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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.

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.