Formal languages
Specify symbols, formation rules and formulas independently of any interpretation.
Side 211Side Studies / Research
Subject
Mathematical Logic
Purpose
Formal languages, proof systems, models, computability and incompleteness studied through the limits and structure of mathematical reasoning.
Structure
05 movesFormal systemV0
Definitions → structures → relations → proof → application
Separate syntax from semantics.
Mathematical logic clarifies what can be expressed, proved, modeled and computed by making the rules of reasoning themselves mathematical objects.
Proof systems
Define derivations and ask which conclusions follow syntactically from axioms under explicit inference rules.
Model theory
Interpret formulas in structures and distinguish truth in a model from provability in a formal system.
Computability
Formalize algorithms and undecidability through equivalent models of effective procedure.
Completeness & incompleteness
Study the boundaries between semantic consequence, formal proof and statements that escape sufficiently strong consistent systems.
Keep the boundaries visible.
truth ≠ provability
consistency ≠ completeness
undecidable ≠ unknown today
Questions that organize the Side.
How can a statement be true in one model and false in another?
What does Gödel incompleteness actually require of a formal system?
Why do different definitions of computability converge on the same class of functions?
What should carry weight here?
Formal derivations and metatheorems require explicit systems and hypotheses; intuitive arguments should be translated into the relevant syntax/semantics distinction.