The constructive emergence of the real numbers

Michael T. Hutchins

Version 1.0 | January 2026

 

Abstract

We develop a constructive framework in which the real numbers arise as the closure of a transfinite diagonal process over stabilized binary growth classes, rather than as a completed totality. Beginning from the algebraic numbers and admissible verification procedures, numbers are classified by their infinite binary tails. A hierarchy of growth-indexed diagonals yields a Cantor-type jump and iterated refinement collapses all tail distinctions at a closure ordinal, producing continuity as a fixed point. The resulting continuum is tail-complete in the Baire ultrametric and connects to Hardy’s orders of infinity and the rigidity results of Roth and Adamczewski-Bugeaud.

 

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citation

Hutchins, Michael T. 2026. “The Constructive Emergence of the Real Numbers.” Preprint, mikehut.com. Version 1.0, January 2026.

DOI: https://doi.org/10.17605/OSF.IO/YBXKM