FOM: Reply to Tragesser on CH and elementary proofs

[jshipman@bloomberg.net]

CH doesn't depend on the apparatus of Set Theory, at least not very far up.
Therefor a proof from some principle "about sets" would not be "elementary" if
this principle involved powerful abstract axioms of "pure set theory" that were
not specifically "about" real numbers and sets of real numbers (especially if
they required infinitely many uncountable cardinals).  Not that such principles
are necessarily problematic, they just go far beyond the statement of the
problem and hence their use on CH cannot constitute an "elementary" argument.
Having said this, I must admit that it was premature to say the independence of
CH (from ZFC+large cardinals) precludes an "elementary" proof.  Further progress
in our understanding of reals and sets of reals may lead us to positively
reevaluate the plausibility of some of the many axioms which do settle CH one
way or the other (e.g. Martin's Axiom, V=L, existence of a real-valued measure
on the continuum, the strong "Fubini" principles in my TAMS paper, etc.), in a
context that is "elementary" relative to CH.  By the way, see Mathias's cited
paper on analysis which has a good perspective on these issues. -- Joe Shipman