Formalizing objective mathematics

More reverse mathematics of the Heine-Borel Theorem

Using the techniques of reverse mathematics, we characterize subsets X ⊆ [0, 1] in terms of the strength of HB(X), the Heine-Borel Theorem for the subset. We introduce W(X), formalizing the notion that the Heine-Borel Theorem for X is weak, and S(X), formalizing the notion that the theorem is strong. Using these, we can prove the following three results: RCA0 `W(X)→ HB(X), RCA0 ` S(X)→ (HB(X)→W...

Formalizing Mathematics using the Lean Theorem Prover

Formalizing Complex Plane Geometry

Deep connections between complex numbers and geometry had been well known and carefully studied centuries ago. Fundamental objects that are investigated are the complex plane (usually extended by a single infinite point), its objects (points, lines and circles), and groups of transformations that act on them (e.g., inversions and Möbius transformations). In this paper we treat the geometry of c...

Modeling the World

Once we create symbols, we can also hypothesize relationships among the symbols which we can later check for consistency with what we really observe in the World. By creating relationships among the symbols of things we observe in the World, we are in effect formalizing our knowledge of the World. By formalizing we mean the creation of rules, such as verbal arguments and mathematical equations,...

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 ordi...