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Concept

Asymptotic

A description of how a quantity behaves in the limit, ignoring everything that stays comparatively small. Two quantities are asymptotic when their ratio approaches one, which permits an error growing without bound so long as it grows more slowly.

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

Three properties that leave the middle, and one that cannot. Four measured curves of the share of random graphs having a property, plotted against the number of points: three first-order properties running to zero or one, and the parity of the edge count sitting on a half throughout.

Nearly always, or nearly never

Toss a coin for every pair of points and ask whether the graph that results has some property. For a property a first-order sentence can state, the answer in the limit is never a genuine probability — it is zero or it is one, and the game is what proves it.

logic · Ehrenfeucht–Fraïssé games Five families of primes, counted below 100,000. Five counting curves on logarithmic axes: every prime, the primes one more than a multiple of four, the twin pairs, the primes one more than a square, and the Mersenne primes. Two of the five families are known to be infinite and three are open questions.

Which infinitudes are proved

The primes never stop, and neither — apparently — do the twin pairs, the primes one more than a square, or the Mersenne primes. Three of those four statements are theorems and one is not, and counting the members of each family tells nobody which.

number · Infinitude of primes

Named alongside it

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

ConjectureCounting argumentDivergenceElementary equivalenceExhaustive searchExpressive powerHeuristicOpen problemPrimeProbabilityQuantifierRandom graph

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