Repercussions from the Richard Paradox: Math Rules


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Bertrand Russell: Image by Kevan Davis

Richard’s Paradox demonstrated that a simple rule to define a set of numbers may lead to a paradox.

Predecessors to the Richard Paradox

In 1905, French mathematician Jules Richard shifted the focus from certain earlier mathematical paradoxes by showing that the definitions themselves might be at fault.

In the very early 1900s, paradoxes in the mathematics of set theory had already been found and published. In 1903, for example, Bertrand Russell’s “Principles of Mathematics” discussed paradoxes that dealt with the ordering of infinite ordinal numbers, or with defining a self-containing set with a condition that both includes and excludes itself.

An English Version of Richard’s Paradox for Real Numbers

Richard built his main paradox in this manner.

Create a set of Real numbers, ‘E’, with the following definition. “E = {e}, where 0 < e < 1 in the Real numbers” and each ‘e’ is further defined by a finite number of words in the English language. Order this set by the number of letters in the string and, within each length, alphabetically.

To create the paradox, form a new Real number ‘f’ in the range “0 < f < 1” and based on the existing set ‘E’ as follows. For each number in ‘E’, let ‘d[n, n]’ be the nth digit of the nth number in ‘E’. If the current ‘d[n, n]’ is 8 or 9, change it to 1; otherwise add one to ‘d[n, n]’. Use the new value of ‘d[n, n]’ as the nth digit of the new number ‘f’.

A Cantor Diagonal: Image by Mike DeHaan

This uses the famous “Cantor diagonal” method to ensure that the new number ‘f’ is not already in set ‘E’; it differs from the nth number in ‘E’ at the nth digit.

However, the definition itself has a finite number of words, so the number ‘f’ should have been included in ‘E’ already.

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