Reverse math
The reverse mathematics , a branch of mathematical logic is trying to determine what axioms are necessary to certain theorems to prove. Reverse mathematics is thus in a sense the reverse of ordinary mathematics, which tries to derive theorems from axioms.
Reverse mathematics was introduced as a mathematical project by Harvey Friedman in 1974 . The idea for this arose from the results of set theory , including the classic theorem that the axiom of choice and Zorn's lemma are equivalent to the Zermelo-Fraenkel set theory ZF .
principle
The basic idea of reverse mathematics is this: You start with a core system of axioms that is too weak to prove a particular theorem, but still strong enough to derive the basic concepts it contains. The goal is now to expand the core system by exactly those axioms that are necessary to prove the theorem.
To do this, axioms are added to the core system and then two proofs are presented. The first proof must show that the theorem can be logically derived from the extended axiom system at all . The second proof must show that no weaker system of axioms is able to prove the theorem. This proof is carried out by showing that every other axiom system that the theorem can prove contains the axiom system at hand.
The approach is closely related to the consideration of the subset relationship of generating systems and minimal generating systems of vector spaces in algebra . What happens there, however, according to set theory within the axiomatic framework, happens here on a logical meta-level with the axioms themselves.
swell
- ^ H. Friedman: Some Systems of Second Order Arithmetic and Their Use. In: Proceedings of the International Congress of Mathematicians. Vancouver, USA, 1974, Vol. 1, pp. 235-242. Or: Canadian Mathematics Congress, Montreal, Québec , 1975.
literature
- Harvey Friedman / Stephen G. Simpson : Issues and Problems in Reverse Mathematics , in: Computability Theory and its Applications, Contemporary Mathematics 257, 2000, 127-144.
- SG Simpson: Reverse Mathematics , Lecture Notes in Logic 21, ASL 2005.
- SG Simpson: Subsystems of second order arithmetic. Perspectives in Mathematical Logic . Springer-Verlag, Berlin, 1999 (chapters 1–4)