Berry’s paradox

We begin with calling a positive integer curious if it can be defined in the English language using no more than 1234 words. Since there are finitely many English words, we see that there are only finitely many curious positive integers.

Define $n_0$ to be: the least positive integer that is not curious.

$n_0$ has just been described in $8\le 1234$ words, therefore, it is curious after all!

The paradox above is called Berry’s Paradox. Berry’s Paradox suggests the advantage of separating the language used to formulate mathematical statements or theory (the object language) from the language used to discuss those statements or the theory (the metalanguage).

Berry’s Paradox can be avoided by the following reformulation:

1. fix the object language, called ${𝐄}^{\ast }$;

2. declare ${𝐄}^{\ast }$ to be different from our metalanguage, which is English here;

3. define a curious positive integer to be one which can be described in ${𝐄}^{\ast }$ using no more than 1234 words of the language;

4. define $n_0$ to be the least positive integer that is not curious.

In the reformulation, we have defined curious positive integers and $n_0$ in English, which is not ${𝐄}^{\ast }$. Thus, we have no basis to conclude that $n_0$ is curious, hence no contradiction arises.

Commonly, ${𝐄}^{\ast }$ is the first order logic. However, it is not often necessarily the case, and ${𝐄}^{\ast }$ above could have been English anyway. We only need to formally distinguish the statements formulating the mathematics from the statements discussing those formulations, i.e., declaring the two classes of statements to be disjunct, perhaps by italicizing the former. Nevertheless, such approach evidently involves more work and is understandably hard to follow.

Title	Berry’s paradox
Canonical name	BerrysParadox
Date of creation	2013-03-22 18:06:03
Last modified on	2013-03-22 18:06:03
Owner	yesitis (13730)
Last modified by	yesitis (13730)
Numerical id	7
Author	yesitis (13730)
Entry type	Topic
Classification	msc 03B99