In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 24 Jan., 09:49, Virgil <vir...@ligriv.com> wrote: > > > > > If there were, one would also have to have a difference between the set > > > > which contains all finite initial segments of |N and |N itself > > > > > Correct. There is no difference. Therefore we can, in mathematics, use > > > only the Binary Tree that contains all finite paths. > > > > But if it contains all finite paths then it must contain nested > > sequences of infinitely many finite paths whose unions are each one of > > those uncountably many infinite paths. > > I do not construct the tree by means of unions.
However you construct them , if yu do it properly, those unions exist.
> Every finite path that > is constructed *replaces* its predecessor (there is always only one > predecessor, because the others had been removed before).
Then let us speak of finite initial sequences of nodes, i.e., a FISON is Finite Initial Set Of Nodes linked by the parent child relation, starting with the root node, having exactly one child node for each of its non-end nodes and terminating at some end node. The number of nodes in a FISON is its length.
> And > obviously a finite path can never produce an infinite path, can it?
It can help!
In a Binary Tree nested sequences of FISONs exist in which each non-maximal FISON is a proper subset of a successor FISON having one more node, the child of the first one's end node.
In a Complete Infinite Binary Tree, there are maximal such nested sequences of FIS's, whose unions are the paths in that tree.
> > It is very easy to replace the Binary Tree by a decimal one.
But WM cannot do it!
> > > that can be applied in mathematics and in Cantor's diagonal argument. > > > We can neither distinguish nor apply by digits more than all > > > terminating decimal fractions. Therefore all that appears in Cantor's > > > list is terminating decimal fractions. Therefore Cantor proves the > > > uncountability of a countable set. > > > > Thus Cantor has= proved, among other things, that an infinite binary > > sequence is not in any list of finite binary sequences, a result which > > is hardly surprising to anyone other than WM. > > Moreover, he has proved that even in the set of all finite binary or > decimal sequences there is always one binary or decimal sequence that > does differs from every other one at a finite digit (hence it is > completely irrelevant with respect to this fact, whether or not the > differing sequence is infinite, i.e., whether or not it has a last > digit). > > > > But Cantor also proved that for any list of infinite binary sequences > > there is an infinite binary sequence not included in it. > > Since everything happens in a finite initial segment, there is no > reason to consider the question whether the number is finite or has > been written with red ink or other irrelevant stuff.
And there is no reason to restrict it to being finite either!
WM thus admits above that there is nothing Cantor does that is not valid. --