The principle of transfinite induction is often stated as the following theorem.
Theorem. Suppose that $A$ is a class of ordinals. If
- (zero) $0$ is in $A$,
- (successor) whenever an ordinal $\alpha$ is in $A$, then $\alpha+1$ is also in $A$, and
- (limit) if $\lambda$ is a limit ordinal and $\lambda\subset A$, then $\lambda\in A$,
then $A$ contains all ordinals.
There are other accounts of transfinite induction that unify the hypotheses into the single statement that whenever all smaller ordinals than an ordinal $\alpha$ are in $A$, then $\alpha$ is in A, and it is considered more elegant to use that formulation when it is possible, but nevertheless many uses of transfinite induction consist in verifying the three properties above.
