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This is the definition of $|A| <= |B| $:

there exists an injective function from $A$ into $B$.

Why it is defined that way and not like the following proposal, which is equivalent under the assumption of the axiom of choice:

there exists a surjective function from $B$ into $A$.

Amit Keinan
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    I'd say that was the definition of $|A|\le| B|$. Also I like $|\emptyset|\le|{0}|$. Don't you? – Angina Seng Jul 25 '20 at 17:36
  • Why shouldn't it be? In injective function seems to imply that $A$ is "smaller" or equal to $B$ (because we can "fit $A$ into $B$") whereas there being a surjective function seems to imply that $B$ is larger or equal to $A$ (because we can "wrap $B$ around $A$"). Given the choice between to equivalent functions I'd choose the one that linguisticly most conveys the concept we want to convey. (Also not having to rely on AoC is a plus... [but that just puts off the issue for later]). – fleablood Jul 25 '20 at 17:45
  • @AnginaSeng: Formally you're correct, of course, but the usual thing is to either require surjection from a subset of $B$ or explicitly allow for $A$ to be empty. – Asaf Karagila Jul 25 '20 at 17:54
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    Amit, the short answer is that the surjection-based relation is not provably antisymmetric. – Asaf Karagila Jul 25 '20 at 17:56
  • There are also a lot of links in my answer to https://math.stackexchange.com/questions/2158581/is-denying-paradoxical-partitioning-equivalent-to-accepting-the-axiom-of-choic, which itself is somewhat relevant to your question. – Asaf Karagila Jul 25 '20 at 18:12
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    Elaborating on Asaf's comments: the Cantor-Schroeder-Bernstein theorem, which is provable without choice, shows that when we have injections from $A$ to $B$ and from $B$ to $A$ then we have a bijection between $A$ and $B$. So the definition of "$\le$" via injections is appropriately antisymmetric: we have $\vert A\vert\le\vert B\vert$ and $\vert B\vert\le \vert A\vert$ iff $\vert A\vert=\vert B\vert$. By contrast, without the axiom of choice we could have two sets each of which surjects onto the other but which cannot be in bijection with each other - a very annoying phenomenon. – Noah Schweber Jul 25 '20 at 22:02

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