The 20 th century choice for the axioms 1 of Set Theory are the Zermelo-Frankel axioms together with the Axiom of Choice, these are the ZFC axioms. This particular choice has led to a 21 th century problem: The ZFC Delemma: Many of the fundamental questions of Set Theory are formally unsolvable from the ZFC axioms. Perhaps the most famous example is given by the problem of the Continuum
... [Show full abstract] Hypothesis: Suppose X is an infinite set of real numbers, must it be the case that either X is countable or that the set X has cardinality equal to the cardi-nality of the set of all real numbers? One interpretation of this development is: Skeptic's Attack: The Continuum Hypothesis is neither true nor false because the entire conception of the universe of sets is a com-plete fiction. Further, all the theorems of Set Theory are merely finitistic truths, a reflection of the mathematician and not of any genuine mathematical "reality". Here and in what follows, the "Skeptic" simply refers to the meta-mathematical position which denies any genuine meaning to a conception of uncountable sets. The counter-view is that of the "Set Theorist": 1 This paper is dedicated to the memory of Paul J. Cohen.