Are Maxwell's Equation's Enough?
                        ________________________________

                                Michael O'Connor

                                Department of Physics and Applied Physics
                                University of Massachusetts Lowell
                                Lowell, MA 01854
                                OCONNORMI@WOODS.UML.EDU
	                        (received July 24, 1994)

Abstract:
_________

	In this paper the overhead transparencies of a recent talk on 
"Inconsistency and Incompleteness in Mathematical Systems" are presented, in 
which the impact of Godel's theorem on a theory of everything (T.O.E.) are 
discussed.  The approach taken is mathematical.  The paper concludes by moving 
the discussion to electrodynamics and Maxwell's equations.  The suggestion that 
Maxwell's equations are not enough is humbly made.

Introduction and Discussion:
____________________________

	Recently I attended a seminar given by Peter Bertone on "Inconsistency 
and Incompleteness in Mathematical Systems." Peter has been formally schooled 
in mathematics and physics.  He was kind enough to share his notes with me and 
I present them here [1].  Unfortunately, I have temporarily lost touch with him 
so I am unable to reconstruct a complete list of the material he surveyed for 
this presentation.  I am quite certain that the discussion was meant to be more 
expository than original.  Some of the material has been freely taken from 
Nagel and Newman's notable book "Godel's Proof," [2] while other material has 
been taken from "Godel, Escher, Bach: an Eternal Golden Braid" by D.R. 
Hofstadter [3].  I also found Edna Kramer's book: "The Nature and Growth of 
Modern Mathematics" [4], as well as Penrose's book: "The Emperor's New Mind"[5],
helpful. 
	Comments which were not in the original notes are denoted by square 
brackets  and each page break is denoted by a row of equal signs. Without 
further ado:


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Inconsistency and Incompleteness in Mathematical Systems

========================================================

A mathematical system consists of:
  1) a set of objects
  2) definitions
  3) axioms
  4) theorems

The axioms implicitly define the "undefined" terms.

Theorems are true propositions derived from the axioms.

========================================================

Example: Arithmetic of Natural Numbers

Axioms:
  1) 1 is a natural number.
  2) The immediate successor of a number is a number.
  3) No two numbers have the same immediate successor.
  4) 1 is not the immediate successor of a number.
  5) Let M be a set of natural numbers with the following properties:
      i) 1 belongs to M
     ii) If x belongs to M, then so does the immediate successor of x.

     Then M contains all the natural numbers.

========================================================

Example: Arithmetic of Natural Numbers (continued)

The set of objects is the natural numbers.

We can define 2 as the "immediate successor" of 1, etc.

"1", "natural number", and "immediate successor of" are the 
undefined terms.

We can prove as a theorem that:
   x + y = y + x  for all x and y which are contained in M

=========================================================

A mathematical system is inconsistent if it contains a proposition
                         ____________
`A' such that both `A' and the negation of `A' are theorems.



An inconsistent system is considered WORTHLESS 
                                     _________

==========================================================

Example: "Intuitive" set theory is inconsistent. 
         [sometimes called "naive" set theory]

Normal classes do not contain themselves as members.
(i.e. the class of mathematicians is not a mathematician)

Non-Normal classes do contain themselves as members.
(i.e. the class of all thinkable things is itself thinkable)

Let N be the class of all normal classes

If N is normal, then N is a member of itself (since N contains all 
normal classes).  But then N is normal since all the members of N  
are normal classes.

Therefore N is normal if and only if N is non-normal.

[This is an example of Russell's paradox.  One variant of the paradox
made popular by Martin Gardner [6] goes as follows: In a certain town
the barber shaves all those men in town, and only those men, who do
not shave themselves. Who shaves the barber?
   This is closely related to the liar paradox (attributed to 
the Cretan Epimenides circa 6th century B.C.).  He is reported to have
said: "All Cretans are liars."]
==========================================================

Example of a Proof of Consistency:

1) Any two members of K are contained in just one member of L.

2) No member of K is contained in more than two members of L.

3) The members of K are not all contained in a single member of L.

4) Any two members of L contain, between them, just one member of K.

5) No member of L contains more than two members of K.


