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. A line, a point, and many parallels. A disc whose lines are arcs meeting the boundary at right angles, showing several lines through one point that never meet a given line.

    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. A tree branching at most 3 ways, to depth 4, and the path through it. A tree drawn level by level, with the nodes that die out faint and a highlighted path that always steps to a node with descendants at the bottom.

    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

All ladders