Is the following statement never true or sometimes it is: $U=\emptyset$?

Question

I've come up with two reasons why I'd think sometimes is true:

We define our universe arbitrarily: Usually we let $U=\mathbb{R}$ but that needn't be so.
$U=\emptyset$ means, owing to the axiom of extensionality,$\forall x\in U(x\in \emptyset)$ and $\forall y\in \emptyset(y\in U)$. The latter is triviual because $y\in \emptyset$ is an absurdity but

If providing for all object $x$ there exists a set $S$ for which $x\in S$ (I don't bear in mind wheter this is a fact or not I just don't think so),

the former is a contradiction.

at present, your question is slightly unclear, as you have not defined what qualities of $U$ you would like there to be! for instance, I suspect you do not mean the axioms of set theory, since any way of making the set $\mathbb{R}$ into a model of ZFC (ie defining a binary relation $\in_\mathbb{R}$ on $\mathbb{R}$ such that $(\mathbb{R},\in_\mathbb{R})\models \text{ZFC}$) would not use the structure of $\mathbb{R}$ in any organic way that I can think of. so it would be helpful if you stated what qualities you are looking for in a "universe" :) — Atticus Stonestrom, May 19 '21 at 19:35
When you say for all object $x$, do you mean for all sets $x$? If so, that statement is a direct corollary of the axiom of pairing. — Rushabh Mehta, May 19 '21 at 19:38
@AtticusStonestrom is there an instance when $U=\emptyset$ is an absurdity? — , May 19 '21 at 19:42
@DonThousand By $x$ I mean elements: if $x$ is not a set, is $U=\emptyset$ sometimes true? — , May 19 '21 at 19:44
@DonThousand Because of the axiom of pairing? So, $U\ne \emptyset$? — , May 19 '21 at 19:53
unfortunately I am still very confused by what you are asking, but, if you are asking whether it is possible to make the empty set into a model of ZFC, then the answer is "no", since every model of ZFC has at least one element. (this is ensured by the axiom of the empty set.) in particular, the empty structure cannot be a model of ZFC, since it has no elements — Atticus Stonestrom, May 19 '21 at 20:52
(and in fact, many formalizations of first-order logic do not even allow there to be an empty structure, in which there is no first order theory that has the empty structure as a model. see eg here for some discussion on this point) — Atticus Stonestrom, May 19 '21 at 20:55
@AtticusStonestrom That answers my questions. Thank you very much! :D — , May 19 '21 at 22:55

Is the following statement never true or sometimes it is: $U=\emptyset$?

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