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In the 800 pages set theory book by Jech, he uncommented starts using

$$Y=\{ u\in X : \phi(u)\}$$

as equivalent to

$$Y=\{ u:u\in X \wedge\phi(u)\}$$

on the first few pages. The fact that, in the first version, the expression in the first space before the colon is a formula bothers me a little.

I wonder if there is are consideration on what the notation $\{ ...\ :\ \ ...\ ...\ ...\}$ is supposed to be doing, such that one of these notation gets forbidden or both are explained. Obviously, it runs through some options, restricted by a condition spacified in the second space.

How to rigorously formalize $\{ ...\ :\ \ ...\ ...\ ...\}$?

Also, if so, why is it used for sets and also for classes?

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I wouldn't say that it's a formula as much as it is a statement. Sometimes it's convenient to put $x \in$ [some other set] (say $X$ in this case) before the colon to give the reader an idea of where to look for such elements. – The Chaz 2.0 Apr 20 '12 at 22:14
@TheChaz: I'm not saying I don't know what it's supposed to mean, but I'm not sure if not just adding one $\wedge$ is worth it. I know that this is quite a bit of nitpicking here. In any case I'd like to know if there are some formal considerations and definition before I come up with some on my own (which might later unnecessarily crash with others). – NikolajK Apr 20 '12 at 22:16
@TheChaz: It is a formula with parameters, and $u$ is the free variables. We gather all the elements which satisfy the formula [and are in $X$, which is our parameter]. – Asaf Karagila Apr 20 '12 at 23:24
up vote 7 down vote accepted

First let me reveal a little secret about set theorists, and mathematicians in general: we hate complicated notations. We define different objects in different ways, but we then abuse the notation to ease the readability. It takes time to get used to, yes, but then it gets better.

First note that this is only a notation, if we agree from the beginning that either of these notational conventions mean the same thing, then they indeed mean the same thing. It is possible (and likely, and also true) that sometimes it makes more sense to use one notation and at other times to use the other one.

Second this is indeed a formula, but this is what we call a bounded formula. We bound ourselves to $X$. This often appears in explicit sentences (and formulae) as:

  • $(\exists x\in X)\varphi(x)$ which really means: $\exists x(x\in X\land\varphi(x))$, or in universal versions:
  • $(\forall y\in Y)\psi(y)$ which really means: $\forall y(y\in Y\rightarrow\psi(y))$.

As you can see it saves up both space and improves the readability, especially when there are three quantifiers or more.

Lastly, sets are classes. Recall that a class is a collection defined by a first order formula, but this formula is allowed to have parameters. So $X=\{v\mid v\in X\}$. Not all classes are sets, but all sets are indeed classes. This is why we often use $\{x\in X\colon\varphi(x)\}$, or $\{x\colon x\in X\land\varphi(x)\}$. It tells us that the collection is a subclass of $X$ and if $X$ is a set then this collection is also a set.

To close this, I'll add a remark that one of my most pedantic teachers told me that he sat a long time ago to verify that you can syntactically add the $\{\ldots\}$ to the language and all that. I cannot recall whether he left it undone; tried to retrace his proof a few years after and then left it undone; or originally tried to retrace the proof given by his teacher and left it undone.

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So the answer in the last part basically means: "Taken as a notation, the $\wedge$ version" is clear and can be explained in two sentences, but if we want to draw it from logic we have to work a whole lot? – NikolajK Apr 20 '12 at 23:44
@Nick: Pretty much, I think. – Asaf Karagila Apr 20 '12 at 23:51

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