Count(q) versus the Pigeon-Hole Principle
نویسنده
چکیده
For each p 2 there exists a model M of I 0 () which satisses the Count(p) principle. Furthermore, if p contains all prime factors of q there exist n; r 2 M and a bijective map f 2 dom(M) mapping f1; 2 A corollary is a complete classiication of the Count(q) versus Count(p) problem. Another corollary shows that the pigeonhole principle for injective maps does not follow from any of the Count(q) principles. This solves an open question Ajtai 94].
منابع مشابه
Count( qq) versus the pigeon-hole principle
For each p ≤ 2 there exist a model M∗ of I∆0(α) which satisfies the Count(p) principle. Furthermore if p contain all prime factors of q there exist n, r ∈ M∗ and a bijective map f ∈ Set(M∗) mapping {1, 2, ..., n} onto {1, 2, ..., n+ qr}. A corollary is a complete classification of the Count(q) versus Count(p) problem. Another corollary solves an open question ([3]). In this note I state and pro...
متن کاملCount(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,. . ....
متن کاملDiscrete Mathematics What is a proof?
The pigeonhole principle is a basic counting technique. It is illustrated in its simplest form as follows: We have n + 1 pigeons and n holes. We put all the pigeons in holes (in any way we want). The principle tells us that there must be at least one hole with at least two pigeons in it. Why is that true? Try to visualize the example of n = 2; therefore, we have 3 pigeons and 2 holes. Let’s try...
متن کاملA Note on Polynomial-size Monotone Proofs of the Pigeon Hole Principle
We see that the version of the pigeonhole principle in which every hole is forced to receive a pigeon (called onto) and the version in which every pigeon is mapped into exactly one hole (called functional) have polynomial-size proofs in the tree-like monotone sequent calculus. The proofs are surprisingly simple reductions to the non-monotone case.
متن کاملGaps between Prime Numbers and Primes in Arithmetic Progressions
The equivalence of the two formulations is clear by the pigeon-hole principle. The first one is psychologically more spectacular: it emphasizes the fact that for the first time in history, one has proved an unconditional existence result for infinitely many primes p and q constrained by a binary condition q − p = h. Remarkably, this already extraordinary result was improved in spectacular fashi...
متن کامل