Beyond Natural Numbers

Intro to Proofs · Axiom Academy

Induction is a chain of falling dominoes — but the chain can start anywhere, run either way, and even branch. One push, and the whole line goes down When you first met induction, the picture was a line of dominoes: knock the first one over ( P(1) ), make sure each one topples the next ( ), and the whole infinite row falls. That single idea is far more flexible than "start at 1 and march right." Watch the chain reaction first — then you'll bend it three different ways yourself. Here the dominoes are the statements , one per natural number. The green domino is the base case you prove outright; each falling domino knocks over its neighbor, so the truth propagates down the line. Press play and watch it cascade. The base case starts it; the implication carries it the rest of the way — forever. The base case doesn't have to be 1 Lots of theorems only kick in past some threshold — "for all ." No problem: prove the base case at n=k instead of n=1 , then the same step takes you the rest of the way. Drag to move the base case k and watch the proven region shift. Everything to the left of k is simply never claimed. Same chain reaction, started farther along — the proof now covers exactly . The chain can fall leftward, too

This is the written version of the interactive lesson above. See the full Intro to Proofs course.