FOM: when we are finite.

[Franklin Vera Pacheco]

    Thesis1: Every infinite object must be given with a rule to build it.

  Let us see some of the results that we can get assert this. First of all 
we can work with finite objects without problem, the naturals can be given 
with the pair (0,+1) "where 0 is the initial element of the set and +1 is 
rule to form the set. 
The interesting case is giving an irrational number, then we must 
define the number with a rule serving to find the decimals of the number.e.g.

     sqrt(2) is given by an initial aproximation (for instance an integer 
i.e. finite) plus an algorithm to get in the k-th step an proximation to 
sqrt(2) exact up to the k-th decimal.

  Asserting this thesis we are not allowed to work with the set of real 
number as we use to do. The reals aren't then all the series of 0s and 1s 
with no infinite queues of 1s. The reals must be,at most, all the series 
of 0s and 1s with no infinite queue of 1s and that is generated by an algorithm.

 (DEF)  We say that an infinite set is numerable if it is well given i.e. 
it's given an initial element and a rule.
 
 Well the next step is to work as Cantor's proof.

   If we have a numerable set of real numbers it can be given as a pair:
 (initial set , algorithm  ) (1). 
 
 But, the initial set is formed by real numbers which are  in general 
pairs of the form: 
(initial aproximation, algorithm) (2)

 Then the algorithm in (1) acts on the pairs in (2) giving the next member 
of the series of reals. 

 Now, giving (1) and (2) we can build an algorithm constructing a real 
number different to every other number of the previous series. (for 
instance using Cantor's anti-diagonal method)

  Hence not all the real numbers are in the previous series.
 
So all the real numbers can not be given in a numerable set. And taking 
into account the notion I gave in a previous message about objects that we 
can say are actual infinites "if they can be given as pairs (initial 
object , rule) ", the reals are not such an actual infinite set.

   

-- 
Franklin Vera Pacheco
45 #10029 e/100 y 104
Marianao, C Habana,
Cuba.
e-mail:franklin at ghost.matcom.uh.cu
tel: (537) 2606043