Chapter 21 Reverse algebra

Abstract

Publisher Summary In this chapter, Reverse algebra is discussed in detail. Reverse algebra is part of a program called “reverse mathematics” whose goal is to answer the main question: “Which set existence axioms are needed in order to prove the theorems of countable algebra?” The set existence axioms are formulated in the context of weak subsystems of the second-order arithmetic. Almost all countable algebra can be developed within the formal system Z 2 of second-order arithmetic. Investigations revealed that the set existence axioms of Z 2 are in fact, too strong. If a theorem of countable algebra is proved from the weakest possible set existence axioms, it will be possible to “reverse” the algebraic theorem by proving that it is equivalent to those axioms over a weaker base theory. The chapter provides examples of how some theorems of countable algebra can be developed in weak subsystems of Z­ 2 , and how to “reverse” them. How the reverse algebra is related to recursive algebra has been discussed. The similarities and differences between reverse algebra and recursive algebra is illustrated by an example.