Subset definition: universal quantifier over which set?

[PeroK]

For the first, we have to consider $x \in \emptyset \Rightarrow x \in \emptyset'$ which can be difficult to swallow, given that there is no such $x$.

Here's why you have to swallow this. Suppose we want to prove, for example, that there is no real $x$ such that $x^2 = -1$.

We might assume such an $x$ exists, reach a contradiction and conclude that no such $x$ exists.

But, wait a minute! If no such $x$ exists, then we effectively chose $x$ initially from the empty set and applied some logic (which is what we don't accept as valid). The logical basis of our proof of the non-existence of such an $x$ by contradiction has been undermined somewhat.

That's why, even if $x$ does not exist, we must be able to proceed logically from the assumed properties of $x$. And, thereby, swallow the concept of vacuous logic.

 

[stevendaryl]

In the language of set theory, $\forall x \Phi(x)$ is the primitive operation. Roughly speaking, it means for any set x, $\Phi(x)$ holds. Restricted quantification is a defined concept: $\forall x \in A ~ \Phi(x)$ is considered shorthand for $\forall x (x \in A \Rightarrow \Phi(x))$.

So there is no implied universal set that is being quantified over.

The issue about $x \in \emptyset \Rightarrow x \in \emptyset'$ is not specific to set theory. It's just the usual interpretation of $\Rightarrow$:
$$(\text{True} \Rightarrow \text{True}) ~ \text{is True}$$
$$(\text{False} \Rightarrow \text{True}) ~ \text{is True}$$
$$(\text{False} \Rightarrow \text{False}) ~ \text{is True}$$
$$(\text{True} \Rightarrow \text{False}) ~ \text{is False}$$

 

[SSequence]

In the language of set theory, $\forall x \Phi(x)$ is the primitive operation. Roughly speaking, it means for any set x, $\Phi(x)$ holds. Restricted quantification is a defined concept: $\forall x \in A ~ \Phi(x)$ is considered shorthand for $\forall x (x \in A \Rightarrow \Phi(x))$.

Regarding this, there is one related point/question that I had in mind for some time (relation to translation of "ordinary mathematical statements" to "sets only").

Very briefly it goes like this:
Suppose we want to re-convert a statement of form:
$\forall x \in \mathbb{R} \, [\, P(x) ]$
to sets only. I think we would go like:
$\forall x [ x \in \mathbb{R} \rightarrow P(x) ]$

=============================================

It seems that such translations (like above) would be good when we are dealing with "collections" which are sets. However, it seems, I think, that in general math one has to deal with "collections" which are not sets [collections too big to be sets]. So consider such a collection $G$. We might write:
$\forall x \in G \, [ \, P(x) ]$

But since the "collection" of objects $G$ is not a set, the primitive translation can't be like the above case for $\mathbb{R}$. Here is how I "think" we might proceed. Make a formula like $Q(x)$ [one free variable] such that $Q(a)$ is true iff the set $a$ is in collection $G$.
Then write:
$\forall x [x \in G \rightarrow P(x) ]$
$\forall x [ Q(x) \rightarrow P(x) ]$

(The $x \in G$ above is meant to be informal)

=============================================

It feels reasonable to some extent. However, is there something else here that one needs to keep in mind? Or perhaps other typical examples or issues that occur in re-translation?

 

[stevendaryl]

Regarding this, there is one related point/question that I had in mind for some time (relation to translation of "ordinary mathematical statements" to "sets only").

Very briefly it goes like this:
Suppose we want to re-convert a statement of form:
$\forall x \in \mathbb{R} \, [\, P(x) ]$
to sets only. I think we would go like:
$\forall x [ x \in \mathbb{R} \rightarrow P(x) ]$

=============================================

It seems that such translations (like above) would be good when we are dealing with "collections" which are sets. However, it seems, I think, that in general math one has to deal with "collections" which are not sets [collections too big to be sets]. So consider such a collection $G$. We might write:
$\forall x \in G \, [ \, P(x) ]$

But since the "collection" of objects $G$ is not a set, the primitive translation can't be like the above case for $\mathbb{R}$. Here is how I "think" we might proceed. Make a formula like $Q(x)$ [one free variable] such that $Q(a)$ is true iff the set $a$ is in collection $G$.
Then write:
$\forall x [x \in G \rightarrow P(x) ]$
$\forall x [ Q(x) \rightarrow P(x) ]$

(The $x \in G$ above is meant to be informal)

=============================================

It feels reasonable to some extent. However, is there something else here that one needs to keep in mind? Or perhaps other typical examples or issues that occur in re-translation?

You're absolutely right. If $G$ is a definable collection--definable via a formula in set theory--then you can use that formula to express quantification over all of $G$. Quantification over the elements of a set is just a special case.