[FOM] Cardinality in the absence of AC

[Mario Carneiro]

On Wed, Jun 5, 2019 at 11:24 PM Timothy Y. Chow <tchow at math.princeton.edu>
wrote:

> Let us work in ZF + "all subsets of reals are Lebesgue measurable" (or
> ZF+LM for short).
>
> Is the following situation possible?
>
> We have three sets, A, B, and C.  There exists an injection from A to B,
> and an injection from B to C.  There *does not exist* a surjection from A
> to B, and there *does not exist* a surjection from B to C.  Nevertheless,
> there exists a surjection from A to C.
>

I believe this is impossible even in ZF. If there is a surjection from A to
C and an injection from A to B, then there is a bijection from A to a
subset B' of B, and the inverse of that bijection produces a surjection
from B' to C, which can be extended to a surjection from B to C if C is
nonempty; but if C is empty then B is also empty because of the injection
from B to C, so the surjection from B to C still exists.

Mario
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