THE METAMATHEMATICS OF STABLE RAMSEY'S THEOREM FOR PAIRS

摘要

In this paper, we are motivated by the related questions "What are the implications between familiar infinitary mathematical principles?" and "What are the finitary consequences of these principles?" We consider principles, such as comprehension, compactness, measure or combinatorics, that assert the existence of sets of natural numbers, or similarly, real numbers. To give two examples, the compactness of the Cantor set says that every infinite subtree of the full binary tree has an infinite path and Ramsey's Theorem for Pairs says that for every partition of the pairs of natural numbers into finitely many pieces there is an infinite set, all of whose pairs belong to the same piece. We want to precisely pose and answer questions such as "Does the infinite Ramsey's Theorem for Pairs follow from the compactness of the Cantor set?" or "What consequences for the finite sets follow from the infinite Ramsey's Theorem for Pairs?"