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Although Creative Objectivism retains much of the power of contemporary mathematics in defining sets of integers it does not allow quantification over all sets. Some statements like the Continuum Hypothesis are not considered meaningful in any absolute sense. However the set of all reals definable in a particular formal system is a meaningful concept. The Continuum Hypothesis may be true false or undecidable within a formal system. Formal systems are perfectly valid objects of study in Creative Objectivism. After all they are TMs for enumerating theorems.
I doubt that any mathematics of practical value cannot be incorporated into Creative Objectivism. The continuum as the limit of finite processes is compatible. The logic of the calculus in developing derivatives and integrals does not present a problem. What would change is how the foundations of those disciplines are formulated.