Logic

Self-reference, diagonalization, and the limits of formal systems.

Self-Reference Paradoxes — Liar, Russell, Berry — three self-reference tricks that power Gödel.
Cantor's Diagonal Argument — Some infinities are bigger than others — and diagonalization is the proof.
Gödel's Incompleteness — Self-reference plus arithmetization yields a true sentence that proves nothing.

Further Topics

Functional Completeness of Connectives — Why a handful of connectives — or NAND alone — express every truth table.
Compactness Theorem (Sentential) — Satisfiable if and only if every finite subset is.
Quantifiers & First-Order Translation — Render English into and , and read quantifier scope.
Models & Satisfaction — When a structure satisfies a sentence, and building models to test it.
Natural Deduction — Prove conclusions from premises with introduction and elimination rules.
Soundness & Completeness — Provable equals valid — Gödel's bridge between syntax and semantics.
Löwenheim–Skolem Theorem — Every satisfiable theory has a countable model, and the Skolem paradox.
Nonstandard Models of Arithmetic — Compactness builds a model of arithmetic with an infinite number.
Decidability & Recursive Sets — Decidable versus merely semi-decidable, and why incompleteness blocks a decision procedure.