Let $\mathfrak{A}$ is a well-ordered set.
Let $f$ is a function which maps a start segment $\{ i\in\mathfrak{A} \,|\, i Let a transfinite sequence $a$ on $\mathfrak{A}$ is 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: 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.