Exploring The Strength Of Ramsey’s Theorem

Reverse mathematics is a hot topic in mathematical logic nowadays. Differing from the ordinary mathematics which always prefers to look for a proof or construct a counterexample, reverse mathematics is more concerned about the proofs of an existing theorem. There are five subsystems of second order arithmetic( RCA0, WKL0, ACA0, ATR0 and Π11-CA0, known as “the big five”) which is designed to sort all of the mathematical theorems. For a specific theorem one first analysis its proofs carefully and then put it into one of the big five. With in-depth study, however, more and more theorems are found that could not be classified to any specific subsystems. Ramsey’s Theorem(RT) and some other mathematical principles deriving from RT are the important examples. The studies on the strength of RT have been one of the emphases for research in reverse mathematics over the past several decades.In this paper we overview the research achievements on the strength of RT from the elementary to the profound under the background of reverse mathematics in rigorous but plain language. Firstly, we tell the historical story of reverse mathematics and the origin of RT while reviewing the foundations of computability. Then we give the formal definition of the second order arithmetic(Z2) and make an introduction to RCA0, WKL0 and ACA0 in which the strengths of them are strictly increasing. Then along the studies of the past 50 years we introduce the studies in the strength of Ramsey’s Theorem for Pairs(RT22). From the pioneering works of Specker and Jockusch in the early 1970 s to Seetapun’s theorem and Liu’s theorem, and to the newest results gained by C hong, Slaman and Yang recently we try to draw an outline of the whole study clearly.On the basis of the previous text and discussed results, we reconsider the definition of the stable coloring- function of Stable Ramsey’s Theorem for Pairs(SRT22). With making some restrictions to the coloring- function such as setting the computability of the coloring- function we gain a new mathematical principles WSRT22, then we try to explore its strengths under the background of reverse mathematics and give some results(such as that WSRT22 is provable in ACA0), and give a sufficient condition of that WSRT22 is no t provable in RCA0.