# [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
(http://home.hetnet.nl/~sufra/foundations%20of%20constructive%20mathematics.pdf),
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...

frank

----- 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
Mathematical

> 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
>
>
>

`