[FOM] Directions for Computability Theory Beyond the Pure Mathematical

S. Spijkerman / F. Waaldijk sufra at hetnet.nl
Sun May 14 14:24:11 EDT 2006

perhaps you will allow me to introduce a related issue that i'm not sure 

i put it in my recent paper 
but i haven't had any really interested responses yet.

the issue is this:

1. we can construct (constructive!) coverings of the recursive interval 
[0,1] which have arbitrarily small classical measure (although they are not 
measurable constructively). in fact we can for any n give a countable 
sequence of intervals (S_n,m)_(m in N) such that the recursive interval 
[0,1] is covered by (S_n,m)_(m in N), and such that also constructively the 
sum of the lengths of the intervals (S_n,m)_(m in N) does not exceed 2^(-n).

2. taking every 10 seconds say a nontrivial measurement from nature (a 
fluctuating natural phenomenon) which in principle yields an infinite 
sequence \alpha, we can easily transform \alpha to be a decimal real number 
in [0,1].

3. letting H_0 be the hypothesis: `the real world is non-computable' 
(popularly speaking), and letting \beta be the uncertainty parameter of say 
2^(-40) , we could start constructing (S_40,m)_(m in N).

4. it seems to me that if we ever discover an m such that \alpha is in 
S_40,m , then we have to discard H_0.

this seems to me a legitimate scientific experiment, which can be actually 
carried out. of course, it is one-sided, but on the other hand any outcome 
would be spectacular i think?

well, i hesitate but since it was printed already...go on and shoot...


----- Original Message ----- 
From: "John Case" <case at mail.eecis.udel.edu>
To: <fom at cs.nyu.edu>
Cc: "John Case" <case at mail.eecis.udel.edu>
Sent: Saturday, May 13, 2006 12:31 AM
Subject: [FOM] Directions for Computability Theory Beyond the Pure 

> Ted Slaman in [Long range goals, COMP-THY Archives, \#13, April 1998]
> nicely mentioned the central theme of his intellectual motivation
> (deriving from the influence of Sacks) for working in
> Computability Theory (CT), a.k.a. Recursive Function Theory,
> namely, definability.  There was also some call around that time
> for posting new directions in CT.  I started to draft such a posting, but
> didn't finish until yesterday.  When I was in Singapore January'06 and
> talking with Sergey Goncharov also visiting there, he invited me to submit
> a book chapter for [Mathematical Problems from Applied Logic.
> New Logics for the XXIst Century II (edited by D. Gabbay, S. Goncharov, 
> and
> M. Zakharyaschev), International Mathematical Series, Springer, to appear
> 2006].   A draft of my intended book chapter, entitled,
> Directions for Computability Theory Beyond Pure Mathematical, is, 
> currently,
> available from http://www.cis.udel.edu/~case/papers/rft-directions.pdf
> (-8 John
> John Case                              Email: case at cis.udel.edu
> Professor                              Phone: +1-302-831-2714
> Computer and Information Sciences Dept. 101A Smith Hall 
> FAX: +1-302-831-8458
> University of Delaware Newark, DE 19716-2586 (USA)             Home: 
> +1-302-836-4888
>                URL:  http://www.cis.udel.edu/~case

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