[FOM] Cardinality in the absence of AC
Noah Schweber
schweber at berkeley.edu
Wed Jun 5 23:49:50 EDT 2019
Hello Tim,
Unless I'm missing something the answer is no, that situation is not
possible - and this is provable in (much less than) ZF alone.
Let f: B->C be injective and g: A->C be surjective, fix b in B and consider
the map h from A to B given by:
- If g(a) is in the range of f, then h(a) is f^{-1}(g(a)) (which is
well-defined since f is injective).
- Otherwise h(a)=b.
This seems to be a surjection from A to B (since f is defined on all of B).
- Noah
On Wed, Jun 5, 2019 at 10:18 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 was led to ask this question after reading an expository article by Alan
> Taylor and Stan Wagon in the April 2019 issue of the American Mathematical
> Monthly, about the fact that in ZF one can prove the existence of an
> injection from R to R/Q and in ZF+LM one can prove the nonexistence of an
> injection from R/Q to R. They interpret this fact as saying that in ZF+LM
> one can prove that there are "more" elements in R/Q than in R or that R/Q
> "outnumbers" R.
>
> It occurred to me that another interpretation could be that what is wrong
> is the definition of "|A| < |B|" as "there is an injection from A to B but
> no bijection from A to B." Perhaps it is better to define "|A| < |B|" as
> "there is an injection from A to B but no surjection from A to B." I
> believe that in the absence of choice, these two definitions are not
> equivalent.
>
> What is not immediately clear to me is whether the "<" relation as I have
> proposed it is transitive. Hence my question.
>
> Tim
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