[FOM] Incompleteness versus the Platonic multiverse

Paul Budnik 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 

Paul Budnik
www.mtnmath.com <http://www.mtnmath.com>

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