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It's said that the primitive concepts of set theory are those of "set" and "membership", then all axioms of set theory must begin with "Let $A$ be a set" or "Let $x\in A$", but they don't. For example, let us consider the subset axiom:

Subset Axiom. Let $\varphi(x)$ be a formula and let $A$ be a set. Then there exists a set $S$ such that for all sets $x$ we have that $x\in S$ if and only if $x\in A$ and $\varphi(x)$.

This axiom begins with "Let $\varphi(x)$ be a formula" but "formula" is not a primitive concept, then, for having sense, it must be a defined concept, but as far as I see we can't define "formula" in terms of "set" and "membership" if I am wrong, please tell me. Now, in the case where we can't define "formula", how is the subset axiom justified from a logical point of view?

In many books, when the subset axioms is introduced, the statement "Let $\varphi(x)$" is used informally, I will appreciate if you recommend me a book on set theory where the concept of "formula" is used formally, where there is a formal definition of what a formula is.

Thank you for your reading.

user3840170
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RataMágica
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    One book that elaborates in quite a bit of detail about the anatomy of formulas (and everything to do with their syntax, such as free variables, bound variables, what an "occurrence" of a variable in a formula means, etc.) is Quine's Mathematical Logic (1940). However due to its age it is quite dated and its notation is hard to read (akin to Principia Mathematica), and the axiomatic set theory it presents is somewhat different than ZFC (it presents a system based on New Foundations). – C7X Feb 11 '24 at 04:54
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    There are many books which define what a “formula” is, but these are usually logic books rather than set theory books. Some examples: Van Dalen, Logic and Structure (terse and concise), Propositional and predicate calculus, Goldrei (good beginner text). In order to study set theory, it helps to have an understanding of logic. – Porky Feb 11 '24 at 14:42
  • Subset Axiom schema. See also Schema – Mauro ALLEGRANZA Feb 12 '24 at 08:18
  • See also First-Order Logic and Set Theory – Mauro ALLEGRANZA Feb 12 '24 at 09:49

2 Answers2

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In formalizations of set theory (e.g. ZFC), this subset axiom you've described in usually called the axiom schema of specification. Note the use of the word "schema" in its name: this is where your confusion stems from. Indeed, formally speaking, this is not a single axiom, but a whole collection of axioms: one for every formula $\phi(x)$. For every formula $\phi(x)$, there is an axiom of set theory which goes "$\forall A \exists S \forall x (x \in S \Leftrightarrow (x \in A \wedge \phi(x)))$". Taken together, all these axioms make up what you call the subset axiom.

Edit: Axiomatic set theory is based on first-order logic, which is the system of language and inferences that are used to do this sort of math. It starts with symbols (including variables, connectives like $\neg$ and $\wedge$, quantifiers $\forall$ and $\exists$, the $=$ symbol, and predicate symbols) which are combined according to certain rules to make formulas. First-order logic also uses rules of inference to deduce some formulas from others. Set theory uses first-order logic with the predicate symbol $\in$; its axioms and theorems are formulas. I suggest reading some more (on Wikipedia or in your favorite textbook) if you're interested in learning more about how first-order logic works.

Edit 2 (re: comments): I think you misunderstand the distinction between the mathematical universe and the syntax used to describe it. The mathematical universe is only composed of sets; it does not contain symbols. Symbols are what make up formulas, and formulas are what we use to describe the mathematical universe and assert things about it.

"Set theory" is, in a way, a particular usage of FOL. The language of FOL is built out variables, connectives, and quantifiers (these are the logical symbols), as well as constants, functions, and predicates (these are the non-logical symbols). (See Wikipedia.) You can't change the logical symbols of the language, but you're free to select what symbols you want to use for constants, functions, and predicates. Set theory is a specific use case of this, where there are no constants or functions, and there is only one predicate: $\in$. If we are given axioms, then we can use them to prove theorems using the FOL deduction system. The things we can prove using the set theory axioms are "theorems of set theory".

Another important note is the domain of discourse. Informally, the domain of discourse is what we quantify over. When we write something like $\exists x (\forall y \neg (y \in x))$, what are $x$ and $y$? We say that they belong to the domain of discourse. In set theory, we say that the domain of discourse is the universe of sets. But, importantly, if we're just using FOL, the domain of discourse isn't something we ever "refer to". If you think of logic as just a game of manipulating strings of symbols, then the domain of discourse doesn't really exist. (It becomes useful when we start talking about model theory, but that's another story.)

