[Fis] Cantor Goedel and Einstein and incompleteness of physics

[Louis Kauffman]

Dear Eric,
Thank you for the paper by NEWTON C. A. DA COSTA. 
Some comments here.

1. The question of the completeness of a physical or biological theory is different from the same question for a mathematical theory since one is always ready to take on more empirical evidence in a scientific theory. Also one is probably willing to change the structure of the language as well. With these changes allowed, I do not see how it is possible to apply the Goedelian or Turing arguments. These arguments depend on fixing the language and being able to enumerate all grammatical statements in the language. So I think we are left to understand that a scientific theory is not going to be known to be complete, and more likely it is known to be incomplete in various ways.

2. By the same token mathematics as an ongoing endeavor is not proved to be incomplete. It is only sharply delineated regions of mathematics called formal systems that are either inconsistent or incomplete (if rich enough to handle arithmetic). Mathematics as a whole, goes on just like science, inventing new languages and sometimes inventing or finding new axioms, often bumping into new phenomena. 

3. It is worthwhile looking closely at the examplar of incompleteness in the Cantor diagonal argument. If X is a set and P(X) is the set of subsets of X. Suppose that 
F:X —> P(X) is a given function. Then for any x in X, either x is in F(x) or x is not in F(x). Cantor forms the set C= { x in X | x is NOT in F(x)}. But then
C can not be equal to F(z) for any z since if z is in F(z) then z is not in C and if z is not in F(z) then z is in C. So C differs from every F(z). This means that there is no way to match all the elements of X with subsets of X.  Note that to show that F is incomplete, we first had to be given F. Then we used F via the set C to show that F is incomplete in that not every subset has the form F(z). The same is the case in all the incompleteness results. You have have pin the formal system to the table and look closely at it in order to show that it has true statements that it cannot prove.

4. All incompleteness arguments are variants of 3. This logic is not a tragedy, not very complicated, worth looking at with some examples until it is clear as 2+2=4.
Then one can discuss what it means for a theory to be complete. 
Best,
Lou


> On Jun 2, 2025, at 10:28 AM, Stuart Kauffman <stukauffman at gmail.com> wrote:
> 
> Thank you Erik. Stu
> 
>> On Jun 2, 2025, at 12:47 AM, OARF <eric.werner at oarf.org> wrote:
>> 
> <jonas0,+p153-6.pdf>
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