On Friday, 20 December 2013 18:55:58 UTC+1, wpih...@gmail.com wrote:
> > No. Counterexample: The potentially infinite sequence of 1's is given by many finite formulas like 1/9 or "0.111..." or "the potentially infinite sequence of 1's behind the point", which are not the sequence but determine the sequence: > > > > So what. There is nothing in the definition of the potentially infinite > > list L that says that two elements of L cannot be the same,
Correct. > > (Note if two formula produce the same value for any n in |N > > they are by definition equal as potentially infinite sequences)
The formulas are different by their letters but identical by their result, just like a and b differ although they may stand for the same number in a = b.
But I wanted to draw your attention to the fact that an infinite sequence, in set theory, has aleph_0 elements, such that it is impossible, by spelling out the terms of the sequence, to express the same as the finite formula does. In the first case you fail to spell out every term. By a finite formula you can succeed. Example: 0.1010101010101010101010 does not determine the next digit. > > All the formulas you mention are also potentially infinite 0/1 > > sequences.
No. They allow to calculate these sequences. I agree that the formulas are same as the limits, if those exist.
> Anything that determines the sequence is a potentially infinite > > sequence (A potentially infinite 0/1 sequence is not the digits but > > the rule which produces them).
That is your personal interpretation. I used another one, in particular with respect to actual infinity. But that is a matter of taste. However, if we agree upon your interpretation, then, in transfinite set theory, there are only countably many elements of any kind.