers, geometric shapes, functions, sets, etc., without mentioning ZFC. (We can first understand a set intuitively as “any unordered collection of anything.”)
However, as is well-known, this runs into issues if we’re allowed to speak of any kind of set we want – for example, if we start talking about the “set of all sets which don’t contain themselves.” This is why axiomatic systems like ZFC were created in the first place: to avoid these kinds of logical contradictions. However, how do we know that ZFC is the “right” one? What would that even mean?
There are multiple components to this:
- Consistency. In order for ZFC to work, it has to actually avoid the issues that motivated its introduction in the first place.
- Intuitive Truth of Axioms. The axioms must be intuitively “true.”
I would consider ZFC to be a valid system, just in the context of how we can talk about sets, if it satisfies these two goals.
For the first point, I have heard a lot about Godel’s Incompleteness Theorem and similar statements, but I do not have the knowledge at this point to interpret or analyze these statements. Nevertheless, given general attitudes of the wider mathematical community as well as proofs of consistency of ZFC based on transfinite induction or the like, this point seems to be resolved.
For the second point, there has been a history of debates among different schools of thought, especially concerning the Axiom of Choice. However, personally, all the statements seem intuitive enough to me (including the Axiom of Choice), especially if we want to be able to form the most general kinds of sets we can. Thus, I would consider this point to also be resolved.
Hence, given both of these points, I would consider ZFC to be a valid system.
Now, we come to another question. It is well-known that set theory can serve as a “programming language” for standard mathematical concepts like numbers, geometric shapes, and so on. But we can now ask the following question: is set theory adequate for all mathematical concepts? How would we define all mathematical concepts, and how would we model this question in a way that we can address?
For this question, I consider math to be the study of all logical systems. Certainly, any adequate axiomatization of the “mathematical universe” should at least be able to define sets, since sets are an intuitively valid mathematical concept that should be modellable, and then this axiomatization would need to account for the same issues that motivated ZFC. Thus, we always need some axiomatization of sets. Then, using mathematical logic, we can define the most general kind of logical system. Incidentally, the most general definition of this must involve sets, since for the general study of mathematical logic itself we must be able to support arbitrary “collections” of arbitrary logical symbols, and if these collections aren’t themselves sets then they’d run into the same logical issues that sets ran into anyway, since collections should intuitively at least support membership. Hence, the mathematical universe can be supported by an adequate axiomatic system for set theory. So the question reduces to: does ZFC allow us to define the most general kinds of sets possible?
We can approach it this way: ZFC is at least valid, so any theory that could define the most general kinds of sets should be able to define the sets that are definable in ZFC. Actually, it seems harder to discuss this question in this post – I’m not sure if this question is somehow formally expressible via mathematical logic, or already investigated to some degree, or whether it will reduce to a circular philosophy discussion that would necessitate more subjective analysis that I don’t want to take part in here. Thus, for now, we can consider ZFC (or a standard replacement like NBG) to be both valid and an adequate basis for the entire mathematical universe.
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EDIT (06/18/2023): Based on thoughts I’ve had when my mind has been thinking about this subject in the background, I want to add some points.
Upon re-reading the above paragraph, I don’t fully understand my logic for concluding that an axiomatic theory of sets is sufficient as an axiomatic theory for all of math. [Note: I went back and edited that paragraph for clarity later on, on 11/05/2023, so that people could read through this post more seamlessly. My confusion on 06/18/2023 was based on something that was confusingly worded. This applies to the rest of this paragraph.] At the very least, sets can serve as a basis for all of standard math, which is sufficient for the general purposes of considering a set theory like ZFC as a basis for standard math. As far as I understand now, the question of adequacy for “all” of math is still open, but I’m not even sure if it’s modellable or answerable in any satisfactory way.
Let’s discuss another point. We mentioned that we can develop math perfectly fine from numbers, shapes, etc. without ZFC. Really, the “inflection point” at which we need a set theory like ZFC is when we start to discuss arbitrary sets. If we discuss specific sets of well-known objects, then we don’t need anything like ZFC to be able to construct sets based on these specific sets, or any kinds of subsets. [EDIT: this is actually not true, as I realized later on; see below.] However, an axiomatic set theory starts to matter (from a logical perspective, not necessarily a practical one) whenever we discuss subjects that involve arbitrary sets. These subjects pop up once we introduce more abstraction. For example, abstract algebra operates on arbitrary sets that satisfy certain rules. Or as another example, topology is based on arbitrary sets that satisfy certain axioms to be called topological spaces. The key part here is that any arbitrary set that satisfies the axioms is in scope of the subject. Thus, the assumptions we make on these sets matter, and that’s when axiomatic set theories matter. Even then, in a practical sense, “most of the time” axiomatic set theories wouldn’t apply since they all need to satisfy a certain list of “reasonable” informal conditions based on what mathematicians would want to be considered as a set, so we can generally implicitly assume these conditions whenever working with arbitrary sets without caring about specifics of the axiomatic set theory we’re working on top of. It’s only for certain results with more technical proofs that such theories would matter.
A consequence of this discussion concerns the philosophical interpretation of independence from ZFC. In particular, independence from ZFC is not necessarily the same as independence from math philosophically, since much of math can be done without talking about arbitrary sets, and thus without the need for an axiomatic set theory like ZFC. It is a valid question as to what “independence from math” would mean philosophically: for “concrete” objects like numbers, we can say that math concerns whatever is true or false about them, not just whatever is true or false within a particular proposed axiomatization. While the latter is definable within mathematical logic, what would it mean to be independent of whatever is true or false about the objects in question? I would argue that this notion isn’t even really definable, and at the very least is not something that is equivalent to independence from ZFC or from any other particular axiomatic system.
