I found two definitions of the Downward Löwenheim-Skolem Theorem:
Def.1: If Γ is consistent, then it has a countable model, i.e. it is satisfiable in a structure whose domain is either finite or countably
infinite.
From Zach et al.
Def. 2: Let B be a model for ℒ and let κ be any cardinal such that |ℒ| ≤ κ ≤ |B|. Then, B has an elementary submodel A of cardinality κ.
From Weiss et al.
What confuses me is that the first definition makes statements of finite models while the second definition makes statements only about models of cardinality |ℒ| ≤ κ, i.e. κ must be at least countably infinite, since "any language ℒ contains infinitely many variables".
Is there an explanation that somehow unifies those two definitions? Or are they fundamentally different?
Looking forward to the responses 😊.
Best Answer
Well the difference is that the first version of the theorem is simply false if there are not more assumptions on the language; and even if the right hypotheses are given, the first version is incredibly weaker than the second one.
Note that, as stated, version 2 is also false, $\kappa$ needs to be assumed to be infinite for it to be true.
With these precisions, let's compare the two versions. For simplicity, and to make the first version true and the second version easier to state, let's assume the language is at most countable.
In this case the first version becomes true, and the second one becomes : if $B$ is an $\mathscr{L}$-structure and $\kappa$ an infinite cardinal, $\kappa\leq |B|$ then $B$ has an elementary substructure of size $\kappa$.
From this we may deduce the first version by simply saying : if $\Gamma$ is consistent, then either it has a finite model, in which case we are done; or it has an infinite model, in which case we may apply version 2 with this model $B$ and $\kappa = \aleph_0$.
Since $B$ is a model of $\Gamma$, any elementary substructure is also a model of $\Gamma$, so we get a countable model and we are done.
Now version 1 is much weaker than version 2 because it does not speak about any cardinals beyond $\aleph_0$, and it does not even speak about elementary substructures; you only get models of a certain theory.
I don't know if this weakness can be quantified, but for instance I wouldn't be surprised if there were models of ZF+(version 1) + $\neg$ (version 2) (note that I have to put ZF and not ZFC because of course ZFC proves version 2 - the existence of these models of course relying on the consistency of ZF) - though I can't be sure, someone more qualified than me should be able to answer.