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I've seen the following statement a few times:

"Let $A$ be a set, then $\emptyset\subseteq A$".

Or, written 'more formally': $$ \forall A\,\, \emptyset\subseteq A $$

My doubt is: I've always seen the quantifier $\forall x$ to mean "for all $x$ elements of some set $S$". However, when talking about all the sets, how do we define this quantification?

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6 Answers 6

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It's just an unbounded quantifier, not constrained to the elements of a fixed set $S$. This is the normal case; the bounded quantifiers, constrained to members of a set, are defined in terms of the basic quantifiers, $\forall x$ and $\exists x$.

$\forall x$ means just that: for all $x$, anything $x$. In the mathematical universe, $x$ ranges over sets, and not, say, people, atoms, oranges, etc. Unless some further conventions are in force constraining $x$ to range over, for example, only reals, $\forall x$ means "for all sets $x$". In order to say "for all $x$ in $S$" you have to bound the quantifier, as in "$\forall x\in S$". However, bounded quantifiers are a shorthand, defined in terms of the basic unbounded quantifiers: $$\begin{align} \exists x\in S\,P(x) &\stackrel{def}\iff \exists x\,(x\in S \land P(x)) \\ \forall x\in S\,P(x) &\stackrel{def}\iff \forall x\,(x\in S \to P(x)). \\ \end{align}$$

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    $\begingroup$ In that case, what prevents the sentence $\forall A\, \emptyset \subseteq A$ from being interpreted in particular as $\emptyset \subseteq \text{John}$, as an example? (Which would make no sense.) $\endgroup$
    – YoTengoUnLCD
    Dec 10, 2015 at 4:53
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    $\begingroup$ Since $\subseteq$ showed up, one ought to infer that we are talking about sets, no? $\endgroup$
    – Cameron Buie
    Dec 10, 2015 at 4:54
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    $\begingroup$ As I said, the variables of mathematics don't range over people, atoms, oranges, or any collection containing John. $\endgroup$
    – BrianO
    Dec 10, 2015 at 4:54
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    $\begingroup$ @BrianO But not all constructions of the reals treat individual real numbers as sets, and there's no necessary reason to do so. "Universal" as the $\forall$ quantifier is, $A$ nevertheless cannot be anything but a set in the statement $\forall A\,\, \emptyset\subseteq A$. Though come to think of it, the question is tagged "elementary set theory," in which case I suppose everything is indeed a set. $\endgroup$
    – Kyle Strand
    Dec 10, 2015 at 18:40
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    $\begingroup$ @KyleStrand In foundational theories where the reals are not sets (primarily type theory), there is no unbounded universal quantifier - all quantifiers are typed, so this doesn't come up. In fact, the unbounded universal quantifier is in some sense the hallmark of first-order set theory, since there is no need to disambiguate the domains of quantification. Any more complicated ontology needs multiple different kinds of quantifiers. $\endgroup$
    – Mario Carneiro
    Dec 10, 2015 at 19:00
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From the ZFC perspective, there's nothing special going on here. In ZFC, everything is a set, and universal quantification is always regarded as quantification over the entire world of sets (the "cumulative hierarchy"). So the sentence $\forall x P(x)$ always means "for all sets $x$, $P(x)$." By convention, the notation $\forall x \in X, P(x)$ is viewed as a convenient shorthand for the more cumbersome $\forall x(x \in X \rightarrow P(x)).$

On the other hand, from the type-theoretic perspective, this is an example of "quantification over type variables," and it is usually treated separately to ordinary quantification. I suggest using Google to learn more.

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In many set theoretical universes, one deals with the situation that all objects (dealt with by the theory) are hereditary sets. That is, every element of a set is, itself, a set. So, when making a set-theoretic statement in such a case, $\forall x$ means "for all sets $x$" (since we aren't talking about anything that isn't a set).

In some cases, though, there are set elements that aren't sets. Such objects are sometimes called "atoms" or "ur-elements." If we're dealing with such a theoretic realm, $\forall x$ means (for example) "for all sets or atoms $x,$" instead.

In general, a given theoretical exploration will be confined to a particular universe of discourse, in which case $\forall x$ will mean something to the effect of "for all $x$ addressed by the theory under discussion."

