Let $\mathfrak{A}$ be a well-ordered set.
Let $f$ be a function which maps a start segment $\{ i\in\mathfrak{A} \,|\, i
Let a transfinite sequence $a$ on $\mathfrak{A}$ be defined by the formula:
$a_c = \Phi f\left(a|_{\{ i\in\mathfrak{A} \,|\, i
where $\Phi$ is a function which maps an element of a poset into an atom under this element.
We have the axiom (in fact a theorem which follows from my definition of $f$ which I don't put here) that provided $a$ conforms to this formula, $f\left(a|_{\{ i\in\mathfrak{A} \,|\, i
Now I am confused: to prove that such $a$ exists, is it transfinite induction or transfinite recursion? The formula $a_c = \Phi f\left(a|_{\{ i\in\mathfrak{A} \,|\, i
My questions:
- What is this: transfinite induction, transfinite recursion, or a combination of these two?
- How to describe all this in terms of transfinite induction and transfinite recursion?
Note: I intentionally ignore the "set of all sets" paradox, because it is not the thing I have trouble with.
Note: I want to define $\Phi$ using axiom of choice as an arbitrary function which maps a non-least element into an atom under this element.