The key example is the proof of the independence of the Continuum Hypothesis, by demonstrating that there is a consistent situation in which this hypothesis fails.
The faith, or fidelity, of the subject is based on the Axiom of Choice as it allows, in the model of forcing the independence of the Continuum Hypothesis, the differentiation of the non-constructible sets from any given constructible or non-constructible set on the basis of a finite amount of information.
Paul Cohen, Set Theory and the Continuum Hypothesis, New York, W A.
Put together, those two results indicate that it's impossible either to prove or to disprove the continuum hypothesis using the standard axioms.
Cohen's demonstration that the continuum hypothesis could be neither proved nor disproved "caused a foundational crisis," Woodin says.
To formalists, it makes no sense to talk about whether the continuum hypothesis is true or false.
To Platonists, the continuum hypothesis feels like a concrete statement that should be true or false.
INFINITE ELEGANCE In the decades that followed Cohen's 1963 result, mathematicians trying to settle the continuum hypothesis ran into a roadblock: While some people proposed new axioms indicating the continuum hypothesis was true, others proposed what seemed like equally good axioms indicating the it false, Woodin says.
A good axiom, he felt, should help mathematicians settle not only the continuum hypothesis but also many other questions about Cantor's hierarchy of infinite sets.
However, Woodin suspected a compromise is possible: There might be axioms that answer all questions up to the level of the hierarchy that the continuum hypothesis concerns--the realm of the smallest uncountably infinite sets.
In a book-length mathematical argument that has been percolating through the set theory community for the last few years, Woodin has proved--apart from one missing piece that must still be filled in--that elegant axioms do exist and, crucially, that every elegant axiom would make the continuum hypothesis false.
If there's a simple solution to the continuum hypothesis, it must be that it is false," Woodin says.