First-Order Logic with Equicardinality in Random Graphs

We answer a question of Blass and Harary about the validity of the zero-one law in random graphs for extensions of first-order logic (FOL). For a given graph property P, the Lindström extension of FOL by P is defined as the minimal (regular) extension of FOL able to express P. For several graph properties P (e.g. Hamiltonicity), it is known that the Lindström extension by P is also able to interpret a segment of arithmetic, and thus strongly disobeys the zero-one law. Common to all these properties is the ability to express the Härtig quantifier, a natural extension of FOL testing if two definable sets are of the same size. We prove that the Härtig quantifier is sufficient for the interpretation of arithmetic, thus providing a general result which implies all known cases of Lindström extensions which are able to interpret a segment of arithmetic.