Projects of Academic Interests

1. Introduction The nature and hierarchy of infinite sets have fascinated mathematicians since Georg Cantor’s foundational work in the late 19th century. Cantor’s revolutionary theory introduced the concept of different sizes, or cardinalities, of infinity - notably distinguishing the countably infinite set of natural numbers, denoted ℕ, from the uncountably infinite set of real numbers, ℝ. This distinction rests on the fundamental insight that while ℕ can be put into a one-to-one correspondence with any of its infinite subsets, no such correspondence can exist between ℕ and ℝ. Cantor’s diagonalization argument formally proves the uncountability of ℝ, establishing a strict hierarchy of infinities with ℝ being strictly “larger” than ℕ. However, classical set theory inherently assumes that every element of ℕ is a finite natural number, possessing a finite representation - for example, a finite string of digits in base 10. This assumption aligns well with our standard, intuitive unders...