(baby) Model theory

This is of course a huge topic -- this is just a temptation/taste for
kiddies.

One may notice a formal similarity between
$X \implies Y$ and $A \subset B$

That is, they're both partial orders.
(reflective, transitive, anti-symmetric)

Every partial ordered set has a representation (model) in a power set
(via $Y \mapsto \set{X : X \leq Y}$; note that this represents it in
 its own power set, just as a group has a symmetric representation by
 its action on itself);

...this is why Venn diagrams work for both subsets and implications.

more generally, given any system of axioms,
a -model- is a set such that its subsets satisfy the axioms.
(make more clear and correct)

From this POV, every ring is a -model- for the axioms of ring theory,
etc.
However, in any -particular- ring, several things are true that are
not true in -general- (maybe it has characteristic $0$, say): thus not
all the models are -equivalent-.

One is often interested in systems of axioms with a unique model,
like $\bR$ (complete ordered field) or $\bC$ (uncountable
algebraically closed characteristic $0$).
