With equal looseness one can say that "the number of real numbers is -1". (More precisely, this is the combinatorial Euler characteristic of R; I think Schanuel was the first to argue for the idea that we should consider Euler characteristic as a kind of cardinality.) Here's the elevator-pitch for chi(R) = -1: The set of real numbers is homeomorphic to an open interval. Every open interval can be written as the disjoint union of an open interval, a point, and an open interval. So if we let x denote the "size" of every open interval, and size is finitely additive, we must have x = x + 1 + x, whose only solution is x = -1. Jim Jim Propp On Fri, Apr 14, 2017 at 8:06 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
As a companion piece to the well-known(-ish!)
1 + 2 + 3 + ... = -1/12
https://terrytao.wordpress.com/2010/04/10/the-euler- maclaurin-formula-bernoulli-numbers-the-zeta-function-and- real-variable-analytic-continuation/
the guru has recently given us to understand that (loosely speaking)
"the number of finite sets is e "
https://terrytao.wordpress.com/2017/04/13/counting- objects-up-to-isomorphism-groupoid-cardinality/
--- should small children be permitted to overhear such seditious material, one wonders?
WFL
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