Matpic
mathematics, in pictures
Ladder

Models — the ladder

2 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. Pone line, one point off it, and 4 lines through the point that never meet itevery arc here meets the rim at a right angle, and every miss was checked for crossingsinside the disc

    Two worlds that both obey the rules

    A statement is independent of a list of axioms when there is a structure satisfying the axioms where it holds and another where it fails. That is not a claim about what nobody has managed to prove — it is a proof that nobody can.

    rung 1 · logic
  2. every node has at most 3 children and the tree reaches level 416 nodes on the bottom row; the marked walk takes a surviving child at every step

    An infinite tree has an infinite path

    A tree that goes on forever, in which every node has only finitely many children, must contain a single branch that goes on forever. The proof is a rule for walking, and the rule is the whole of why finite information can decide an infinite question.

    rung 2 · logic

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