The statement, "Any two aleph-1-dense subsets of the reals are order isomorphic."
A subset X of R is called aleph-1-dense if between any two real numbers, there are exactly aleph-1 elements of X. On the one hand, Sierpinski used a diagonalization argument (working within ZFC) to construct kappa pairwise non-isomorphic suborderings of R each of density kappa, where kappa is the cardinality of R, so the Continuum Hypothesis implies that this statement fails. On the other hand, James Baumgartner used a clever forcing argument to build models of ZFC where the size of R is aleph-2 and any two aleph-1-dense suborderings of R are isomorphic.
See "All aleph_1 dense sets of reals can be isomorphic," James E. Baumgartner, Fundamenta Mathematicae v. LXXIX (1973), pp. 101-106.