Feasible Mathematics II pp 1-14 | Cite as

# On the Existence of modulo p Cardinality Functions

## Abstract

Suppose that *A* = 〈*A*, …〉 is a first-order structure with a finite number of finitary relation and function symbols, and *p* is a prime number. We say that a function *μ* with mod *p* values is a mod *p* cardinality function if it is defined on the first-order definable subsets of *A*, *A* ^{2}, … and it satisfies the following basic properties of the usual notion of cardinality. It is invariant under one-to-one first-order definable maps, it is additive with respect to disjoint union, it is multiplicative with respect to direct product, and the cardinality of singletons is 1. We show that if there is a first-order definable ordering of the universe then the existence of a mod *p* cardinality function is equivalent to the following statement: There are no two first-order definable equivalence relations Ф and Ψ on a (first-order definable) subset *X* of *A* ^{i} for some *i* = 1,2,… with the following properties: (1) each class of Ф contains exactly *p* elements, and (2) each class of Ψ with one exception contains exactly *p* elements, the exceptional class contains 1 element. (We will refer to this statement as the modulo *p* counting principle.)

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