Some amazing math

There has been a video on the net claiming that the infinite sum 1+2+3+4... = -1/12.
Initially I thought this was nonsense.
But just read an amazing blog post (in Hebrew) explaining the matter.

I thought it was nonsense because I thought we already know that 1+2+3+4.. is equal to infinity,
and infinity is clearly not equal to -1/12.

But it is not true that we know this.
What we know is that when embedding the rational numbers into the real numbers,
and taking distances between them as real numbers - for example the distance between 4 and 1 is 3,
then the sum 1+2+3+4+.. is just not defined! It does not `get close' to anything.
(For an example of `getting close', look at 1+1/2+1/4+1/8+1/16... we see that this sequence `gets close' to 2, and indeed you can prove in mathematics it is equal to 2.)

What they prove is that in any embedding of the rational numbers to a metric space such that
1+2+3+4+.. is defined, then it would have to equal -1/12.

A simpler example given in the post:
Suppose S= 1+2 +4+8 +16... (all powers of 2)
suppose we are in a world where S is really well defined.
Then we can write

1+2S = 1+ 2*(1+2+4+8+16..) = 1+2 +4+8 +16+32.. = S

So 1+2S =S and S = -1

The amazing thing is that there is  a mathematical world called the 2-adic numbers
where this sequence S indeed equals -1.