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Topic: Dispute over Infinity Divides Mathematicians
Replies: 6   Last Post: Dec 8, 2013 7:11 AM

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Sam Wormley

Posts: 607
Registered: 12/18/09
Dispute over Infinity Divides Mathematicians
Posted: Dec 5, 2013 5:42 PM
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Dispute over Infinity Divides Mathematicians

> From Quanta
> Original story here
> 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. 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 second-smallest ? 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.?

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