[FOM] (no subject)

[Adam Epstein]

Hello, all. I'm a complex analyst with recreational interests in FOM.


Here's someting someone once told me, and which I've finally
reconstructed.


Consider the following assertion:

* There exists a non-Borel subset of the real line which can be expressed
  as the union of some linearly ordered (with respect to set inclusion) 
  family of countable sets.




Assertion * is in fact true, with a very quick proof in ZFC (I haven't
checked whether Choice is truly necessary, but never mind that).

The odd thing is that the proof takes the following form:

 "If the Continuum Hypothesis holds then ... whence *. On the other hand,
  if the Continuum Hypothesis does not hold, then ...' whence *."

This looks contrived, but it isn't. The proof exhibits two
possible ways of trying to construct such a set, and notes that the first
approach works iff CH holds, and that the second works iff CH fails.


Most regular subscribers are probably familiar with this.



The same person also told me that there is a theorem in number
theory whose proof goes as follows:

"If the Riemann Hypothesis then ... while if the Riemann Hypothesis fails
then ...' "


Can anyone supply details about this one? 




Adam Epstein