Weak Cardinality Theorems for First-Order Logic

Kummer’s cardinality theorem states that a language is recursive if a Turing machine can exclude for any n words one of the n+ 1 possibilities for the number of words in the language. It is known that this theorem does not hold for polynomial-time computations, but there is evidence that it holds for finite automata: at least weak cardinality theorems hold for finite automata. This paper shows that some of the recursion-theoretic and automata-theoretic weak cardinality theorems are instantiations of purely logical theorems. Apart from unifying previous results in a single framework, the logical approach allows us to prove new theorems for other computational models. For example, weak cardinality theorems hold for Presburger arithmetic.