[FOM] extramathematical notions and the CH
Joe Shipman
JoeShipman at aol.com
Tue Feb 5 00:45:36 EST 2013
With careful coding one can formalize all of the physics in some subsystem of second order arithmetic, but when so formalized the physics becomes opaque and all intuition is lost. It makes much more sense to allow real numbers, functions on real numbers, operators on such functions, and so on for some finite number of levels.
Zermelo Set Theory is enough; so are weaker systems like Maclane Set Theory. It is probably possible to be even weaker and use versions of type theory that are conservative over ACA_0 for arithmetical sentences, such as Feferman's system W, but I'm not aware how much of physics has been shown to be representable in W (in contrast, it's trivially obvious Zermelo Set Theory suffices and not too hard to see that Maclane Set Theory will also do to develop physics).
-- JS
Sent from my iPhone
On Feb 3, 2013, at 6:39 PM, Sam Sanders <sasander at cage.ugent.be> wrote:
Dear All,
I second Nik Weaver's claim that the Mathematics used in Physics can be
formalized in systems far weaker than ZFC.
However, according to the FOM community, what are weak systems still sufficient to
formalize the Math used in Physics?
RCA_0 ?
RCA_0^* ?
I\Delta_0 +EXP?
Best,
Sam
> If indispensability arguments show
> anything, it is that we have experimental confirmation of the consistency
> of the weakest formal systems in which our physics can be formalized, not
> of systems which are much stronger than what is needed.
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