Continuum Hypothesis within L using Upward Lowenheim-Skolem Trying to understand the proof of why $\mathit{CH}$ holds within $L$, I've seen that it strongly relies on downward Lowenheim Skolem to define a structure with an $\aleph_0$ cardinality. My question is whether we could use upward lowenheim skolem to define a similar structure with $\aleph_1$ cardinality (or superior), and then see if that structure holds in $L$. Thank you very much in advance.
 A: Generally, set theory focuses on well-founded structures, because we can then use a Mostowski collapse to convert them to transitive sets/classes which have nice absoluteness results.  This idea works with with downward Löwenheim–Skolem: if $N$ is an elementary substructure of $M$ and $M$ is well-founded, then so is $N$ (any infinite $\in^N$-decreasing sequence is $\in^M$-decreasing too).  In the case of $\mathrm{L}$, if we have an elementary substructure of $\mathrm{L}_{\omega_2}$, we get a well-founded structure which can be "collapsed" to a transitive set with nice properties (i.e. being a "small" level of $\mathrm{L}$).
But this is an issue with upward Löwenheim–Skolem: if we start with a well-founded $N$ and $N$ is an elementary substructure of $M$, there's no reason to think $M$ will be well-founded: $M$ might have an infinite $\in^M$-decreasing sequence of elements not in $N$.  We actually always know this can happen if $N$ is infinite: if we take the elementary diagram (so in the language adding constant symbols for every element of $N$)
$$T=\{\phi:\phi\text{ is a formula with parameters in }N\wedge N\models \phi\}\text{.}$$
We know that there are ill-founded models $M\models T$ by compactness (since $N$ is infinite).  But any such model has $N$ as an elementary substructure of $M$. So regardless of whether $N$ is well-founded or not, we have no control over whether its elementary extensions are well-founded, and this means we can't translate these elementary extensions to transitive sets like levels of $\mathrm{L}$.
A: JunderscoreH gave an excellent answer. But let me give another perspective.
The use of the downwards Löwenheim–Skolem theorem (+ the condensation lemma) is meant to show that every subset of $\omega$ was added in some countable stage of the hierarchy. Therefore $L\models\mathcal P(\omega)\subseteq L_{\omega_1}$, and therefore the Continuum Hypothesis holds.
But the upwards theorem is not enough to control the idea "every subset was captured at a countable stage". As Skolem's paradox teaches us, first-order logic is really not strong enough to distinguish countable and uncountable.
For example, suppose that $A\subseteq\mathcal P(\omega)$ and $|A|=\aleph_2$, then $L[A]$ the continuum hypothesis fails. Now look at $L_{\omega_2}[A]$, and consider any elementary submodel of size $\aleph_1$, it will satisfy $V=L[A']$ for some $A'\subseteq A$, and it will satisfy "There is no bijection between $\omega_1$ and $\mathcal P(\omega)$", but nevertheless the structure has size $\aleph_1$, and it is elementary.
So unless you a priori know that $\sf CH$ holds, you can't quite prove this way. In summary, the goal of using downwards Löwenheim–Skolem is a sort of "divide and conquer", which is incompatible with the upwards version of the theorem.
