Matpic
mathematics, in pictures
Concept

Consistency

The property of a set of assumptions from which no statement and its own negation can both be derived.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

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.

logic · models S1S2S3S4S5S6S7S1S2S3S4S5S6S7Rthe sets that do not contain themselvesrow i, column j is marked when set i contains set j — the diagonal asks whether a set contains itselfthe row beneath is the complement of the diagonal, and it is not one of the rows above it

A list that cannot contain itself

The set of all sets that do not contain themselves is not a set. The argument is the diagonal again, applied to a table whose rows and columns are the same objects, and it destroyed the foundations of mathematics in a postcard.

logic · diagonalisation φ10110101000φ21101010000φ30011111000φ41010100000φ50001001000φ60111110000φ71110011001φ8010100000110011001neweach row is a sentence, and whether it says yes to each numbered question; the markedsquares are the diagonalthe row underneath is the sentence that answers each question the opposite way to theone asked about it — so the answers cannot all be given by a sentence on the list

The sentence that says it has no proof

Number every sentence and every proof, and a formal system can talk about itself. Then the diagonal is available one more time, and what it builds is a sentence that is true exactly when it is unprovable.

logic · diagonalisation

Named alongside it

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

Diagonal argumentSelf referenceArithmetisationAxiomComprehensionFixed pointFormal systemGeodesicIncompletenessIndependenceInterpretationMembership

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