Formalizing Scientifically Applicable Mathematics in a Definitional Framework

Abstract

In [Arnon08, A framework for formalizing set theories based on the use of static set terms.] a new framework for formalizing mathematics was developed. The main new features of this framework are that it is based on the usual first-order set theoretical foundations of mathematics (in particular, it is type-free), but it reflects real mathematical practice in making an extensive use of statically defined abstract set terms of the form { x | p(i) }, in the same way they are used in ordinary mathematical discourse.In this paper we show how large portions of fundamental, scientifically applicable mathematics can be developed in this framework in a straightforward way, using just a rather weak set theory which is predicatively acceptable and essentially first-order. The key property of that theory is that every object which is used in it is defined by some closed term of the theory. This allows for a very concrete, computationally-oriented interpretation of the theory. However, the development is not committed to such interpretation, and can easily be extended for handling stronger set theories (including ZF).

Keywords

• MKM
• Scientifically Applicable Mathematics
• safety relations