At the End of Infinity

TL;DR

This paper introduces a new infinite number, λ, to address inconsistencies in Cantor's infinity within calculus, providing a precise way to quantify infinity and resolving several classical mathematical paradoxes.

Contribution

It proposes a novel infinite number λ with defined arithmetic properties, challenging traditional set theory and resolving longstanding paradoxes.

Findings

λ allows precise quantification of infinity

No set is uncountable under λ

Resolves Riemann Rearrangement and Banach-Tarski paradoxes

Abstract

From 1873 to 1897, Georg Cantor worked on developing set theory, and despite a strong initial resistance, it rapidly became accepted as the foundation of mathematics. In this work, however, we'll demonstrate that Cantor's use of infinity is inconsistent with Calculus. Since both the cardinal and the ordinal numbers are to blame, we'll introduce the new infinite number $\lambda$ to remedy this situation. By developing its arithmetical properties for each basic operation, we'll show that infinity can be quantified with precision, and that no set, however large, is uncountable. Moreover, by working with the number $\lambda$, we'll resolve several long-standing paradoxes in mathematics, such as the Riemann Rearrangement Theorem and the Banach-Tarski paradox.