Richard's Paradox

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Paradox

Certain expressions of natural language define real numbers unambiguously, while other expressions of natural language do not.

For example:

The real number whose integer part is $17$ and whose $n$th decimal place is $0$ if $n$ is even and $1$ if $n$ is odd

defines the real number $17.1010101 \ldots = 1693/99$.

However, the phrase:

the capital of France

does not define a real number.

Neither does the phrase:

the smallest positive integer not definable in under sixty letters

because the expression does not define a number unambiguously (see Berry's Paradox).


Thus there exists an infinite list of English phrases of finite length that define real numbers unambiguously.

Let this infinite list of phrases be ordered linearly:

$(1): \quad$ by increasing length
$(2): \quad$ then order all phrases of equal length in lexicographic order, for example, using the ASCII code, such that the phrases can only contain codes $32$ to $126$).

This ordering is declared to be the canonical form of the sequence of the real numbers.

Thus we have an infinite sequence of the corresponding real numbers:

$r_1, r_2, \ldots$

Now define a new real number $r$ as follows.

The integer part of $r$ is $0$.
The $n$th decimal place of $r$ is:
$1$ if the $n$th decimal place of $r_n$ is not $1$
$2$ if the $n$th decimal place of $r_n$ is $1$.


The preceding text on this page is an expression in the English language that unambiguously defines a real number $r$.

But because of the way $r$ was constructed, it cannot equal any of the $r_n$.

Thus, despite being unambiguously defined, it is at the same time undefinable.

This contradiction is known as Richard's paradox.


Resolution

This is an antinomy.

The definition above of $r$ as refers to the definability itself, in natural language, of a real number.

If it were possible to determine which expressions in English actually do define a real number, and which do not, then the paradox would go through.

Thus the resolution of Richard's paradox is that there exists no way to unambiguously determine exactly which English sentences are definitions of real numbers.

That is, there exists no way to describe in a finite number of words how to tell whether an arbitrary English expression is or is not a definition of a real number.

Hence the root of the contradiction is the assumption that it is possible to unambiguously define every real number using a finite length expression in natural language.

$\blacksquare$


Also see


Source of Name

This entry was named for Jules Antoine Richard.


Sources

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