On 24 Jan., 09:32, Virgil <vir...@ligriv.com> wrote: > In article > <395bea71-7f73-49dc-94d0-5a422088d...@z8g2000yqo.googlegroups.com>, > > WM <mueck...@rz.fh-augsburg.de> wrote: > > The following is copied from Mathematics StackExchange and > > MathOverflow. Small wonder that the sources have been deleted already. > > > How can we distinguish between that infinite Binary Tree that contains > > only all finite initial segments of the infinite paths and that > > complete infinite Binary Tree that in addition also contains all > > infinite paths? > > WM also distinguishes between the union of all finite initial segments > of |N and |N itself,
Is the set of all terminating decimal expansions of real numbers of the unit interval not a set? Is it poorly defined? Do not all its elements exist in ZF? (Not even AC is necessary.) Is it impossible to distinguish a non-terminating decimal like that of 1/3 from all terminating decimals?
The infinite set T of all terminating decimals t_i exists. With respect to ithe indexed digits of the t_i we can do everything that is done in a Cantor-list, since we cannot distinguish, by indexed digits, whether we work in T or in |R.
Of course there might be some messages from the God of matheology or some clairvoyance that is not accessible to non-matheologians. But in mathematics, there is no difference.
> which is the unary equivalent of his binary tree
Of course. That is what I have been arguing for nearly ten years now. In order to finish actual infinity of |N you need more than the natural numbers, which supply only potential infinity. But in the unary tree the difference is too small (1 element versus infinitely many) to be recognized by most mathematicians. Therefore if have finally considered the Binary Tree. Here the difference is uncountably many versus countably many. And that should be possible to recognize by many mathematicians, not by all though.