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Hugh Woodin: On the necessity of infinite truth


Is there in the final analysis a conception of the transfinite which is as clear and unambiguous as our conception of the integers? The ubiquity of independence results in Set Theory during the 40 years since Cohen's proof of the (formal) unsolvability of the Continuum Hypothesis has severely challenged any hope that such a view might emerge and indeed it has cast some doubt on the possibility of any meaningful conception of the transfinite. I shall discuss some recent developments in the context of this question and argue that not only is there no evidence against an unambiguous view of the transfinite (a view which answers the question of the Continuum Hypothesis as well as all of the other known unsolvable questions of Set Theory), there is strong evidence that there is such a view. Moreover this could evolve exactly as Gödel suggested, through the understanding of Strong Axioms of Infinity.