Episode #488 from 2:17
Infinity & paradoxes
Some infinities are bigger than others. This idea from Cantor at the end of the 19th century, I think it's fair to say, broke mathematics before rebuilding it. I also read that this was a devastating and transformative discovery for several reasons. So one, it created a theological crisis, because infinity is associated with God, how could there be multiple infinities? Also, Cantor was deeply religious himself. Second, there's a kind of mathematical civil war. The leading German mathematician, Kronecker, called Cantor a corrupter of youth and tried to block his career. Third, many fascinating paradoxes emerged from this, like Russell's paradox, about the set of all sets that don't contain themselves, and those threatened to make all of mathematics inconsistent. Finally, on the psychological and personal side, Cantor's own breakdown. He literally went mad, spending his final years in and out of sanatoriums, obsessed with proving the continuum hypothesis. So laying that all out on the table, can you explain the idea of infinity, that some infinities are larger than others, and why was this so transformative to mathematics?
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Why this moment matters
Some infinities are bigger than others. This idea from Cantor at the end of the 19th century, I think it's fair to say, broke mathematics before rebuilding it. I also read that this was a devastating and transformative discovery for several reasons. So one, it created a theological crisis, because infinity is associated with God, how could there be multiple infinities? Also, Cantor was deeply religious himself. Second, there's a kind of mathematical civil war. The leading German mathematician, Kronecker, called Cantor a corrupter of youth and tried to block his career. Third, many fascinating paradoxes emerged from this, like Russell's paradox, about the set of all sets that don't contain themselves, and those threatened to make all of mathematics inconsistent. Finally, on the psychological and personal side, Cantor's own breakdown. He literally went mad, spending his final years in and out of sanatoriums, obsessed with proving the continuum hypothesis. So laying that all out on the table, can you explain the idea of infinity, that some infinities are larger than others, and why was this so transformative to mathematics?
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