A sketch of the proofs of the upward and downward Löwenheim-Skolem theorems, set to the tune of White Rabbit by Jefferson Airplane.
One way makes it larger
and the other makes it small
If you want to find a model
Of any size at all
Go ask L-S
Leopold and Thoralf
If you have two nested structures
Then to show that they agree
On each first-order question
You could ask conceivably
Just find a witness
For every backwards $\exists$
To make this happen, add a function
For every existential
If there’s a witness, just point to one
If not, then return null
There’s a substructure
Of size $\omega$ plus $\mathcal{L}$
If logic and proportion
Have fallen to their knees
When we add some distinct constants
and every sentence about these
The contradiction was with us already
Compactly, QED