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.