The point is, "set theory" just refers to the particular case of FOL where we have one 2-ary predicate $\in$, and we assume the set theoretic axioms. The primitive concept of membership appears as the predicate $\in$; the primitive concept of set doesn't really appear at all: it's just the name we give to things that refer to the domain of discourse.

So, that's the mathematical universe. It's things we call sets, which we make statements about using a particular kind of FOL.

But, if we want to start talking about FOL itself, we're firmly outside of the mathematical universe. (You might call it "meta-mathematical".) The description of FOL involves variables and symbols and formulas and such, but we're no longer talking about sets; we are talking about the form of our language. By its nature, we can only talk about such things informally. Of course, you could use a formal logical language to talk about FOL (and in fact we do when we do model theory!), but from a foundations perspective, this just creates an infinite regress problem.

So, whenever you're building foundations for math, the logical system (FOL or another) has to be built "from scratch". Formulas and symbols are not defined in terms of "set" or "membership"; you can only give somewhat informal definitions.

To go back to your original question: the subset axiom as you have written it is not "an axiom". Here is perhaps a better way of phrasing it. Given any formula $\phi(x)$ (of FOL), we have the following axiom of set theory.

Subset Axiom for $\phi(x)$. Let $A$ be a set. Then there exists a set $S$ such that for all sets $x$, we have that $x \in S$ if and only if $x \in A$ and $\phi(x)$.

For any formula $\phi(x)$, say for instance $\forall y ¬(y \in x)$, there is a corresponding subset axiom. And, for any $\phi(x)$, we can formalize the subset axiom for $\phi(x)$ as a sentence in FOL: it is $\forall A \exists S \forall x (x \in S \Leftrightarrow (x \in A \wedge \phi(x)))$. However, you cannot formalize "for any formula $\phi(x)$, the subset axiom for $\phi(x)$ holds" as a sentence in FOL.

Fundamentally, in your statement of the subset axiom, "let $\phi(x)$ be a formula" and "let $A$ be a set" are very different, despite seeming very similar. The latter can be formalized into part of an FOL sentence ($\forall A$), while the former cannot. Perhaps this was the source of your confusion.

