Why does the Downward Löwenheim-Skolem Theorem guarantee that a first-order logic theory $T$ with a model of infinite cardinality must have a model of cardinality of $\aleph_0$? For all proof I have read, it is stated that a $T$ with an infinite model must have a model of a cardinality of $\max\{\aleph_0, \kappa\}$, in which $\kappa$ is the cardinality of the language $L$ and is assumed to be countable. However, if my knowledge is correct, languages for first-order theory may be uncountable, then in case $\kappa>\aleph_0$, would not $T$ only be proven to have a smaller uncountable model of cardinality $\kappa$, but not to have a model of cordiality of $\aleph_0$?
Such as the proof provided in [1]:
Theorem. If a set of sentences $T$ is consistent, in a language $L$ of cardinality $\kappa$, then $T$ has a model of cardinality $\leq\{\aleph_0,\kappa\}$.
Corollary (The Downward Löwenheim-Skolem Theorem). Suppose a language $L$ has cardinality $\kappa$, and a set of sentences $T$ has a model of cardinality $\kappa'>\kappa$, then $T$ has a model of cardinality $\max\{\aleph_0,\kappa\}$, assuming $\kappa'$ is infinite.
Proof. Enlarge the language $L$ to $$L'=L\cup\{C_\alpha:\alpha<\max\{\aleph_0,\kappa\}\},$$ where the $C_\alpha$'s are $\max\{\aleph_0,\kappa\}$ many new constant symbols. Set $$T'=T\cup\{C_\alpha\neq C_\beta:\alpha\neq\beta\}.$$ $T'$ has a model of cardinality $\kappa'$, and $T'$ has a model of cardinality $\leq\max\{\aleph_0,\kappa\}$ ($\kappa<\aleph_0$ $L'$-cardinality $\aleph_0$). $T'$ has no models of cardinality $<\max\{\aleph_0,\kappa\}$ since the $C_\alpha$'s have to have distinct interpretations. Therefore, $T$ must have a model of cardinality $\max\{\aleph_0,\kappa\}$. $\square$
Sorry if this question looks rather silly. I just started learning the theory part.
Thanks for all answers in advance, as I may not be able to reply later.
It doesn't.
If you additionally assume the the language is countable, then you get a countable model of $T$. But in general the best we can do is a model of size $\max(\aleph_0,|L|)$.
When people talk about Löwenheim-Skolem, they often say it can be used to produce a countable model, but they're either talking about a context where the language is countable, or they're implicitly assuming countable language for simplicity.