Consider the following statements about a given set $X$ in in $\mathsf{ZF}$:

(1) There is $x_0\in X$ such that there is a bijection $\varphi: X\to X\setminus\{x_0\}$.

(2) There is an injective map $\iota:\mathbb{N}\to X$.

It is easy to see that (2) implies (1) in $\mathsf{ZF}$, but are they equivalent?

Consider the following statements about a given set $X$ in in $\mathsf{ZF}$:

(1) There is $x_0\in X$ such that there is a surjective map $\varphi: X\setminus\{x_0\}\to X$.

(2) There is an injective map $\iota:\mathbb{N}\to X$.

It is easy to see that (2) implies (1) in $\mathsf{ZF}$, but are they equivalent?