Presburger arithmetic
Named by 2 essays across one field — each of them below, with the objects they name alongside it.
Arithmetic with addition alone
Over the real numbers, a quantifier's shadow is described by inequalities. Over the whole numbers with addition and multiplication, a shadow can be any set a computer can list. In between lies arithmetic with addition and no multiplication, and there the shadows are always the same kind of thing: a finite exception, then a pattern that repeats. The whole numbers made from coins worth 6, 9 and 20 are every number from 44 on; the squares, which need multiplication, never repeat at all.
A machine that carries one bit
Write three numbers in base two, one above another, and read the columns from the right. Whether the bottom number is the sum of the top two can be checked with one bit of memory — the carry. That two-state machine is the whole of addition, and machines can be combined, negated and made to guess a missing number. So every sentence about whole numbers built from addition can be decided by building a machine for it, and the same machines can also handle 'x is a power of two', which addition alone cannot say.
Named alongside it
The objects these essays reach for when they reach for this one.
Decision procedureQuantifierBinaryCompletenessDecidabilityFinite automatonFrobenius numberIncompletenessModular arithmeticPeriodicityQuantifier elimination