Berry’s paradox

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

Define $n_0$ to be: the least positive integer that is notcurious.

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

The paradox above is called Berry’s Paradox. Berry’sParadox suggests the advantage of separating the language used to formulatemathematical statements or theory (the object language) from the languageused 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 ourmetalanguage, which is English here;

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

4. define $n_0$ to be the least positive integer that is notcurious.

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

Commonly, ${𝐄}^{\ast }$ is the first order logic. However, it isnot often necessarily the case, and ${𝐄}^{\ast }$ above couldhave been English anyway. We only need to formally distinguish thestatements formulating the mathematics from the statementsdiscussing those formulations, i.e., declaring the two classes ofstatements to be disjunct, perhaps by italicizing the former.Nevertheless, such approach evidently involves more work and isunderstandably hard to follow.