FOM: Infinite sequences in the physical world

[JoeShipman@aol.com]

Richman:

>> In particular, if our best theories of physics lead us to conclude
>> that a certain reproducible experimental procedure will generate a
>> sequence that is mathematically definable but not computable, then
>> CT-phys will be false,

>I'm not sure what this means. "Reproducible" seems to refer to an
>actual procedure that can be carried out. Although we are constantly
>going back and forth between our mathematical models and the real
>world, isn't it some sort of categorical mistake to talk about a
>procedure of this sort generating an infinite sequence of positive
>integers?

Not at all.  The sequence doesn't have to be infinite to be 
metamathematically significant.  The idea is that the reproducible 
experimental procedure, will, in theory, generate integers until you run out 
of raw materials or die of old age or the sun turns into a red giant or 
whatever, which will at any given time represent a finite initial segment of 
a mathematically definable nonrecursive sequence f.  Even if you never have 
the completed infinite sequence, at some point you have the value x of f(N) 
for some N large enough that ZFC does not settle the truth value of the 
mathematical assertion "f(N)=x".  (Because if contrariwise ZFC did settle all 
such questions then f would be recursive after all.)

-- Joe Shipman