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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?

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    $\begingroup$ Oh yeah, I just realized where I learned this from: mathoverflow.net/questions/104913/… should this be closed as a duplicate? $\endgroup$
    – Asaf Karagila ♦
    Commented Mar 30, 2015 at 7:51
  • $\begingroup$ Now the question has changed and I removed my comment. With this version, the answer is no, as is shown in Asaf's answer. $\endgroup$
    – Mohammad Golshani
    Commented Mar 30, 2015 at 7:51
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    $\begingroup$ (2) is strictly stronger than (1), but equivalent to the dual version of (1) ("1-1 map from $X$ into $X\setminus \{x_0\}$"). The dual version of (2), on the other hand (surjection from $X$ onto $\omega$) follows from (1) but is strictly weaker $\endgroup$
    – Goldstern
    Commented Mar 30, 2015 at 10:27

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No, they are not equivalent.

It is a nice theorem that if there exists an infinite Dedekind-finite set (which is a set which satisfies the negation of (2)), then there is one which satisfies the first condition.

If $D$ is a Dedekind-finite set, then $S(D)$ which is the set of all injective finite sequences from $D$ is also Dedekind-finite (because the sets are injective, every collection of them is uniformly enumerated, so if there was a countable infinite set of these sequences, their union would be a union of uniformly enumerated sets, which would be a countably infinite subset of $D$).

Now simply consider the projection from $S(D)\setminus\{\varnothing\}$ onto $S(D)$ where you remove the last coordinate of the sequence.

You might be interested in the following paper:

Truss, J. Classes of Dedekind finite cardinals. Fund. Math. 84 (1974), no. 3, 187–208. PDF

In which the author takes seven definition of finiteness (proposed by Azriel Levy) and investigates the relations between them. Your first condition is one of the properties considered there.

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