As in the title: the existence of which sets is implied by the axioms of $\mathsf{ZF}$? For example one such set would be the empty set whose existence is demanded by the Axiom of the Empty Set.
But are for example all ordinals also present in every model because their existence follows from the axioms? I presume the answer must be no as we can have countable models of $\mathsf{ZF}$ but there are uncountably many ordinals.
Or do ordinals just happen to be present in any standard model because of the structure of the model?
In response to Trevor's last paragraph: Trying to make my question mathematically precise, I would like to ask: apart from $P(x)=$ "$x$ is the empty set", which sets we know intuitively are present in every model. For example, we know there exists an infinite set. But can there be a model where the smallest infinite set is already uncountable so that there are no natural numbers $\omega$? And similarly, can there be a model where there is no set representing the real numbers?
Thank you for your help.
Second edit: After reading Trevor's edit I would like to rephrase my question as follows:
For which sets $s$ that we know, like e.g. empty set, $\omega$, $\mathbb R$, ordinals, does $M$ think that $s$ exists? Trying to write it as a formula: For which $s$ does $M \models \exists x (x = s)$.