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Concept

Frame — where it appears

An ordered list of independent directions at a point, one for each dimension. Two frames at the same point are compared by the determinant of the matrix carrying one to the other, and the sign of that determinant is the whole of what an orientation is.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

Which axioms hold on which frames. A table of frames against modal axioms, each cell decided by checking the axiom under every valuation.

The axiom is the shape of the graph

Add one operator meaning necessarily and the choice of which axioms to accept stops being a matter of taste. Each candidate axiom is true of exactly those worlds-and-arrows diagrams whose arrows have a stated property, and a logic is a class of graphs.

logic · Modal logic A frame carried round a Möbius band. A flat rectangle whose ends are about to be joined, with a pair of arrows carried along it — one along the band and one across it — and the sign of the frame at each station.

Orientation is a sign

Carry a pair of arrows once round a loop and compare them with the pair they started as. The comparison is a determinant, its sign is the whole of the answer, and no room, no side and no normal vector appears anywhere in the statement.

topology · Orientability Axioms, the conditions on the arrows they answer to, and the one that answers to none. A table of modal axioms with the property of the accessibility relation each corresponds to, every row decided by sweeping all relations on up to four worlds.

The axiom with no property of the arrows

Each axiom of modal logic can be matched by hand to a condition on the arrows between worlds. There is a recipe that does it for a whole class of axioms, and there is an axiom the recipe cannot reach — not because nobody has looked, but because no condition on the arrows defines it at all.

logic · Modal logic The frames on which provability makes sense, and the two that are refused. Six small frames marked by whether Löb's axiom is valid on them: transitive frames with no cycles accept it, and any frame in which a world can reach itself does not.

Necessity that means provable

Read the box as "the theory proves" and one modal logic stops being a proposal about what necessity might mean. It becomes a complete description of what a formal system can prove about its own proofs — and its frames run forward, compose, and stop.

logic · Modal logic Every section of the Möbius band crosses the zero section. The Möbius band drawn flat, with 3 continuous sections; zeros per section: 1, 3, 1.

Every section of the band must vanish

Read the Möbius band as a line standing over each point of a circle, and choose a point on each line continuously: a section. On a cylinder a section can stay away from zero all the way round. On the band it cannot — every section crosses zero an odd number of times — and that single fact is what orientability means for a bundle. There are exactly two line bundles over a circle, told apart by one sign, and two Möbius bands added together make the plain cylinder's twin.

topology · Orientability

Named alongside it

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

Kripke modelModal logicAccessibilityAxiomDeterminantExhaustive searchOrientabilityOrientationTransportVector bundleDefinabilityExpressive power

All concepts