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Proof that 0 is infinity (lemmy.nz)

submitted 3 weeks ago* (last edited 3 weeks ago) by xiao@sh.itjust.works to c/Math_history@sh.itjust.works

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1 = 1

1 = (-1) + 2

1 = (-2) + 3

1 = (-3) + 4

1 = (-4) + 5 ... And so on...

By adding term by term to all these equations, we get: 1 + 1 + 1 + 1 + 1 + 1 + ... = 1 + (-1) + 2 + (-2) + 3 + (-3) + 4 + (-4) + 5 + ...

In the expression on the right, all the terms cancel out in pairs, giving: 1 + 1 + 1 + 1 + 1 + 1 + ... = 0

The expression on the left, consisting of an infinite sum of terms equal to 1, tends to infinity. Thus, 0 is equal to infinity.

And yet 0 is not equal to infinity. So where is the mistake?

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[–] SwingingTheLamp@piefed.zip 9 points 3 weeks ago

My thoughts...

Gonna have to say that, like in Cantor's Theorem, the fault lies in the handwaving phrase, "and so on." It's a mental shortcut that leads us astray. If it actually means going on to infinity, then the right side of the equation isn't zero. Rather, it never converges on a sum— for every negative number, there will always be a next term, a positive integer with greater magnitude.