I am trying to figure out how one constructs limit ordinals that are larger than $\omega$. In my book, the Axiom of Infinity (to create a set $X$ that contains the natural numbers) followed by the Separation / Comprehension Schema (with the formula $\varphi(n):= n \text{ is a natural number }$) is used to carve $\omega$ from $X$…the author calls this set $\omega$ the least limit ordinal.
I have read that, starting at $\omega$, you would (informally) apply the successor function an infinite number of times. This yields $S(\omega) = \omega \cup \{\omega\}=\omega+1$, $S(S(\omega))= \omega+1\ \cup \ \{\omega+1 \}=\omega+2$, etc.
Is the process as straightforward as defining the following set:
$T = \{ \omega + n\,|\, n\in \omega\}$
And then taking the union $\bigcup T$? (Knowing that such a union yields the supremum \ least upper bound of the set $T$)
If the above construction is the standard construction, then my once concern is, "From which set am I selecting the $\omega + n$ elements?".
The strategy I am familiar with is invoking the Separation / Comprehension schema to carve away subsets. But I think that must not be what I am doing above…because that would presuppose the existence of $\omega + \omega$ (or at least some ordinal that is larger than $\omega + \omega$).
So what is the method that I am actually using to create $T$ in the first place?
Best Answer
You're right that Separation doesn't do the job. Instead, this is exactly what the axiom (scheme) of Replacement does!
We first write down a formula $\varphi(x,y)$ in two free variables which intuitively says "$y=\omega+x$." We then prove that for each $x\in\omega$ there is a unique $y$ such that $\varphi(x,y)$ holds - at which point Replacement lets us form $$\{y:\exists x\in\omega(\varphi(x,y))\}=\{\omega+n: n\in\omega\}.$$ Applying the Union axiom then finishes the job.
Meanwhile, to see that Separation won't suffice, it's a good exercise to check that $V_{\omega+\omega}\models\mathsf{ZC}$ (= the $\mathsf{ZFC}$ axioms except Replacement). Since $\omega+\omega\not\in V_{\omega+\omega}$, this means that $\mathsf{ZC}$ alone can't prove that $\omega+\omega$ exists. (See here for further discussion of this point.)