To Settle Infinity Dispute, a New Law of Logic

To Settle Infinity Dispute, a New Law of Logic
By: Natalie Wolchover
November 26, 2013
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In the course of exploring their universe, mathematicians have occasionally stumbled across holes: statements that can be neither proved nor refuted with the nine axioms, collectively called “ZFC,” that serve as the fundamental laws of mathematics.
Most mathematicians simply ignore the holes, which lie in abstract realms with few practical or scientific ramifications.
But for the stewards of math’s logical underpinnings, their presence raises concerns about the foundations of the entire enterprise.

“How can I stay in any field and continue to prove theorems if the fundamental notions I’m using are problematic?” asks Peter Koellner, a professor of philosophy at Harvard University who specializes in mathematical logic.

Chief among the holes is the continuum hypothesis, a 140-year-old statement about the possible sizes of infinity.
As incomprehensible as it may seem, endlessness comes in many measures: For example, there are more points on the number line, collectively called the “continuum,” than there are counting numbers.