在无限的尽头

摘要：

从1873年到1897年，康托致力于发展集合论，尽管最初的阻力很大，但它很快就被接受为数学的数学基础。然而，在这项工作中，我们将证明康托对无穷大的使用与微积分不一致。由于基数和序数都是罪魁祸首，我们将引入新的无限数$λ$来弥补这种情况。通过对每个基本运算的算术性质的发展，我们将证明无穷大可以被精确量化，并且没有一个集合，无论多么大，是不可数的。此外，通过使用数字$λ$，我们将解决数学中几个长期存在的悖论，如黎曼重排定理和巴拿赫-塔斯基悖论。

英文摘要：

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 $λ$ 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 $λ$, we'll resolve several long-standing paradoxes in mathematics, such as the Riemann Rearrangement Theorem and the Banach-Tarski paradox.

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