Halfway Up To the Mathematical Infinity: On the Ontological and Epistemic Sustainability of Georg Cantor's Transfinite Design

TL;DR

This paper critically examines Cantor's concept of mathematical infinity, arguing that its ontological and epistemic foundations are compromised by outdated reductionist scientific ideologies, despite its historical significance.

Contribution

It challenges the sustainability of Cantor's transfinite set theory by analyzing its philosophical and scientific underpinnings, proposing a reevaluation of its foundational assumptions.

Findings

Cantor's transfinite design is rooted in 19th-century reductionist science.

Mainstream set theory has relied on ad hoc axiomatizations of Cantor's ideas.

The ontological and epistemic basis of Cantor's infinity is fundamentally compromised.

Abstract

Georg Cantor was the genuine discoverer of the Mathematical Infinity, and whatever he claimed, suggested, or even surmised should be taken seriously -- albeit not necessary at its face value. Because alongside his exquisite in beauty ordinal construction and his fundamental powerset description of the continuum, Cantor has also left to us his obsessive presumption that the universe of sets should be subjected to laws similar to those governing the set of natural numbers, including the universal principles of cardinal comparability and well-ordering -- and implying an ordinal re-creation of the continuum. During the last hundred years, the mainstream set-theoretical research -- all insights and adjustments due to Kurt G\"odel's revolutionary insights and discoveries notwithstanding -- has compliantly centered its efforts on \emph{ad hoc} axiomatizations of Cantor's intuitive transfinite design. We demonstrate here that the ontological and epistemic \emph{sustainability} of this design has been irremediably compromised by the underlying it peremptory, Reductionist mindset of the XIXth century's ideology of science.