On the Existence of modulo p Cardinality Functions

  • Miklós Ajtai
Conference paper
Part of the Progress in Computer Science and Applied Logic book series (PCS, volume 13)


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.)


Pairwise Disjoint Congruence Relation Propositional Formula Peano Arithmetic Pigeonhole Principle 
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Copyright information

© Birkhäuser Boston 1995

Authors and Affiliations

  • Miklós Ajtai
    • 1
  1. 1.K/53IBM Almaden Research CenterSan JoseUSA

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