# Description
A subfield of math that studies the foundations of mathematics. It rigorously formulates the notions of a mathematical language, formal logic, and proofs. Using these formalisms, it proves results about Logical Soundness and Logical Completeness of these systems.
I found it odd to talk about math fundamentally, and talk about set theory, using set theory. It felt very circular. The way I’ve come to understand it is that we are really proving things about these formal structures (e.g. Languages and Models). Because these structures apply to various different mathematical theories, we have now proven things about those theories.
# TODO
- Some examples of Theorys and Elementary Classes - See Marker - An Invitation Mathematical Logic Book examples 1.33-1.42
- Logical Consequence
- Universal Statements can be proved by proving for a generic element - Lemma 1.48 in Marker - An Invitation Mathematical Logic Book
- Language Homomorphism - 2.1 in Marker - An Invitation Mathematical Logic Book
- Language Embedding
- Language Substructure
- Language Isomorphism
- Elementary Equivalence
- Elementary Embedding
- Tarski-Vaught Theorem
- Downward Lowenheim-Skolem Theorem
- Formal Proofs…