Recently on Marilyn's discussion boards...
From robert 46:"The point " seems to be trying to get across... is that if we view Cantor's algorithm as sequential, and we have constructed the first n items (call it s) of the anti-diagonal,
there will always be later strings in the list starting with s (actually infinitely many of them depending on the set being examined), and therefore the algorithm somehow fails to "examine" those strings.
However, there are at least two errors in Robert's reasoning:
1. Cantor's method isn't sequential; it constructs the anti-diagonal all at once, just like I showed by using maps/functions."
This is no more than an appeal to the miraculous. It does not in any way nullify what happens during a sequential application of the Diagonal method. Rather, it attempts to sweep-under-the- rug all the dirt which the sequential application brings to light.
"2. Even if it were sequential, once it gets to those "missing strings", the constructed anti-diagonal thus far still not equals those strings, and we therefore conclude that it will never actually be equal to any string of any row, and is hence not part of the enumeration."
As the sequential application progresses, it produces strings missing from the portion of the list so far examined. Indeed some of these missing strings can be found in the continuing examination, but other missing strings are produced faster than older ones are found- so the count of them is always increasing. ~D will not be produced until the creation of these intermediate missing strings ceases. Clearly, however, the production of the intermediate missing strings never ceases because the sequential examination is endless. Thus an endless process cannot produce ~D, but will produce a monotonically increasing count of missing strings.
Clearly, if the list was preloaded with all strings of finite significant characters (sfsc) then the missing strings must be in the list somewhere. Yet the monotonically increasing count of them informs us that the sequential Diagonal method cannot find them all. Yet it is invalid to infer from this increasing count of missing sfsc that the list is incomplete, and thereby the sfsc comprise an uncountable infinite set. We "know" that the sfsc are a countable infinite set because they can be produced by a sequential process. So the alternative conclusion is that the sequential Diagonal method is unable to examine the entire list to confirm that it is complete, and thereby confirm that the sfsc are a countable infinite set.
The hypothesized miraculous production of ~D is irrelevant to this prior problem of the sequential Diagonal method failing to confirm that the sfsc are a countable infinite set.