At the End of Infinity

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.