2

I am writing an introductory text to set theory (it's just for me though, to see if I'm able to do that and practice mathematical writing). Would it be okay if I enunciated the axiom schema of separation like this?

For any formula $\varphi$ in the formal language of set theory such that $b$ is not free in $\varphi$, and for any $a$, there exists $b$ such that, for all $x$, $x\in b \iff x\in a\land\varphi$.

My question is about the parameters. In this case, I am not writing them down explicitly (that is, I am not writing $\varphi(x,u_1,\dots,u_n)$,) which is in fact intentional (I would like not to write them explicitly.) Is this way still as correct and rigorous, or is it a problem?

Also, when I write $\varphi$, I think there should be a "for all [parameters]," but I am not sure how to place it and whether I should place it at all.

If I wrote $\varphi(x)$ without any parameters would it still be correct? After all, if my proposition is $\varphi(x,u,v)$ I can just consider the proposition $\psi(x)\equiv \forall u\forall v(\varphi(x))$, so the parameters aren't strictly necessary, right?

Elvis
  • 1,543
  • Check out „classic set theory“ by Derek Goldrei: I had the same aspirations as you and the Book gave me plenty of inspirations – Curiosity Dec 24 '24 at 12:14

1 Answers1

2

No, you can't just replace the formula with its universal closure under its parameters and retain the same meaning. For instance if $\varphi(x,u)$ is $x=u$, then separation bounded by $a$ gives the set $\{x\in a: x=u\}$ which is equal to $\{u\}$ if $u\in a$ or $\emptyset$ otherwise. Whereas $\{x\in a: \forall u \; u = x\}=\emptyset$.

As to the stylistic aspect, I wouldn't definitively say it's wrong to omit the parameters, but it would probably be beneficial to include them for clarity, as you are confused on this point. Namely, the set $b$ that you get depends on the parameters in the formula as well as on the bounding set $a$. Really, the axiom schema takes the form $$ \forall \vec p\forall a \; \exists b \forall x(x\in b\leftrightarrow x\in a\land \varphi(x,\vec p))$$ for any $\varphi(x,\vec p)$ where $b$ is not free.

You can write it out in English, i.e. "for any formula $\varphi(x,\vec p)$ and any values of the parameters $\vec p$..." which is fine, but I don't like conflating the schematic quantification over $\varphi$ with the internal quantification over sets.

spaceisdarkgreen
  • 67,151
  • Yeah, you're right about the quantification part, but it might be a problem to write $\vec p$ for the parameters, as I have not defined $n$-tuples yet at the moment of the statement of the axiom. At the same time, I don't like to write $\varphi(x,p_1,\dots,p_n)$, as it seems to rely on some form of natural numbers, which also have not been defined yet. – Elvis Dec 23 '24 at 22:33
  • 2
    @Elvis My intention with $\vec p$ was just as an abbreviation for $p_1,\ldots, p_n.$ You are not obligated to define the natural numbers (in set theory) before writing things like $p_1,\ldots, p_n$... this is using the natural numbers in the metatheory to describe formulas, not using the natural numbers as developed formally in set theory. – spaceisdarkgreen Dec 23 '24 at 22:39
  • Then what are $1,\dots,n$? How do you formalize this concept? – Elvis Dec 23 '24 at 22:49
  • 1
    To OP: see point 3 here about "metatheory natural numbers" to which spaceisdarkgreen refers. There is also question about whether parameters are necessary in specification or replacement, or can we prove parameter-ful versions from the parameter-less versions. The answer is "depends on the other remaining axioms". If the other axioms have equivalences (as should be standard), not implications, then "yes", otherwise "no". The standard reference is "Parameters in Comprehension Axiom Schemas of Set Theory" by Azriel Levy. To spaceisdarkgreen: Hi! – Linear Christmas Dec 23 '24 at 22:50
  • @Elvis They are the natural numbers 1 through n, surely you've heard of them. Don't formalize it. Or formalize it however you like. The point is if you're trying to develop ZFC formally and you're holding out on writing numbered lists of variables in your formula because you haven't 'gotten to' the natural numbers in the ZFC development yet, you are making the error of conflating theory and metatheory. – spaceisdarkgreen Dec 23 '24 at 22:56
  • @spaceisdarkgreen yes, I have heard of them, I'm asking for their definition. – Elvis Dec 23 '24 at 22:57
  • @Elvis "Don't formalize it. Or formalize it however you like." Do you ask for a definition of the natural numbers when they come up when you study real analysis or algebraic topology? – spaceisdarkgreen Dec 23 '24 at 22:58
  • @spaceisdarkgreen no, because I know their formal definition already. In this case, I don't. What is it? – Elvis Dec 23 '24 at 23:01
  • @Elvis "I know their formal definition already" "In this case I don't". These seem in conflict with one another. You either know what they are or you don't. – spaceisdarkgreen Dec 23 '24 at 23:06
  • @spaceisdarkgreen I know their definition within ZFC. I know how to build them from the axioms of ZFC, but in this case I am enunciating an axiom of ZFC, so I haven't defined the natural numbers yet and I can't use the definition I'm used to. It seems like you're trying to avoid defining the natural numbers right now, and I don't really understand why. – Elvis Dec 23 '24 at 23:14
  • @Elvis I highly doubt a course on real analysis or algebraic topology would be keen to give a formal definition of the natural numbers either. There is no need for such a thing and if you think the definition in ZFC somehow imbues them with meaning they didn't have before, you are probably not thinking clearly. – spaceisdarkgreen Dec 23 '24 at 23:21
  • @spaceisdarkgreen a course on real analysis won't give a definition of natural numbers because it assumes such concept to be already defined. But still, this concept does have a formal definition. I'm just asking you what definition of natural numbers you are using, and you're responding with "but you already know what they are," which is not an answer. Also, no, I clearly don't know what they are, because the definition I'm used to cannot be used here. So, will you answer my question or not? If not, this discussion is pointless. – Elvis Dec 23 '24 at 23:27
  • @Elvis It is certainly pointless (or misguided might be a better word) to try to give a completely noncircular definition of the natural numbers or other things useful for describing/manipulating formal syntax. But it might not be pointless to point you to this question and its answers. – spaceisdarkgreen Dec 23 '24 at 23:43
  • @Elvis If you like, also, one can imagine the different variables as some distinguishible fragments of syntax, or even just have two characters $x$ and $'$ and have the different variables be $x',x'', x''', x'''',\ldots$, but ultimately there's going to be some "perceived circularity" somewhere in the description / parsing / etc. – spaceisdarkgreen Dec 24 '24 at 00:14
  • @spaceisdarkgreen thanks for the source. It was very interesting, and a bit depressing, to read. Also, I have seen $x',x'',x''',\dots$ used, but I don't think that really solves the problem, since you do need an underlying notion of natural number to make sense of the dots. – Elvis Dec 26 '24 at 03:38