The Structure and Quantity of Real Numbers V.2

This work presents intuitions, concepts, and important theorems concerning the foundations ofmathematics which, in principle, refute Cantor’s fundamental theorems regarding the uncountabilityof the real numbers and his diagonal argument. The results demonstrate that rational and irrationalnumbers are distributed in a highly regular and interleaved pattern along the real number line, leadingto the conclusion that the sizes of these two sets are exactly the same and that irrational numbers aremerely “unattainable theoretical limits of increasingly precise approximations by rational numbers”.Should the presented results prove correct, not only would Cantor’s theorems on the uncountabilityof the real numbers be refuted, but so too would other pivotal results of modern mathematics. Theseinclude the hierarchy of infinities in set theory, the distinction between decidable and undecidablelanguages in computability theory, as well as other significant findings in logic and mathematicalanalysis. Under this framework, the Continuum Hypothesis would hold vacuously, as its antecedentcondition would be false. Interpretations and Refutations of Cantor's Diagonal Argument are provided.