[FOM] Incompleteness and Physics

[joeshipman@aol.com]

It's really very simple. Suppose a physically measurable quantity is, 
according to a physical theory (which one believes in for physical 
rather than mathematical reasons), a real number X that is DEFINABLE in 
ZFC but not recursive. Then some mathematical sentence of the form "the 
nth bit of X is 1" or "the nth bit of x is 0" is independent of ZFC; 
however physical experiments might nonetheless give us KNOWLEDGE of the 
nth bit of X, and therefore of the truth of a ZFC-independent statement.

The fact that you can then assume an axiom about the value of that bit 
is totally irrelevant -- the issue is not that there is no theory which 
can define the value of a certain finite set of bits, it is that (if 
you accept this hypothetical physical theory) there is a bit which 
ALREADY has a definition in ZFC but which can't be proved in ZFC to 
have a specific value.

-- JS


-----Original Message-----
From: Vaughan Pratt <pratt at cs.stanford.edu>
To: Foundations of Mathematics <fom at cs.nyu.edu>
Sent: Sun, 19 Oct 2008 5:34 pm
Subject: Re: [FOM] Incompleteness and Physics




joeshipman at aol.com wrote:
> On the other hand, Godel Incompleteness also implies an influence of
> physics on mathematics -- as Godel pointed out, we might come to
> believe in new mathematical axioms because they made physics work. 
This
> will be the case if, according to some mathematized physical theory,
> there exists a definable but noncomputable real number that is
> measurable within the theory -- ZFC-proofs could only settle the 
value
> of finitely many bits of such a number, possibly fewer than can be
> measured.

Given that every finite set of natural numbers is representable in ZFC,
no matter how precisely one measures a given quantity there will always
be a ZFC sentence that can "account" for it.

If there is a notion of "physical theory" that rules out certain finite
subsets of N that as dyadic rationals can't be an approximation (to 
that
accuracy) of any fundamental constant of any physical theory, that 
would
be extremely interesting.  If not, I have difficulty understanding what
it would mean to have a finite measurement outside the scope of all
ZFC-based theories.

Vaughan Pratt
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