Given a set $X$ and a cardinal $\kappa$ such that $\kappa<|X|$, can I always find a subset of $X$ which has cardinality $\kappa$?
If that's the case, then there must be a subset of the reals which has cardinality $\aleph_1$ ( even though we cannot prove there's a bijection between it and the reals themselves ) is there an explicit example of such a subset?
On
Under $ZFC$ every set is equinumerous with some cardinal and can be well ordered, so you can just well order it and take the first $\kappa$ items for a subset of size $\kappa$.
It is consistent with $ZFC$ that the reals are of size $\aleph_1$, in which case any subset larger than $\aleph_0$ has size $\aleph_1$. If the reals are larger than $\aleph_1$ they have a (many) subset(s) of size $\aleph_1$ but we cannot display one. If we could and we could show it was smaller than the reals we would know the reals were at least $\aleph_2$.
By definition, the cardinality of set $Y$ is smaller than the cardinality of set $X$ exactly if there exists an injection from $Y$ into $X$. Obviously the image of $Y$ in $X$ then has the same cardinality as $Y$. Thus yes, if $k<|X|$, then $X$ has a subset of cardinality $k$.
And this indeed means that $\mathbb R$ has a subset of cardinality $\aleph_1$. But we cannot explicitly give one. Or more exactly, we cannot give an explicit set of which we can prove that its cardinality is $\aleph_1$. Of course since it is consistent that $\mathbb R$ has cardinality $\aleph_1$, just $\mathbb R$ itself might be such an example. But we cannot prove or disprove it.
Note that my first paragraph is independent of the axiom of choice. However, the second paragraph isn't, as without the axiom of choice it is consistent that $|\mathbb R|\ngeq\aleph_1$.