Ordinals in set theory are well-ordered by $\in$ or equivalently $\subset$. If we define all ordinals greater or equal to $\omega$ as infinite ordinals. Is it true that every infinite ordinal is equinumerous to its successors.
Basically my question is the proof or refutation of the following statement:
Given infinite ordinal $\alpha$. Does there exist an injection from $\alpha^+$ to $\alpha$.