[FOM] Incompleteness versus the Platonic multiverse
paul at mtnmath.com
Wed Jun 12 13:44:45 EDT 2013
The mathematical multiverse view is "that there are many distinct
concepts of set, each instantiated in a corresponding set-theoretic
universe <http://arxiv.org/abs/1108.4223>." For example the continuum
hypothesis may be true in some of these universes and false in others.
This philosophical view leads to an approach for exploring mathematics
that is similar to an approach that stems from the more conservative
view that infinity is a potential that can never be fully realized.
My version of this view sees infinite collections as human conceptual
creations that can have a definite meaning even if they cannot exist
physically. The integers and recursively enumerable sets are examples.
In this view, infinite sets, are definite things only if they are
logically determined by events that could happen in an always finite but
potentially infinite universe with recursive laws of physics. This
includes much of generalized recursion theory, but can never include
absolutely uncountable sets. "Logically determined" is a philosophical
principal that can be** partially defined rigorously
<http://www.mtnmath.com/axioms/formalPdf.pdf>, but will always be
expandable. In this view uncountable sets can be definite things only
relative to specific countable (as seen from the outside) models.
Just as Gödel proved that any formal system embedding basic arithmetic
must be incomplete in provability, Cantor's uncountability proof plus
the Löwenheim-Skolem theorem prove that any sufficiently strong formal
first order system must be incomplete in definability. One can always
define more reals. In this philosophical view uncountable sets are
guides to how mathematical can be expanded. Thus at different stages or
paths of development one might assume different and conflicting axioms
about uncountable sets.
For more about this see www.mtnmath.com/phil/incomplete.pdf
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