TY - JOUR

T1 - Formalizing scientifically applicable mathematics in a definitional framework

AU - Avron, Arnon

AU - Cohen, Liron

N1 - Funding Information:
This research was partially supported by the Ministry of Science, Technology and Space, Israel.

PY - 2016/1/1

Y1 - 2016/1/1

N2 - 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 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).

AB - 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 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).

UR - http://www.scopus.com/inward/record.url?scp=84957073392&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84957073392

VL - 9

SP - 53

EP - 70

JO - Journal of Formalized Reasoning

JF - Journal of Formalized Reasoning

SN - 1972-5787

IS - 1

ER -