Richard’s (logical) paradox

In 1905, Jules Richard, a French logician, invented a very curious paradox. In the English language, some sentences define positive integers and others do not. For example, “The year of the Declaration of Independence” defines 1776, whereas “The historical significance of the Declaration of Independence” does not define a number. So what about this sentence: “The smallest number that cannot be defined by a sentence in the English language containing fewer than 20 words.” Observe that whatever this number may be, we have just defined it using a sentence in the English language containing only 19 words. Oops.

A plausible way out is to say that the proposed sentence does not actually define a specific number. However, it ought to. The English language contains a finite number of words, so the number of sentences with fewer than 20 words is itself finite. Of course, many of  these sentences make no  sense, and many of those do make sense don’t define a positive integer — but, that just means that we have fewer sentences to consider. Between them, they define a finite set of positive integers, and it is a standard theorem of mathematics that in such circumstances there is a unique smallest positive integer that is not in the set. So on the face of it, the sentence does not define a specific positive integer.

But logically, it can’t.

Possible ambiguities of definition such as “A number which when multiplied by zero gives zero” don’t let us off the logical hook. If a sentence is ambiguous, then we must rule it out, because an ambiguous sentence doesn’t define anything. Is the troublesome sentence ambiguous, then? Uniqueness is not the issue:  there can’t be two distinct smallest-numbers-not-definable- (etc.), because one must be smaller than the other.

One possible escape route involves how we decide which sentences do or do not define a positive integer. For instance, if we go through them in some kind of order, excluding bad ones in turn, then the sentences that survive depend on the order in which they are considered. Suppose that two consecutive sentences are:

  1. The number in the next valid sentence plus one.
  2. The number in the previous valid sentence plus two.

These sentences cannot both be valid — they would then contradict each other. But, once we have excluded one of them, the other one is valid, because it now refers to a different sentence altogether.

Forbidding this type of sentence puts us on a slippery slope, with more and more sentences being excluded for various reasons. All of which strongly suggests that the alleged sentence does not, in fact, define a specific number — even though it seems to

With thanks to Prof Ian Stewart for this gem from his Cabinet of Mathematical Curiosities,

Nalin Pithwa

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