Diagonalization is a technique first used by Georg Cantor, a German mathematician. He used it to show that the real numbers can not be put into one-to-one correspondence with the natural numbers, thereby demonstrating the real numbers are not countable. This method can also be applied in other contexts, to show that two sets can't have a correspondence. For example, it can be used to show that no set can be in 1-1 correspondence with the set of all of its subsets.

Proof of the non-countability of real numbers

First, we create a 1-1 correspondence between the entire real line \mathbb{R}\, and the open interval (0, 1)\,. This function:

y = \frac{\tan^{-1}(x)}{\pi} + \frac{1}{2}

maps the entire real line to the open interval (0, 1)\,. Its inverse:

x = \tan(\pi(y - 1/2))\,

maps the open interval to the entire real line.

This means that the real numbers are in 1-1 correspondence with the natural numbers if and only if the open interval (0, 1)\, is in correspondence.

We will now use proof by contradiction to show that this open interval has no such correspondence, and thus it, and the real line as a whole, are uncountable.

Assume the numbers in this open interval are in a 1-1 correspondence with the natural numbers. Then we can make an (infinite) sequential list of them, like this:






Where a_{ij}\in\{0,1,2,3,4,5,6,7,8,9\}

Construct the number,

a=0.a_{1}a_{2}a_{3}a_{4}\dots, where

a_{i}=1 when a_{ii}\neq1 and a_{i}=2 when a_{ii}=1.

Therefore a is not in the list, so we have a contradiction and our assumption is false, the numbers in [0,1] are not countable. Therefore \mathbb{R} is uncountable.[1]

Diagonalization and the existence of God

Some have cited diagonalization as a formal challenge to Saint Anselm's ontological argument for the existence of God. In summary, Anselm argued that there must be a greatest idea and what could be greater than God? Therefore God exists.[2]

However, diagonalization argues that no greatest idea can exist: quite bluntly, God is infinite, therefore He can be diagonalized to produce an even greater infinite.[3]


  1. A. N. Kolmogorov, Introductory Real Analysis. ISBN 978-0486612263.
  3. Topo-philosophies: Plato's Diagonals, Hegel's Spirals, and Irigaray's Multifolds, Arkady Plotnitsky. In After Poststructuralism: Writing the Intellectual History of Theory Tilottama Rajan, Michael James.
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