Completeness and Soundness: The Twin Pillars of Logic
Why first-order logic is the only system where proof and truth perfectly coincide
Model Theory Basics: When Sentences Have Multiple Interpretations
The same axioms, interpreted in different structures, yield different truths — and that is a theorem.
Writing Mathematics: From Scratchwork to Polished Proof
Why the messy path from scratchwork to polished proof is a distinct craft every mathematician must learn
Gödel's Incompleteness: The Limits of Formal Systems
How Gödel proved that mathematical truth forever exceeds what any formal system can demonstrate
Ordinal Numbers: Counting Beyond Infinity
How mathematicians formalized position in infinite sequences and built arithmetic beyond the finite
Mathematical Definitions: How Precision Enables Progress
Why getting definitions exactly right isn't pedantry—it's the engine that drives mathematical discovery forward.
Functions as Proofs: The Curry-Howard Correspondence
The profound discovery that writing programs and proving theorems are the same activity in different notation
Cardinal Numbers: Measuring the Size of Infinity
Not all infinities are created equal—rigorous proof reveals a vast hierarchy of infinite sizes
Axiom of Choice: The Most Controversial Principle in Mathematics
Why this seemingly obvious principle forces mathematics to accept objects it can never describe
Universal and Existential Quantifiers: The Language of Mathematical Statements
Two symbols that transform vague claims into mathematical certainty—and why their order changes everything.
Uniqueness Proofs: Proving There Can Be Only One
Master the logical techniques that prove mathematical objects are one of a kind
The Well-Ordering Principle: Why Natural Numbers Behave Nicely
The deceptively simple property that makes natural numbers uniquely suited for rigorous mathematical reasoning.
Diagonalization: Cantor's Revolutionary Proof Technique
The proof technique that revealed infinity's hierarchy and formal reasoning's hard boundaries
Proof by Cases: Dividing and Conquering Mathematical Problems
Transform impossible proofs into manageable pieces by mastering the logic of exhaustive case analysis
Proof by Contradiction: Harnessing the Power of Impossibility
Learn why assuming the impossible and reaching absurdity creates mathematical certainty when direct proof fails.
Mathematical Induction: The Domino Principle Made Rigorous
Master the two-step technique that proves infinite truths with finite reasoning—and avoid the traps that derail most beginners.
Why Direct Proof Remains the Most Powerful Tool in Mathematics
Master the transparent logic of direct proof and discover why building arguments step-by-step creates mathematical certainty nothing else can match.
Contrapositive Proof: The Hidden Logical Equivalence
Master the logical flip that transforms stubborn theorems into straightforward proofs by working backward from what you want to disprove.
Strong Induction: When Regular Induction Falls Short
Master the proof technique that handles recursive dependencies reaching back beyond the immediate predecessor.
Existence Proofs: Constructive vs. Non-Constructive Approaches
Discover why some proofs hand you what exists while others only prove it must be hiding somewhere—and why this difference matters.
The Pigeonhole Principle: Simple Counting, Profound Conclusions
How the most obvious fact in counting becomes a proof technique for establishing existence when direct construction fails completely.