Reverse mathematics is a relatively new program in mathematical logic to the foundations of mathematics. The program was founded by Harvey Friedman in 1976. Its basic goal is to determine which axioms are required to prove theorems of mathematics assess the relative logical strengths of theorems from ordinary non-set theoretic mathematics.Its defining method can be described as “going backwards from the theorems to the axioms”, in contrast to the ordinary mathematical practice of deriving theorems from axioms. The reverse mathematics starts with a base theory- core axiom and the language framework in spite of it is too weak to prove most of the theorems, but still able to provide the definitions necessary to state these theorems.
For example, to study the theorem “Every bounded sequence of real numbers has a supremum” it is necessary to use a system based on real numbers (speak of real numbers and have sequence of them).Reverse mathematics is formed not only to study possible axioms for set theory but mainly for possible axioms of ordinary theorems.
This theory has wide application in the fields such as study the reverse mathematics for property of any finite character, to study set theory, deriving theorems, study its behaviour in the context of second-order arithmetic, and for the quantifier structure of the formula defining the property. Reverse mathematics is a potential application and has advantages in the field of Applied Mathematics.
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