Natural Science Forum / Physics / Research / August 2008

The EPR paradox argues that quantum mechanics cannot describe all
elements of reality, and so is incomplete.   Bohr replied, in part,
that it describes all actually measured elements of reality, which is
all one can expect (and disagreed that EPRs example constituted
elements of reality, as they weren't actually measured).

I am wondering if a basic property of set theory actually forces any
fundamental description of measurement to be like quantum mechanics.
Here goes:

(1) Suppose that there some complete set S of elements of reality for
our universe (whatever they are!).   Each of these elements will
either have or not have any given physical property P.  The subset S_P
of S, corresponding to those elements which do have property P, can
therefore be taken as a representation of property P.

(2) Now, suppose one can make measurements (maybe the position of some
particle).  The possible measurement results are then surely elements
of reality (S is assumed to be complete, and hence contain
'everything' real).  Hence these results can be indexed by a subset of
S.  So, what "properties" can be measured?

(3) The answer is:  a vanishingly small number of possible
properties.  Note from (2) above that the set of possible results has
cardinality at most that of S.  But from (1) the set of properties has
cardinality equal to that of the power set 2^S of S (i.e., to that of
the set of subsets of S).  S is always a proper subset of 2^S, as
Cantor first showed.

In the unlikely case that S has a finite number of elements, N say,
then the number of properties corresponding to elements of reality are
2^N.  Hence only a fraction of possible properties N/2^N can be
determined by any measurement.

This is quite reminiscent of complementarity in QM, where only some
properties can be measured at any given time.