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Warning: XY problem

My recent question here revolved (among other things) on whether the intersection of all elements in an empty set of sets is a matter of definition or convention.

When working in a Topological space (X,T), I suggested that for $S \subseteq T$, the intersection is defined by (definition A): $$ \bigcap S := \bigcap_{A\in S} (A) =\{x\in X \ \vert\ \forall A \in S: x \in A\}. $$
In which case the intersection of the empty set follows from the definition.

Another user disagreed and said (admittedly, I can't quite follow) that (Edit: "Definition" B):

The definition would say that $\forall x : x \in \cap \emptyset $, while of course there is no set that contains everything; the notion that we restrict to elements of X, so that X is"everything", is a convention.

Question: is $\forall x$ always abuse-of-notation, and simply shorthand for $\forall x \in P$ for some implicit P? or is $\forall x$ actually meaningful without any (even implied) restriction to some set over which we quantify?

Because it looks to me like this statement first denies the existence of "the set that contains everything", then immediately proceeds to quantify over elements of that set. But, I'm a student and unsure of my reasoning.

user019828
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  • We certainly can write down $\forall x: P(x)$ for some logical formula $P$. But, ZFC axioms don't allow us to turn this logical sentence into a set of sets that satisfy this formula. Earlier versions of set theory did, but they induced several logical fallacies, including Russell's paradox. – Rushabh Mehta Jul 13 '21 at 18:50
  • The notation $\cap S$ is not entirely clear without further elucidation. The only time you see intersection or union to the left of something is for constructs analogous to summation and product, but for union and intersection. – Justin Benfield Jul 13 '21 at 18:51
  • Intersection $\cap$ in set theory is a binary operation. It requires two inputs to have meaning. There are some shorthand notations in use for more complex constructs involving intersection (and likewise for union), but what you've given is at least unfamiliar to me. – Justin Benfield Jul 13 '21 at 18:58
  • If you are using $\bigcap$ in analogy with how $\Sigma$ is used, you've failed to quantify over what collection you are intersecting (notice the indexing on summations and that it can only be suppressed when it has been otherwise communicated what collection of quantities are being summed). – Justin Benfield Jul 13 '21 at 19:09
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    See https://en.wikipedia.org/wiki/Axiom_of_union#Relation_to_Intersection for a discussion of something very much like your definition. At least when it comes to union, something like $\bigcup A$ makes perfect sense and is common in the literature. It seems a corresponding notion for intersection needs one more ingredient: the set the elements will come from. – Charles Hudgins Jul 13 '21 at 19:14
  • @JustinBenfield, attempted correction. I had intended to communicate that it is over elements of S, as the set builder definition suggests. – user019828 Jul 13 '21 at 19:15
  • Now your definition does not make sense, you cannot speak of an $x \in A$ when $A$ is an element of $S$, because $S \subseteq X$ and $x \in X$. It would make sense if the intersection was over the $S$'s in $T$. – Justin Benfield Jul 13 '21 at 19:19
  • Sorry I misread your definition for $S$. – Justin Benfield Jul 13 '21 at 19:24
  • @CharlesHudgins, Thanks that was helpful. However notice that my definition has the form ${x \in X: P}$, not ${x | P}$. That makes all the difference doesn't it? – user019828 Jul 13 '21 at 19:42
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    @user019828 Exactly. That was what I was trying to say in my comment. We should instead write something like $\bigcap_X A = {x \in X : \forall y \in A, x \in y}$. Note that we need the set $X$ for the definition to make sense (at least in ZFC). – Charles Hudgins Jul 13 '21 at 20:44

2 Answers2

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The definition you've given is entirely valid (in the context at hand). To determine the contents of $\cap S$. You take each $x \in X$ and ask if, for every $A \in T$ is $x \in A$? The answer will always be no because one such $A$ is the empty set (by requirements of the definition of $T$), and thus in the end, the set of $x$'s one obtains must be empty.

Justin Benfield
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  • I naturally prefer my own definition, but the question is whether definition B is actually either meaningless or equivalent. – user019828 Jul 13 '21 at 18:58
  • I don't see any definition B in your question, but the issue the other poster raised is a serious problem. You cannot have a set of all sets without running straight into Cantor's Paradox (and Russell's for that matter), and with your definition, $\cap \emptyset$ would have to contain all mathematical objects, as the requirements of the definition you gave would be vacuously satisfied for any object. – Justin Benfield Jul 13 '21 at 19:05
  • His statement isn't a definition, it is a critique of your (proposed) definition for $\cap S$ (and very valid one). – Justin Benfield Jul 13 '21 at 19:12
  • I'm confused.. Your answer says "The definition you've given is entirely valid", and your comment says he has a a very valid critique of it. which is it? – user019828 Jul 13 '21 at 19:18
  • Your definition makes sense, but it leads to a serious problem. This gets right to the heart of why 'naive' set theory isn't good enough and why you need to take an axiomatic approach (it's one of the important things you learn in a first course on sets, which is typically part of a set theory and topology class). – Justin Benfield Jul 13 '21 at 19:22
  • Let us continue this discussion in chat. – user019828 Jul 13 '21 at 19:24
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My OP is rooted in a misunderstanding, I now realize, and thus somewhat beside the point. What the comment I referenced was hinting at is that:

  1. In axiomatic set theory intersection of a collection of sets is only defined for a non-empty collection.
  2. ...because otherwise expressions like $\{x | True \}$, the set of all sets, which lead to a Paradox, creep in.
  3. The book I was working with addressed this directly by adopting the convention that all sets under discussion are restricted to be subsets of $X$. That amounts to using Definition A rather than the standard definition from ZFC [1] [2].
  4. For clarity, you have to explicitly state the use of this convention, because it is different from the standard definition in set theory.
  5. According to [3], when used in a logic context, $\forall x:P(x)$ always implies that x is "Limited to the Domain Of Discourse"

[1] Paul Halmos, Naive set theory, Springer-Verlag 1974, pp15.

[2] Wikipedia:Axiom_of_union#Relation_to_Intersection

[3] Wikipedia:Quantifier_(logic)

user019828
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