ON THE TRANSFINITE LENGTH OF THE SEQUENCE OF DIGITS IN THE EXPANSION OF A TRANSCENDENTAL NUMBER AFTER THE DECIMAL POINT

We prove that among the transcendental numbers there are different transfinite cardinalities in the total of digits that compose them in their expansion after the decimal point- if we assume as consistent Cantor's diagonal process. Some transcendentals can be made up of a total of digits ℵ0 after the decimal point, while another by ℵ1 or ℵ2, or ℵ figures. So, we proceed to mathematically construct such cardinalities. Which in turn forces us to also deduce the existence, within the class R of the real numbers, of Rα infinite subsets of these numbers with ℵα cardinalities different from each other, and we likewise construct them. We find the logical inadmissibility of identifying, as inadvertently in the current theory, a strictly proper subset R1 of R with R itself. And we obtain for R a theorem that is comparable to Cantor's Theorem about power sets. Except that this new theorem leads us to resize the currently accepted cardinalities of the power sets Pα. Establishing that none of them reach the number of elements of R. And for every α the cardinality of each Pα is equal to the cardinality of each Rα, that is, in both cases, אα. Consequently, the real numbers r ∈ R have a greater wealth than has so far been conceptualized in Set Theory and Mathematical Analysis. Based on all this, it is also proved that we need to reformulate the Continuum Hypothesis, assuming Cantor's diagonal method is consistent