                       K
                      / \
                   L /   \ L      [This is called a model]
                    /     \
                   K _____ K
                       L
                       
[The system of axioms is consistent if Euclid's geometry is consistent.
 Absolute proofs of consistency are much more involved.]

[This example is from Nagel and Newman (see ref.) pp. 16-17]


==========================================================

The axioms of a mathematical system are said to be complete 
                                                   ________
if every true statement expressible in the system is
formally deducible from the axioms.



If not every true statement that is expressible in the system 
is deducible, the axioms are incomplete .
                             __________

A system is sufficiently rich if at least arithmetic can
                         ____
be developed within it.  [e.g. calculus contains arithmetic]

==========================================================

Godel's First Theorem
   Any sufficiently rich, consistent system will contain at least
   one undecidable statement `A' where neither `A' or `not A' are provable
   as theorems within the system. [`A' may well be true]

Godel's Second Theorem
   The consistency of a sufficiently rich, consistent system cannot
   be proven by methods which can be expressed or reflected in the
   system.

An undecideable statement is different from an unsolvable problem.

[Adding the statement `A' or `the negation of A' does not help because
 now we have a new system which is subject to Godel's theorem.]

===========================================================

Example of an Undecidable Statement:

The continuum hypothesis is an undecidable statement in the system
of Zermelo-Fraenkel set theory.

Aleph(null) is the cardinality of the set of integers.
[Aleph is the first letter of the Hebrew alphabet]

One can generate a sequence of infinite cardinals:
  Aleph(null), Aleph(one), Aleph(two), Aleph(three), ...

Where the cardinality of Aleph(i+1) is equal to 2 to the power of Aleph(i)
and Aleph(null) is the smallest cardinal.

Let C denote the cardinality of the real numbers.
        C = 2 to the power of Aleph(null)        (Cantor)

Continuum Hypothesis:  C = Aleph(one)
   There is not a set with cardinality between Aleph(null) and
   2 to the power of Aleph(null) (i.e. C).

[Paul Cohen proved that the continuum hypothesis is independent
 of the the other axioms, therefore undecideable]

=============================================================

Physical Theories are clearly math. systems.

A Theory of Everything (i.e. T.O.E.) must be consistent.

The consistency of a T.O.E. cannot be proven from within the theory.

Any T.O.E. will not, in principle or in practice, 
be able to derive every truth in nature.

_____________________________________________________________

Godel's Theorems also place limits on the capacities of
artificial intelligence.

_____________________________________________________________

=============================================================

Conclusion:
___________

	My thoughts reside not so much on theories of everything but more on 
the germane world of electrodynamics.  In electrodynamics we use Maxwell's 
equations as our axioms - they also implicitly serve to define the "undefined" 
terms (the fields).  If it can be shown that our current formalism of 
electrodynamics is sufficiently rich and consistent, then their exists at least 
one proposition which is undecideable.  What's a physicist's to do?  (We should 
be able to show that electrodynamics is consistent by using a model (not based 
on vector calculus or exterior differential calculus - i.e. the current 
formalism of electrodynamics), much like we used the Euclidean triangle above 
to show the consistency of the classes K and L above. I doubt the sufficiently 
rich criteria will present much difficulty either -- for a mathematician 
that is.)
	I think we have to examine the nature of any and all of these 
undecideable problems, determine the measure of the set of undecideable 
problems, examine alternative formalisms of electrodynamics and acknowledge 
the limitations of using mathematical systems for describing the physical 
world.  Not a small order, but look on the bright side, it might mean some of 
us could get jobs.

References:
___________

[1] Peter Bertone (private communication).
[2] E. Nagel and J. Newman, "Godel's Proof" (New York University Press, 1958).
[3] D.R. Hofstadter, "Godel, Escher, Bach: an Eternal Golden Braid,"
    (Basic Books, New York 1979).
[4] E.E. Kramer, "The Nature and Growth of Modern Mathematics," 
    (Princeton University Press; Princeton, New Jersey 1981).
[5] R. Penrose, "The Emperor's New Mind," (Oxford University Press,
    New York 1989).
[6] M. Gardner, "Aha! Gotcha," (Scientific American, Inc. 1975;
    W.H. Freeman and Company, 1982).