I am well familiar with the principle of mathematical induction. But while reading a paper by Roggenkamp, I encountered the Principle of Transfinite Induction (PTI). I do not know the theory of cardinals, and never had a formal introduction to Set theory or Cardinals theory. The concept of PTI was amusing to me so I looked it up.
I googled and open Wikipedia which states it as "Let $P(\alpha)$ be a property defined for all ordinals $\alpha$. Suppose that whenever $P(\beta)$ is true for all $\beta < \alpha$, then $P(\alpha)$ is also true (including the case that $P(0)$ is true given the vacuously true statement that $P(\alpha)$ is true for all $\alpha\in\emptyset$). Then transfinite induction tells us that $P$ is true for all ordinals."
To understand it, I definitely had to read what an ordinal is. So another link on Wikipedia says: "A set S is an ordinal if and only if S is strictly well-ordered with respect to set membership and every element of S is also a subset of S." (due to von Neumann)
Now these definitions do not make it clear to me what PTI means. I know every set can be well ordered, i.e. given any set $S$, we can find a well ordered set $A$ such that $S$ can be written as $$S=\{s_\alpha: \alpha\in A\}$$ which I understand has been used in it.
Can some explain what PTI says, along with explaining what an ordinal is (which is comprehensible to a non-set theorist) and how it can be used by using some beginner cases to illustrate it.
Thanks!