Paper rejected twice, once because it proved two true statements imply each other, once because it proved two false statements imply each other?

Question

I remember a while back finding an anecdote about a paper in set theory from the early 20th century establishing an equivalence between two concepts (I think, but I’m not sure, that it was the axiom of choice and the well-ordering principle). The story goes that the paper was rejected from one journal with the comment that it was not interesting that two true statements imply one another, and rejected from another journal on the grounds that it was not interesting that two false statements imply one another.

I’m having trouble locating which paper this was. Google isn’t turning anything up with the search queries I tried, and ChatGPT hallucinated a nonexistent paper and an invalid Wikipedia link.

Any ideas what paper this is?

Answer 1

You are talking about Alfred Tarski's famous result:

The axiom of choice is equivalent to the statement "For every infinite set $X$, there is a bijection $f\colon X\to X^2$".

The paper was submitted to Comptes rendus de l'Académie des Sciences, originally Lebesgue had rejected the paper since no one would be interested in the equivalence between two false statements; but, as the story goes, suggested Fréchet as a more sympathetic editor. Unfortunately, Fréchet rejected it since the equivalence between two true statements would not be of any particular interest to the readers.

Tarski never submitted to Comptes again after that.

(See also on Wikipedia.)