[FOM] Cardinality in the absence of AC

[Timothy Y. Chow]

I wrote:
>       can we construct sets A and B such that
>
>       1. in ZFC we can prove that |A| = |B|, but
>       2. in ZF+LM we can prove that B outnumbers A?

On Fri, 7 Jun 2019, Floris van Doorn wrote:
> You can take
> A = { X in powerset(R) | X is not Lebesgue measurable }
> B = powerset(R)

Good point!  So I guess my real question is a vaguer one---are there 
examples that violate one's intuition about cardinality, the way the R vs. 
R/Q example violates one's intuition?

Tim