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How many real numbers exist? New proof moves closer to an answer (2021)
For 50 years, mathematicians have believed that the total number of real numbers is unknowable. A new proof suggests otherwise.
In October 2018, David Asperó was on holiday in Italy, gazing out a car window as his girlfriend drove them to their bed-and-breakfast, when it came to him: the missing step of what’s now a landmark new proof about the sizes of infinity. For all the expansiveness of Martin’s maximum, in order to simultaneously permit all those products of forcing (while satisfying that constancy condition), the size of the continuum jumps only to a conservative $latex\boldsymbol{\aleph}_{2}$— one cardinal number more than the minimum possible value. Ever since Gödel and Cohen established the independence of the continuum hypothesis from ZFC, infinite math has been a choose-your-own-adventure story in which set theorists can force the number of reals up to any level — $latex\boldsymbol{\aleph}_{35}$, or $latex\boldsymbol{\aleph}_{1000}$, say — and explore the consequences.
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