Should I worry about the following appeIarance of circularity in ZFC set theory? In constructing the universe of sets, you start with the empty set and then keep taking the power set over and over.
But this process continues through all the ordinals. The ordinals are there in the beginning to allow the construction of the universe, but in the end we discover that ordinals are certain kinds of sets.
What gives?
Dave
On
Logically, you just prove things about the cumulative hierarchy from the ZFC axioms and it's not circular.
Conceptually, the way I like to think of it, the iterated powerset operation and the ordinals that you use to iterate build each other as you go. It's only by applying powerset to $V_\omega$ that we get an uncountable set. Using the well-orderings that you have, Replacement then translates those into ordinals to keep iterating along. As you go, you build bigger and bigger cardinalities and this allows the construction to continue forever.
There is no circularity.
When a universe is given, all the sets there exists. Including the ordinals. However when we are given a universe of $\sf ZFC$ we can prove that the sets $V_\alpha$ form a strictly increasing and definable hierarchy, and every set is a member of some $V_\alpha$.
On the other hand, though, suppose you are given a universe where all the axioms of $\sf ZFC$ hold except the axiom of regularity, in that case you can actually say that you construct the universe of $\sf ZFC$ using the $V_\alpha$'s, but these sets and the ordinals already exist, you just show that $V=\bigcup_{\alpha\in\sf Ord}V_\alpha$ is definable in the given universe, and that it satisfies all the axioms of $\sf ZFC$.