Count(q) Does Not Imply Count(p)

I solve a conjecture originally studied by M. Ajtai. It states that for different primes q, p the matching principles Count(q) and Count(p) are logically independent. I prove that this indeed is the case. Actually I show that Count(q) implies Count(p) exactly when each prime factor in p also is a factor in q. 1 The logic of elementary counting “She loves me, she loves me not, she loves me,. . .” The final answer does not depend on the order in which the leaves are pulled of. Every child who is familiar with the process of counting knows that. The underlying logical principle states that a set A has a well-defined cardinality modulo 2. Yet, the Count(2) principle can fail in quite strong systems of Arithmetic [2],[3]. Similarly for the counting principle modulo p (=Count(p)) where she can be in p states of mind. This is very difficult to visualise. In 1962 Cohen invented the famous technique of forcing. He used the method to show the independence of the continuum conjecture. Inspired by these ideas Ajtai showed that the elementary pigeon-hole principle need not hold in strong systems of Arithmetic [2]. Ajtais result was a major break through. The main novelty was the mixture of forcing and powerful probabilistic techniques. The Count(q) versus Count(p) problem has various formulations and variants. The most famous variant is from circuit complexity theory [13]. It asks (in the ∗This work was initiated at Oxford University England. †Basic Research in Computer Science, Centre of the Danish National Research Foundation.