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Richard's Paradox

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This one is cool!


The paradox begins with the observation that certain expressions in English unambiguously define real numbers, while other expressions in English do not. For example, "The real number whose integer part is 17 and whose nth decimal place is 0 if n is even and 1 if n is odd" defines the real number 17.1010101..., while the phrase "London is in England" does not define a real number.

Thus there is an infinite list of English phrases (where each phrase is of finite length, but lengths vary in the list) that unambiguously define real numbers; arrange this list by length and thendictionary order, so that the ordering is canonical. This yields an infinite list of the corresponding real numbers: r1, r2, ... . Now define a new real number r as follows. The integer part of r is 0, thenth decimal place of r is 1 if the nth decimal place of rn is not 1, and the nth decimal place of r is 2 if the nth decimal place of rn is 1.

The preceding two paragraphs are an expression in English which unambiguously defines a real number r. Thus r must be one of the numbers rn. However, r was constructed so that it cannot equal any of the rn. This is the paradoxical contradiction.

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It is cool, indeed.

It is of the same type as Both involve English phrases that describe numbers.

One form of Berry's Paradox asks:

What is the smallest number not specifiable using fewer than twenty-three syllables?

Let's say that number is 1,777,777.

One mil-ion sev-en hun-dred se-ven-ty sev-en thou-sand sev-en hun-dred sev-en-ty se-ven -- that's 23 syllables.

And there doesn't seem to be a smaller number that requires that many.

But that number is specified by the phrase in red above. And that phrase contains fewer than 23 syllables.

So 1,777.777 is specifiable by fewer than twenty-three syllables.

The usual work-around for paradoxes of this type is to segregate statements that use verbal descriptions from statements that are numerical.

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