1
vote
0answers
58 views

Generalization of simple and transfinite induction

Definition For a set of ordinals $\boldsymbol\alpha$ and ordinals $\gamma$, $\beta$, let $$\boldsymbol\alpha \xrightarrow[\gamma]{}\beta$$ symbolize the proposition that ...
5
votes
1answer
113 views

Induction over all ordinal numbers.

I'm starting to learn set theory. I've already learned transfinite induction and I started to wonder if we could do the same on a well-ordered proper class, like the class of all ordinal numbers. It ...
4
votes
0answers
111 views

Transfinite induction and valuation rank

In Engler's valued fields, exercise 3.5.2 goes as follows Construct valuations on $\Bbb{C}$ of rank $\kappa$ for every cardinal $\kappa \leq 2^{\aleph_0}$. The idea behind this (for any ...
2
votes
2answers
170 views

$\wp^{(\omega)}(A)=?$

Let $A$ be a set, $$\wp^{(0)}(A)=A$$ $$\wp^{(n+1)}(A)=\wp(\wp^{(n)}(A))$$ But what sense does $\wp^{(\alpha)}(A)$ make where $\alpha$ is a limit ordinal number? The most natural way is let ...