The Mazur swindle

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We start off by considering the following infinite sum:

  \[ S=1-1+1-1+1-1+1-1…  \]    

It’s called Grandi’s series, after the Italian mathematician, philosopher, and priest Guido Grandi (1671-1742).

If we group its terms this way

  \[  S=(1-1)+(1-1)+(1-1)+(1-1)+... \]    

we are led to believe that $S$ should be equal to 0 because each individual bracket is equal to 0. Nothing stops us, however, from grouping its terms in a different way, for example

  \[  S=1+(-1+1)+(-1+1)+(-1+1)+... \]    

which suggests that $S$ should be equal to 1, as all the brackets that come after the initial 1 equal 0.

This suggests that $S$ is equal to both 0 and 1, in other words, that 0=1! Since this isn't the case, something must be wrong with our reasoning. The problem lies with the fact that infinite sums of numbers can't be treated in the same way as finite sums — see here to find out more.

There are settings, however, where an argument akin to the one above does work. This how Mazur explained his swindle to us:

Suppose you have (what you think of as infinitely many different) mathematical objects labelled by numbers

  \[ [-\infty ], .., [-2], [-1], [0], [+1], [+2],..., [+\infty ] \]    

(Yes: even $[\pm {\infty }]$)

Moreover, suppose that you could combine them like train cars, following the naive addition of the numbers in their labels, and that these combined objects are also objects in your collection. For example, $[1]+[1]=[2]. $

Also, suppose you could even get infinitely long trains, so that, for example:

  \[ [0] = [0] + [0] + [0] + [0] + … \]    

Then, if the kind of manipulations we did to the sum S above are allowed in the setting we are looking at, we can prove that

  \[ [0]=[+1], \]    

which also means that

  \[ [+2]=[1]+[1]=[0]+[0]=[0], \]    
  \[ [+3]=[1]+[1]+[1]=[0]+[0]+[0]=[0],  \]    

and so on.

This implies that all the objects we were looking at are actually the same. There aren’t infinitely many objects at all, but only one!

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