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Doing infinitely many things — page 5

Sums that never end, subdivisions that never stop, and the care required to make either of them mean something.

Hilbert's polygons, stage by stage. Hilbert's polygons at stages 1, 2, 3 and 6, each visiting the centres of 4, 16, 64 and 4,096 squares in order, from the bottom-left corner to the bottom-right.

A path that fills a square

A pen moving continuously for one unit of time can pass through every point of a square. Hilbert's polygons, each threading four times as many squares as the last, move no point more than 0.4507 × 2⁻ⁿ from where the previous one put it — and that one estimate, through the theorem that uniform limits stay continuous, is the whole proof. What the limit pays is exactness of a kind: it repeats some points four times, and its points can never be closer than the square root of their parameters allows.

A list of sentences each denying all the later ones. Fourteen sentences, each saying that every later sentence is false, drawn as a grid of which sentences each one mentions — every cell above the diagonal and none on or below it — beside the only consistent assignment of truth values, in which the last sentence alone is true.

Logic

A liar without a loop

Every paradox of truth seemed to need a sentence that talks about itself, directly or round a loop. Stephen Yablo found one that does not: an infinite list in which each sentence says that all the later ones are false. Cut it off anywhere and exactly one assignment of truth values works — all false but the last; keep it infinite and none does. A finite system with no loop always has exactly one, so the paradox lives in the one place a finite picture cannot draw: a list with no last sentence.

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