to show that there is no injection from a finite successor of finite ordinal to itself

Question

im trying to prove this proposition let $\alpha \in \omega$ so i want to show that there is no injection (not necessarily keeps order) from $\alpha +1$ to $\alpha$ without using cardinality or the axiom of choice... I did try some different directions with the order type and contradiction, but didn't work

will be happy for help!

Answer 1

The proof is by induction on $\alpha$.

And it uses the following lemma:

If there exists an injection from $X$ to $Y$, and $x\in X$ and $y\in Y$, then there is an injection $f$ such that $f(x)=y$.

The proof is not difficult, and I'll leave it to you.

From here the inductive argument is pretty straightforward.