Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Topic: Construction of the Binary Tree
Replies: 25   Last Post: Oct 12, 2017 3:50 PM

 Messages: [ Previous | Next ]
 Martin Shobe Posts: 1,469 Registered: 3/11/12
Re: Construction of the Binary Tree
Posted: Oct 6, 2017 7:48 PM

On 10/6/2017 3:22 PM, WM wrote:
> Am Freitag, 6. Oktober 2017 15:57:54 UTC+2 schrieb Dan Christensen:
>> On Friday, October 6, 2017 at 4:35:58 AM UTC-4, WM wrote:
>>> A countable set can be constructed by using always half of the remaining time for the next step. An uncountable set cannot be constructed such that uncountably many elements can be distinguished. So it is possible to construct |N and with it all its subsets. But these subsets cannot be distinguished unless it is indicated which elements are to combine. Therefore we find:
>>>
>>> - The Binary Tree can be constructed because it consists of countably many nodes and edges.
>>>
>>> - The Binary Tree cannot be constructed because it consists of uncountably many distinct paths.
>>>

>>
>> Should be easy enough to a construct binary tree on N.
>>
>> Construct: L: N --> N such that L(n)=2n+1 (the left node of n)
>> Construct: R: N --> N such that R(n)=2n+2 (the right node of n)
>>
>> So, we have L(0)=1, R(0)=2, L(1)=3, R(1)=4, L(2)=5, R(2)=6, .....
>>
>> Does this result in a contradiction? If so, please demonstrate exactly how using ONLY the rules of arithmetic on N, the axioms of set theory and the rules of logic.

>
> There are aleph_0 steps completing all paths, all are distinct, each one is completed by its own step. If so, then there are not more than aleph_0 paths.

None of the paths are completed at any step.

Martin Shobe