# set theory – Dedekind-“finiteness” for arbitrary limit cardinals

In $$mathbf{ZF}$$, it is possible for a set $$A$$ to be infinite but not to admit a countable set. In other words, for any $$alphainomega$$, there is an injection from $$alpha$$ into $$A$$, but there is no injection from $$omega$$ into $$A$$. If we replace $$omega$$ by a successor cardinal $$kappa^+$$ in the above statement, any $$A$$ of cardinality $$kappa$$ works, so there exists an example in $$mathbf{ZFC}$$. However, for limit cardinals, the question seems unclear to me.

Therefore: If $$kappa>omega$$ is a limit cardinal and $$A$$ is a set such that for any $$alphainkappa$$, there is an injection from $$alpha$$ into $$A$$, does it follow in $$mathbf{ZF}$$ that there is an injection from $$kappa$$ into $$A$$? Additionally, does the existence of such an $$A$$ imply the existence of a Dedekind-finite set?