Reverse Mathematics in Bishop’s Constructive Mathematics

Informal Constructive Reverse Mathematics

Constructive Mathematics

Constructive Mathematics , in

The first part of the paper introduces the varieties of modern constructive mathematics, concentrating on Bishop’s constructive mathematics (BISH). It gives a sketch of both Myhill’s axiomatic system for BISH and a constructive axiomatic development of the real line R. The second part of the paper focusses on the relation between constructive mathematics and programming, with emphasis on Martin...

Indecomposability of R and R\{0} in Constructive Reverse Mathematics

It is shown that —over Bishop's constructive mathematics— the indecomposability of R is equivalent to the statement that all functions from a complete metric space into a metric space are sequentially nondiscontinuous. Furthermore we prove that the indecomposability of R \ {0} is equivalent to the negation of the disjunctive version of Markov's Principle. These results contribute to the program...

Semantic Completeness of First-Order Theories in Constructive Reverse Mathematics

We introduce a general notion of semantic structure for first-order theories, covering a variety of constructions such as Tarski and Kripke semantics, and prove that, over Zermelo Fraenkel set theory (ZF), the completeness of such semantics is equivalent to the Boolean Prime Ideal theorem (BPI). In particular, we deduce that the completeness of that type of semantics for non-classical theories ...