@ughaibu,
Quote:There can be, at most, a countable infinity of statements generated by any algorithm. So, you cannot produce an uncountable infinity. If you think that you can, them you haven't understood the matter.
If you assume that there can be a countable infinity of statements generated by an algorithm then it follows that you generate the uncountables as well. And I will show you how:
Cantor's Diagonal argument is used to show that there exists numbers beyond a countably infinite set. He claims that a set must have a one-to-one correspondence with the set of natural numbers to be countable. In his argument he creates a pairing of real numbers (in decimal form) to the positive integers. He then shows that alot of numbers were not accounted for in the pairing between the two sets (the reals and the natural numbers). These numbers are called uncountable real numbers.
Here is the setup to Cantor's Diagonal argument:
n=1 -> . a
1 a
2 a
3 a
4 ...
n=2 -> . b
1 b
2 b
3 b
4 ...
n=3 -> . c
1 c
2 c
3 c
4 ...
n=4 -> . d
1 d
2 d
3 d
4 ...
.
.
.
where n is the natural number that corresponds to some decimal real number given by the sequence . a
1 a
2 a
3 a
4 ... The letters a, b, c, etc... are just to show that the number in that position goes with the other numbers with the same letter.
For instance the decimal . a
1 a
2 a
3 a
4 could be the real number .00000.... where a1 is the first zero, a2 is the second zero and so forth. Each row contains one distinct combination of the countably infinite set of the real numbers. Here is a link to a wikipedia page describing this, you can also find clear and easy to understand demonstrations of the proof on youtube:
The proof goes on to show some numbers exist that were not apart of the countably infinite set of real numbers given. This is done by the diagonal argument.
n=1 -> .
a1 a
2 a
3 a
4 ...
n=2 -> . b
1 b2 b
3 b
4 ...
n=3 -> . c
1 c
2 c3 c
4 ...
n=4 -> . d
1 d
2 d
3 d4 ...
.
.
.
By making the decimal S = . a1 b2 c3 d4 ... we can also define a decimal real number that does not contain these values in those positions. The "anti-sequence" of S is S' = . s1 s2 s3 s4 ... where {s1 ≠ a1, s2 ≠ b2, s3 ≠ c3, s4 ≠ d4, ...} Because S' is defined to be different than each combination in the original list we can see S' cannot be on the previous list of countably infinite numbers.
However, and to Ughaibu's claim that the sequences of uncountable numbers cannot be generated by an algorithm, the algorithm for generating uncountable numbers is given in the proof. It is simple: S' = . s1 s2 s3 s4 ... where {s1 ≠ a1, s2 ≠ b2, s3 ≠ c3, s4 ≠ d4, ...}. So an algorithm can be made to use all the acceptable values for s1, s2, s3, s4,... producing all the possible combinations of such numbers as well. Of course this is still assuming infinites to powers of infinities of time to perform the calculation in.