During model constructions we always have cardinality restrictions on the models (ex: $|M| \leq |L| + \aleph_0$ or $|M| \geq |L| + \aleph_0$). Why?

Question

I realize it has something to do with the number of formulas (with parameters potentially) but I can't articulate the idea exactly.

An obvious example:

In the statement of the Lowenheim Skolem Theorem, given a model $M$ and a set $A \subset M$ and a cardinality $\lambda \leq |M|$, we say we can find an elementary submodel $N$ of $M$, such that $A \subseteq M$, and $|N| \leq |A| + |L| + \aleph_0$, rather than just sayind $|N| = |A|$. What is the exact thing that we're accounting for when we add the extra $|L| + \aleph_0$?

More generally (and I would like to know the general "motivation" because I can get the indiviual cases like in the proof of the Lowenheim Skolem theorems) why we seem to require padding, and why the padding is (almost always) $|L| + \aleph_0$.

Answer 1

This is usually due to some information or construction from formulas, and it's pretty easy to calculate that there are $|L|+\aleph_0$-many $L$-formulas. I'll address the specific case of the Löwenheim–Skolem. When taking an elementary submodel $N$ of a model $M$,

1. One reason for requiring $|N|\ge |L|$: our language could have a lot of constant symbols.

If our language $L$ consists just of $\kappa$-many constant symbols $c_\alpha$ for $\alpha<\kappa$ and $M$ thinks all of these constants are different, then any elementary submodel must also think all of these constants are different and must therefore have at least $\kappa=|L|$ many elements.

2. One reason for requiring $|N|\ge\aleph_0$: Infinitely many members of $M$ are definable.

If we think about $\mathbb{N}$, for example, any elementary submodel $N$ (in the finite language of $\{0,1,+,\cdot\}$) is going to need to contain $0$ and $1$. We can from there define $2=1+1$, $3=2+1$, etc. so that these will need to be in $N$. So it's not hard to see then that $N$ will need to be infinite (and in fact $\mathbb{N}$ itself).