basic lossy notes on model theory

model theory, starting here:

purpose: to explore semantics explicitly by creating a formalized model and exploring its formal properties (explore semantics by investigating syntactical elements of a *corresponding language*

vs. proof theory: study of proofs as syntax, versus models, which study semantics

s-homomorphism:

Löwenheim–Skolem theorem: if theory has an infinite model, it has infinitely many for every infinite cardinal number k.

Absoluteness

finite model: model with finite domain, etc. (e.g. an undirected graph, where A edge B is the relation, which is defined by the interpretation, and nodes are the domain)


isomorphism theorems:

categorical: only 1 model can work, up to isomorphism (only applies for finite models)

first order logic: has there exists, for all, and negation on variables...

"satisfies": Tarski's definition of truth

first order theory: a set of sentences

structure (aka univeral algebra): {domain, signature, "interpretation" linking domain and signature}

signature: models the non-logical symbols of a language, both functions and relations (-, +, x, etc.)
logical symbols: true, false, and, or, not, etc.

domain / carrier / universe: elements

"interpretation": "and this is what the relation means"...i.e. explicit definition of the symbols by functionally defining it on the domain.... for example, can define "is an element of" ... this links in the semantics, before that the symbols are just symbols