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. Every finite path that is constructed *replaces* its predecessor (there is always only one predecessor, because the others had been removed before). And obviously a finite path can never produce an infinite path, can it? > > More cannot be > > > distinguished by nodes. It is the same set that contains all possible > > bit-sequences and is isomorphic to the set of all decimal fractions > > Sets of bit sequences, being binary, are not the same as sets of decimal > sequences.
It is very easy to replace the Binary Tree by a decimal one. If you have trouble to calculate the decimal from the given binary, then simply start with decimal paths in a decimal tree. > > > 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 sequence > 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.