This is from Devlin's "Joy of Sets," page 68, in a discussion that $\omega+\omega$ exists (using ZF axioms):
Let $f:\omega\rightarrow V$ be defined by $f(n)=\omega+n$.
By the Axiom of Replacement, the collection $E=\{f(n)|n\in\omega\}$ is a set.
Previously, the existence of $\omega$ was established.
I have several questions:
$1$) Wherein is the issue that Replacement is needed? $\omega$ was established and well as the natural numbers.
$2$) What is being replaced with what?
$3$) Is there a general rule regarding the use of Replacement along the lines such as when you have a 'collection' that is ...(maybe not seen as a set, e.g., too large), then... (here I really need help).
I have seen some instances where things are spelled out, but would like to be able to know when to and how to do things on my own.
Thanks
We can see the SEP's entry on Set Theory, and sepcifically the Supplement listing the axioms of $\mathsf { ZF }$ :
In a nutshell, what we have to do is to define a formula $\phi(x,y)$ that corresponds to the function $f$ above :
Then we have to prove that:
Having done this, we use the Axiom schema to get: