Content deleted Content added
Laplace777 (talk | contribs) |
Laplace777 (talk | contribs) |
||
Line 619:
Cantor attempts to construct a new real number that would not be present on a list containing ALL real number (by initial assumption). Let's name SPECIAL NUMBER (SP) the number he is creating and let's decompose the steps taken:
a) he decide that his SP will be a real number. Now, even though it is not yet constructed and define (and could many different potential (different) numbers) the thing we know for sure is that it will be a real number. Therefore AUTOMATICALLY from his initial complete real number list assumption, it MUST, it IS present in the enumeration list (because the list is assumed to contain all real and the number is known to be a real)
b) now he contruct the number by defining successively its digits. Let's say he define the first n decimal (to start). After that nth defined decimal we can say at this point that he has a "partially constructed" number that is different from the first n element of the list. He still has not defined fully his number (but we still know that whatever it will be, it will be a real number and IS CURRENTLY on the full pre-existing list because it is a real and the list is supposed to contain all the real)
c) he continue forever to construct his number. At any point, as n increase, all we can observe is:
1) the SP will never be identical to the first n number on the list
2) the SP will always be in the rest of the list (because we have assume that the list contains ALL reals, and we know that are still not completely constructed number is a real.
d) as n grow to infinity, 1) and 2) always remain as is= true. There is no convergence/changes whatsoever occuring relative to those statements, there is no less number remaining in the list (=still an infinity), and most importantly there is still an infinite quantity of number after the first n, one of which by assumption will be identical to SP - IF IT EVER GET CONSTRUCTED. And that is the point: the construction of the number does not converge toward "fully constructed" because there still remains an infinity of decimal (just as many as when we started) to add.
e) therefore there do not appear to be any contradiction and the proof by absurdio would according to this logic fail.
To shed more light. Let's compare to what happens when we deal with other sequences: Let's take for example the geometric infinite sequence 1, 1/2, 1/4, 1/8 .... and calculate the sum S. In that case:
a) as we travel the sequence there still remain as many element in the sequence to travel than when we begin (=infinity). We never reach the last element. This is akin to our SP number that never got constructed in the proof
b) the sum S does however converge toward 2. This is akin in the cantor argument to the statement boolean variable (to use computer jargon):
i) the SP is not present in the first n number=TRUE
ii) the SP is present somewhere in the numbers after the first n number=TRUE
In both i) and ii) the statements remains constant=TRUE (i.e. converge to themselve) throughout the travelling in the list and we can comfortably say tha the limit of each of those statement = TRUE. Therefore no contradiction with the initial assumption.
| |||