Formalizing Scientifically Applicable Mathematics in a Definitional Framework

In [3] 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 |φ}, in the same way they are used in ordinary mathematical discourse. In this paper we outline how the fundamental scientifically applicable mathematics is straightforwardly developed in this framework in the usual way, using a rather weak predicatively acceptable set theory. The key property of the theory is that every object which is used 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 ).