Our discussion also implies that no statement that doesn’t involve arbitrary sets can be independent of an axiomatic set theory. This implies in particular that a problem like the Riemann Hypothesis can’t possibly be independent of ZFC. More generally, any statements about numbers, shapes, and other “concrete” or “specific” objects can’t be independent of ZFC.
Sidenote: earlier I had misunderstood the axioms of ZFC — specifically, I had confused them with von Neumann ordinals. As I understand now, von Neumann ordinals are a particular “implementation” of numbers in terms of sets within ZFC, while ZFC itself doesn’t specify anything about how concepts like numbers, etc. are to be expressed in terms of sets.
Back to our discussion: if ZFC isn’t fundamental to all of math, what does that mean for the philosophy of a statement like Godel’s Incompleteness Theorem? I haven’t yet studied this topic in detail; furthermore, now I’m not sure that I agree with or understand the non-circularity/validity of statements involving an axiomatic system proving its own properties. (In particular, I’m not sure I agree with or understand the validity of using mathematical logic to study set theory, given that mathematical logic operates on sets that are axiomatized via set theory.) Is Godel’s Incompleteness Theorem not actually that provocative a statement? What does it really mean or imply?
At the very least, I will say this: it is imperative that in education we divorce mathematical logic from foundations of math. Justifying mathematical logic concepts based on head-scratching, possibly circular applications to set theories like ZFC seems to be an approach that, while it makes sense to have arisen given the historical motivation of some of the development of mathematical logic, ultimately hides a lot of the power of mathematical logic and paints it as more difficult, technical, and pedantic than it actually is.
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EDIT (07/03/2023): it’s true that we don’t need axiomatic set theory until we come to arbitrary sets. It’s not true that the application of mathematical logic to such theories is in question.
For example, for ZFC, the axioms are intuitively and fundamentally valid, thus we can assume them without question in developing subjects like proof theory and model theory. We can then use these mathematical logic subjects to study the axioms of ZFC. There is no contradiction here, since it starts from the inherent truth of the ZFC axioms. It’s like using numbers, which inherently satisfy properties like commutativity, associativity, and so on, in developing proof theory and model theory, and then using proof theory and model theory to study properties of numbers as axioms. No contradiction in that whatsoever.
So mathematical logic for set theory isn’t in question, and we can use ZFC without reservation while doing this.
I still don’t understand however the concept of a theory’s “provability of itself” (as in Godel’s Incompleteness Theorem discussing theories that can’t establish their own consistency) — I would need to take a deeper dive into that for me to comment further on it.
Of course, axiomatic set theory still isn’t equivalent to foundations of math, since as we discussed previously there’s a lot of math not dealing with arbitrary sets; thus, the application of mathematical logic to axiomatic set theory should be separated from foundations of math.
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EDIT (11/05/2023): As I noted above with some “EDIT” tags, some of the claims I made earlier are not true. For example, it is not true that axiomatic set theory only applies to theories involving arbitrary sets. As a direct counterexample, the existence of non-measurable sets for the real numbers is dependent on choice of set-theoretic axioms.
In general, an axiomatic set theory defines rules for constructing sets, often from known sets; thus, the types of subsets we can talk about, even for concrete sets, very much depends on the axiomatic set theory we are working in. Furthermore, sets are used in proofs, even as intermediaries for statements that don’t seem to involve sets. As a result, it is very much possible that statements in say number theory (like the Riemann hypothesis) can depend on choice of axiomatic set theory.
In fact, diving deeper into the appearance of sets as intermediaries in proofs, sets are “inherently understood” objects too, and they have to be used implicitly in general in mathematical arguments, any time we involve a concept of “grouping” or “collection” or the like. Thus, in general any mathematical proof would be expected to involve sets implicitly as much as it can involve something like numbers or functions implicitly, and the choice of axiomatic set theory then clearly influences what proofs we are able to devise. (For example, what is true or false in constructivist math would be very different from what is true or false in non-constructivist or “standard” math.)
One point I made that is still true is that independence from math is not equivalent to independence from a particular axiomatic system like ZFC. In fact, it doesn’t even seem possible to define what independence from math would mean, given that math deals not just with axiomatic systems but also “inherently understood” mathematical objects like numbers and shapes, which are not just models of specific sets of axioms. (As a very concrete example of the difference, there exist statements like the Paris-Harrington theorem which are true about numbers but unprovable from the Peano axioms. Of course, it isn’t necessary to have such examples for the point to stand, but it’s topical and a fun tidbit to mention here.)
Expanding on this, I still claim that set theory can’t be sufficient for formalizing the entire mathematical universe — or, more precisely, it can’t be proven to be sufficient for formalizing the entire mathematical universe. This is because it can only be proven to yield a specific set of axioms as consequences, and no “inherent object” is just a model of a set of axioms. (For example, the fact that ZFC with the von Neumann ordinal definitions can prove Peano arithmetic doesn’t mean that it can formalize the positive integers.) I would say that it is meaningless to ask whether any axiomatic set theory can formalize the entire mathematical universe, or even whether it can formalize a specific set of “inherent objects” like the positive integers, just as I would say it is meaningless to ask whether a statement is independent from math.
