The Contradiction Method

Intro to Proofs · Axiom Academy

LESSON The Contradiction Method Assume the opposite of what you want to prove, derive a logical impossibility, and conclude the original statement must be true. Every proof by contradiction follows the same elegant pattern. You don't argue for the statement directly — you assume it is false , then let logic walk you straight into an impossibility. Assume the negation, reach a contradiction (⊥), conclude the original The most common type: manipulate equations until you reach a mathematical impossibility like 0 = 1 , or a fraction that can't be in lowest terms. 3. Definitional Contradictions Sometimes the contradiction comes from violating a definition : proving something is simultaneously even AND odd, or both positive AND negative. 4. Self-Referential Contradictions The most mind-bending type: the assumption contradicts itself directly , constructing the very thing it claimed couldn't exist. Proof by contradiction is especially powerful when a direct attack gives you nothing to grab. Assuming the opposite often hands you a concrete object to manipulate. Proving irrationality: showing numbers like √2, √3, log₂3 aren't rational Proving infinitude: showing there are infinitely many primes, infinitely many solutions, etc. Proving impossibility: showing certain equations have no solutions Proving uniqueness: assume two distinct objects exist, derive a contradiction When direct proof seems hard: sometimes assuming the opposite gives you more to work with

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