Matpic
mathematics, in pictures
Concept

Model — where it appears

A structure giving each symbol of a formal language a meaning, so that every sentence of it becomes true or false. Exhibiting one is how a statement is shown unprovable from a set of axioms, since a derivation would have to hold in every model.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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.

logic · models 3 rounds on chains of 4 and 5. Two chains of dots with pebbles placed in turn, and the transcript of a play: Spoiler picks an element of one chain, Duplicator answers in the other, and the pebbles must keep the same order.

A game that decides what can be said

Two players take turns pointing at elements of two structures; if the second can survive k rounds, then no sentence with k quantifiers tells the structures apart — a statement about infinitely many formulas, settled by a finite search.

logic · ef games

Named alongside it

The objects these essays reach for when they reach for this one.

AxiomConsistencyDecision procedureElementary equivalenceExhaustive searchExpressive powerGeodesicIndependenceInterpretationParallel postulateQuantifierQuantifier order

All concepts