[FOM] Platonism and Formalism - Message from Joe Shipman
Karlis Podnieks
Karlis.Podnieks at mii.lu.lv
Fri Oct 10 01:50:28 EDT 2003
This is a personal communication. Posted to FOM with kind permission of Joe
Shipman.
Best wishes,
Karlis.Podnieks at mii.lu.lv
Dear Professor Podnieks,
Thanks for reading my old posts with such attention!
Along the same lines, I have also speculated on FOM that mathematics could
have developed with RVM (there is a countably additive real-valued measure
on the continuum) which does not contradict AC but is incomparably stronger
than ZFC and settles CH. The "Continuum Problem" would not be seen as a
problem at all. Instead of Cohen's work, it is Solovay's that would be
considered of extreme metamathematical importance, because it would show the
independence of RVM just as Cohen showed the independence of AC.
I am convinced that if general relativity had been developed BEFORE the
atomic theory instead of after, mathematicians would when confronted with
the Banach-Tarski paradox have chosen to reject the assumption that
Euclidean space is homogeneous and isotropic rather than rejecting AC or
rejecting the concept of a measure on ALL subsets of the continuum -- they
would have concluded that you cannot assume rotational symmetry, but kept
the primordial intuition that every set of points has a "mass". And nothing
subsequent would shake that MATHEMATICAL conviction -- even when the atomic
theory was discovered, so that it was no longer plausible that physical
space was infinitely divisible, RVM would have been found so fruitful and
powerful (among other things proving Con(ZFC)) that there would have been no
reason to abandon it!
No set theorist has ever given a satisfactory reason, to my mind, that RVM
should be considered false; it's just unfashionable. There are hundreds of
mathematically interesting statements (about sets contained in some
finitely iterated powerset of omega) independent of ZFC; what mathematically
interesting statements are independent of ZFC+RVM?
(you may post this to FOM if you want; I don't want to repeat myself but
don't mind being cited!)
-- Joe Shipman
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