Cantor demonstrated that the set of algebraic numbers is countable, while the set of real numbers is uncountable, leading to the conclusion that transcendental numbers must also be uncountable. This indicates that almost all real numbers are transcendental, as the countable set of algebraic numbers is negligible in comparison. Critics of Cantor's work questioned the concept of different sizes of infinity, which contributed to the controversy surrounding his findings. The discussion also touches on the implications of set theory in understanding the nature of numbers, particularly the distinction between countable and uncountable sets. Overall, Cantor's framework established a foundational understanding of the existence and prevalence of transcendental numbers.