Prove that the set T of transcendental numbers (numbers that do not satisfy some polynomial equation of positive degree with rational coefficients) has the power of the continuum, i.e. has cardinality c. Here's what I have: Since T is uncountable, then |T|>alephnull . Also, since T is a subset of R , then |T| not> c . Thus, by the Continuum Hypothesis, we must have |T|=c . But is there a way to get a proof without using the Continuum Hypothesis, by showing directly a bijection between T and R?