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Dispute over Infinity Divides Mathematicians
Posted:
Dec 5, 2013 5:42 PM


Dispute over Infinity Divides Mathematicians > http://www.scientificamerican.com/article.cfm?id=infinitylogiclaw
> From Quanta > Original story here > https://www.simonsfoundation.org/quanta/20131126tosettleinfinityquestionanewlawoflogic/ > > 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 140yearold > 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. Beyond the continuum > lie larger infinities still ? an interminable progression of evermore > enormous, yet all endless, entities. The continuum hypothesis asserts > that there is no infinity between the smallest kind ? the set of > counting numbers ? and what it asserts is the secondsmallest ? the > continuum. It ?must be either true or false,? the mathematical > logician Kurt Gödel wrote in 1947, ?and its undecidability from the > axioms as known today can only mean that these axioms do not contain > a complete description of reality.?



