A Machine Proof of the Filter-Method Construction for Real Numbers

This paper presents a machine verification of a real number theory where real numbers are constructed using concepts related to filters. The theory encompasses a special filter, namely the non-principal arithmetical ultrafilter whose existence can be proven with the Continuum Hypothesis, to establish several non-standard number sets: ${}^{*}\mathbf{N}$, ${}^{*}\mathbf{Z}$ and ${}^{*}\mathbf{Q}$. The set of real numbers, $\mathbf{R}$, is subsequently obtained by the equivalence classification of a specific subset of ${}^{*}\mathbf{Q}$. The entire theory is thoroughly formalized, with each detail verified to ensure rigor and precision. The verification is implemented using the Coq proof assistant and is grounded in the Morse–Kelley axiomatic set theory. This work contributes a new selection of foundational material for the formalization of mathematical theories.