This paper constructs a series of transcendental subsets of real numbers by creating a construction method based on infinite sets of transcendental numbers. It also proposes the concept that infinite sets can be divided into one-dimensional, two-dimensional, and multi-dimensional sets according to the number of degrees of freedom required for the expansion of their elements. For example, the set of natural numbers, the set of even numbers,and the set of odd numbers belong to one-dimensional infinite sets, while rational numbers belong to two-dimensional infinite sets, and so on. When the number of degrees of freedom is infinite, it becomes an infinite-dimensional set, and several infinite-dimensional sets can constitute a hyperdimensional set.This paper proves that the set of real numbers belongs to the hyperdimensional set, and itscardinality is far greater than the infinite power set of the cardinality of the set of naturalnumbers; between the set of natural numbers and the set of real numbers, there exist infinitely many number sets. The result of this paper disproves the continuum hypothesis.