Abstract

The paper proves that the "Champernowne constant" (0.1234567891011121314 … where all natural numbers are consecutive digits of a decimal fraction) is a rational number, but only strictly within (Peano) arithmetic due to the axiom of induction. Combined with the previous well known results proved to be a transcendent real number in both (Peano) arithmetic & (ZFC) set theory, it is demonstrated to be a "Gödelian real number", rational in (Peano) arithmetic, but irrational (transcendent) in (Peano) arithmetic & (ZFC) set theory: for realizing the Gödel (1931) dichotomy about the relation of arithmetic to set theory ("either incompleteness or contradiction"). The method used initially for the Champernowne constant can be generalized to any computable real number. Then, the (uncountable) set of all noncomputable reals can be defined by their representability by more than one computable real. One can speak of the "relativity of rationality / irrationality" (or computability / non-computability) in Skolem's (1922) manner or even it to be continued to the "relativity of P / NP" in the "P vs NP problem: a conjecture visualized by the computability or non-computability of reals.