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  • $\begingroup$ I think my confusion is at the point where 'the given theoretical exploration is confined to a universe of discourse': Shouldn't the universe of discourse be a set? And as a set containing all sets doesn't exist, I don't understand what could be considered as the universe in this case. $\endgroup$
    – YoTengoUnLCD
    Dec 10, 2015 at 4:56
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    $\begingroup$ Not at all. It suffices that the universe of discourse be a class--that is, a clearly-defined collection of objects, which may be too "large" to be a set. Every set is certainly a class (the class of all its members), but the converse need not hold, as you aptly point out. $\endgroup$
    – Cameron Buie
    Dec 10, 2015 at 4:58
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    $\begingroup$ @YoTengoUnLCD In set theory the universe of discourse is a class (the class of all sets) not necessarily a set. The axioms of ZFC (specifically, Extensionality) disallow atoms: anything with no elements is $\emptyset$. You have to tweak the axiom(s) to allow atoms, or "urelements". $\endgroup$
    – BrianO
    Dec 10, 2015 at 5:01
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    $\begingroup$ Note that even when you have ur-elements, the statement $\forall x,\emptyset\subseteq x$ is still true, because it expands to $\forall xy,y\in\emptyset\to y\in x$ and the antecedent is false. At least with $\sf ZFA$ and other tweaks of set theory for atoms, the $\in$ relation is well-formed when applied to atoms - it is just that $x\in a$ is always false when $a$ is an atom. The real modification is to equality, where an atom satisfies $a\ne\emptyset$ even though $x\in a\iff x\in\emptyset$, i.e. extensionality is violated. $\endgroup$
    – Mario Carneiro
    Dec 10, 2015 at 19:05
  • $\begingroup$ @Mario: Excellent point! Of course, in set theory with ur-elements, we fix Extensionality by saying that $y=z$ if for all $x$ we have $$x\in y\iff x\in z$$ and $$y\in x\iff z\in x.$$ $\endgroup$
    – Cameron Buie
    Dec 10, 2015 at 19:10
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$\forall A$ means "for all $A$" or more completely "for all objects $A$ that the theory we are working in is talking about". If the theory is set theory (say, we started off by writing down the axioms of ZFC), then our $A$ is a set; when working in other theories it may instead mean that $A$ is a natural number or that $A$ is a real number. However, this doesn't not happen by magic or even by convention, it happens by the very fact that we accept the appropriate axioms to hold for our thingies.

So by the very fact that we work with $\forall A\forall B\exists C\colon ((C\in A\leftrightarrow C\in B)\leftrightarrow A=B)$ and $\forall A\forall B\exists C\forall D\colon(D\in C\leftrightarrow (D=A\lor D=B))$ and so on as axioms, it happens that our thingies behave exactly as sets are supposed to behave.

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There is nothing wrong with your statement, though I would insert some punctuation:

$\forall A: \emptyset \subset A$

or

$\forall A (\emptyset \subset A)$

In algebra and analysis, the quantified variable may be an arbitrary placeholder, an element of a set, a set or a function.

If the universally quantified variable is an element $x$ of a set $S$, then the quantifier can be written as:

$\forall x \in S:\space \cdots$

or as

$\forall x:[x\in S \implies\cdots$

Similarly if the universally quantified variable is a subset $x$ of $S$, then the quantifier can be written as:

$\forall x \subset S:\space\cdots$

or as

$\forall x:[x\subset S \implies\cdots$

If the existentially quantified variable is an element $x$ of a set $S$, then the quantifier can be written as

$\exists x\in S:\space\cdots$

or as

$\exists x: [x\in S \land \cdots$

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Without restriction $\forall A$ actually means "for all $A$", while with restriction like $\forall A\in B$ it means "for all $A$ that's a member of $B$".

The only counter intuitive thing here is that in formal set theory it's that everything is a set (ie $\forall A: A\mbox{ is a set}$).

Another thing to know is that the initial universal quantifier is normally omitted and only restrictions on the variables are mentioned. This is useful if you don't like formal set theory and don't accept that everything is a set. Then by this you can write only "if $A$ is a set, then $\emptyset\subseteq A$". If accepting that everything is a set you already accepted the premise and it could even formally be written as compact as $\emptyset\subseteq A$.

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