In ZFC, are all sets hereditary? This a very elementary question, but I am quite new to set theory. I came across this quote on the Wikipedia page for "Hereditary Set".

In formulations of set theory that are intended to be interpreted in the von Neumann universe or to express the content of Zermelo–Fraenkel set theory, all sets are hereditary, because the only sort of object that is even a candidate to be an element of a set is another set.

Is this quote saying that all sets in ZFC are hereditary sets?
How does one then construct a set such as ℝ?
Thank you
 A: To answer your first question, I would refer you to DanielWainfleet and Noah Schweber's comments to your question, or Andrés E. Caicedo's comment to this answer.
As to the second, "how do you make $\mathbb{R}$ in ZFC?", you repeatedly take equivalence classes of smaller sets.
First, you begin with the empty set. You define the successor operator as $S(x) = x \cup \{x\}$. So, $0 = \varnothing$, $1 = S(0)$, $2 = S(1)$, etc. This gives you $\mathbb{N}$. The next step then is to define $\mathbb{Z}$, which will be "equivalence classes of $\mathbb{N} \times \mathbb{N}$" where $(a, b) \sim (c, d)$ if $a + d = c + b$.  After that, we define $\mathbb{Q}$ by equivalence classes of $\mathbb{Z} \times \mathbb{Z}$, where $(a, b) \sim (c, d)$ if $ad = bc$.
Constructing $\mathbb{R}$ now that we have $\mathbb{Q}$ is more complex, and is typically done by considering equivalence classes of Cauchy sequences, or by considering Dedekind cuts. You can view these constructions in all of their gory details by reading the appendix to chapter 1 of Rudin's "Principles of Mathematical analysis" and by doing the final exercises in chapter 3 of the same book.
I have skipped a lot of details along the way of course, most notably defining what, exactly, addition and multiplication are in $\mathbb{N}$ and $\mathbb{Z}$, and that these satisfy the typical axioms we expect of addition and multiplication.