Sambo
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  • Then what is the formal definition of formula? – RataMágica Feb 11 '24 at 05:05
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    @RataMágica Formulas and first-order logic come "below" set theory, in that axiomatic set theories such as ZFC rest on top of first-order logic. In each language of first-order logic, there is a definition for what a formula is, which is a string of characters $($, $)$, $\land$, $\lnot$, $\lor$, variable symbols, relation symbols, and predicate symbols, which follows the rules for being a wff of that logic. But this is conceptually "below" set theory, and more general, as it also applies for formalizing what a formula is w.r.t. some other axiomatic systems like Peano arithmetic. – C7X Feb 11 '24 at 05:10
  • About set theory resting on FOL, this answer sounds relevant: https://math.stackexchange.com/a/146487/1030967 – C7X Feb 11 '24 at 05:10
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    @RataMágica See my edit. Don't forget that you can accept an answer if you believe it resolves your question. – Sambo Feb 11 '24 at 19:24
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    @RataMágica: To follow up Sambo’s general explanation with examples: A formula here means any statement in the logical language we’re using, possibly with free variables — so like “$\exists y,\ y \in x$”, “\forall z, (z \in y \leftrightarrow (z = x_1 \lor z = x_2))”, or in prose, “$x$ is inhabited”, “$y$ is the pair ${x_1,x_2}$”. So (1) each individual instance of the axiom schema doesn’t involve the concept of ‘formula’; and (2) once you’re setting out axioms in a formal language, you’re already talking about formulas (possibly under some other name). – Peter LeFanu Lumsdaine Feb 12 '24 at 11:16
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    Strictly speaking, the definition of "formula" doesn't matter to set theory. My understanding is that The first-order logic definition could be written $∀∀∃∀(∈⇔(∈∧()))$, and the definition $(x)$ is no more part of set theory than the definition of $\forall$ or $\exists$. – chepner Feb 12 '24 at 14:29
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    @chepner First-order logic doesn't allow you to quantify over formulas $\phi$. For that, you'd need some kind of second-order logic. I'd say that, although the definition of a formula isn't "part of" set theory, it still matters to set theory, precisely because there is a comprehension axiom for each formula. – Sambo Feb 12 '24 at 15:28
  • @C7X: Your comment here could serve as a very direct answer to the OP's question. – Lee Mosher Feb 12 '24 at 15:33
  • @Sambo I did forget that crucial distinction between first- and second-order logic. But set theory doesn't care what $\phi$ means, right, only that there is a separate axiom for each $\phi$? – chepner Feb 12 '24 at 15:49
  • @LeeMosher Thanks! I have now posted it as an answer. – C7X Feb 12 '24 at 21:52
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    @chepner What specifically is a definition of formula? If it is a time for which strings are considered wffs, on a lower level (first-order logic) there are some properties the collection of wffs must have for set theory to work properly, for example "if $\phi$ and $\psi$ are formulas then $\phi\implies\psi$ is a formula". In this way the choice of formal language is important, and the difficulty of working in a set theory where these kinds of properties are not the case is why well-established languages like first-order logic are what are usually used. – C7X Feb 12 '24 at 23:01
  • @Sambo, then are "Let $A$ and $B$ be a set. Then there exists a set $C$ such that for all sets $x$ we have that $x\in C$ if and only if $x\in A$ and $x\in B$" and "Let $A$ and $B$ be a set. Then there exists a set $C$ such that for all sets $x$ we have that $x\in C$ if and only if $x\in A$ and $x\notin B$" both axioms of ZFC and not corollaries of the subset axioms? and to avoid writing one by one of these axioms we use the trick "Let $\varphi(x)$ be a formula". Therefore this use of formulas in the subset axiom is like "e.g." when we write. – RataMágica Feb 13 '24 at 05:21
  • @Sambo, and then the use of "Let $\varphi(x)$ be a formula" in the subset axioms doesn't have a formal and rigorous justification, rather it is just a trick to simply writing, am I right? – RataMágica Feb 13 '24 at 05:23
  • You are correct that the two statements you've listed are both axioms of ZFC; we might describe them as "instances" of the axiom schema of comprehension. (I should note that, in your examples, $B$ is a free variable. I didn't mention that in my answer, but Wikipedia gives more detail.) – Sambo Feb 13 '24 at 05:28
  • I wouldn't say that "let $\phi(x)$ be a formula" has no formal and rigorous justification. However, I think what you're getting at is that it isn't part of the formal first-order language used as a basis for ZFC: it isn't part of the (formal) axiom itself. You might say that it's part of the meta-language. I think that this can be circumvented to some extent by using second-order logic instead of first-order, but I'm not too sure. – Sambo Feb 13 '24 at 05:31
  • @Sambo, Now I have another understanding of the issue posed in my question. The original problem was that I wanted to formally justify the use of "Let $\varphi(x)$ be a formula". This was problematic because, as I said, in set theory we have only two primitive concepts (set and membership). Therefore, we have to define "formula" in terms of "set" and/or "membership". – RataMágica May 16 '24 at 14:45
  • @Sambo, But as you mentioned, axiomatic set theory is based on first-order logic (FOL). Of course, "set" and "membership" are the primitive concepts of set theory (at least for ZFC), but they are not the primitive concepts of FOL. Thus, the mathematical universe is not composed only of sets as I initially thought; it also includes other entities such as symbols. We can use the concept of symbol to define what a formula is, and "symbol" itself is a primitive concept of FOL, a theory which is more fundamental than set theory. – RataMágica May 16 '24 at 14:46
  • @RataMágica See my edit. I may have rambled a bit, but I hope it helps. – Sambo May 16 '24 at 18:08
  • The mathematical universe is not composed only of sets, there are numbers, figures, etc. Besides that, I think your answer solves the problem, and as a comment, I have read that Homotopy Type Theory can solve the infinite regress problem – RataMágica May 30 '24 at 22:13
  • @RataMágica Happy to help. To answer your latest comment: the point of set theory is that it specifies a mathematical universe in which everything is a set. Numbers, figures, and everything else are described as specific types of sets (e.g. 0={}, 1={0}, 2={0,1}, etc). As for Homotopy Type Theory, from what I've learned about it, I don't think it solves that infinite regress problem, though I'd be happy to be pointed to a source that disagrees. – Sambo May 31 '24 at 02:44
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Formulas and first-order logic come "below" set theory, in that axiomatic set theories such as ZFC rest on top of first-order logic. In each language of first-order logic, there is a definition for what a formula is, which is a string of characters $($, $)$, $\land$, $\lnot$, $\lor$, variable symbols, relation symbols, and predicate symbols, which follows the rules for being a wff of that logic. But this is conceptually "below" set theory, and more general, as it also applies for formalizing what a formula is w.r.t. some other axiomatic systems like Peano arithmetic, or the group axioms.

About set theory resting on FOL, this answer sounds relevant.

C7X
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