## A new point of view on Cantor's diagonalization arguments

Hurkyl,

Can you use a matrix of aleph0 x 2^aleph0 as an input?

All you can do is first choose your unique 01 path until some finite place, and then it is easy to find this finite 01 sequence and its opposite in infinitely many places in the above matrix.

(By the way why did you move my thread to theory development?)
 Recognitions: Gold Member Science Advisor Staff Emeritus You said it was a list. (which, by definition, has only aleph0 rows) And yes, if you output this list, I don't see why I cannot use it as an input. I moved it here because you're not doing mathematics. You may be intent on studying the topics that mathematics likes to study, but you're not doing it in a mathematical fashion. I don't remember the circumstances, but you seemed to prefer theory development to philosophy, so I move your posts here once I think it's clear that you don't want to do things in a mathematical fashion.
 Hurkyl, Please look again on this tree and tell me exactly how to you want to use it as an input. Code:  {..4,3,2,1,0}=Z* 2 2 2 2 2 ^ ^ ^ ^ ^ | | | | | v v v v v /1 1 / \0 1 /\ /1 / 0 / \0 1 |\ /1 | \ 1 | \/ \0 / 0 | \ /1 | 0 | \0 ..1 | /1 | 1 | / \0 \ 1 | /\ /1 | / 0 |/ \0 0 \ /1 \ 1 \/ \0 0 \ /1 0 \0 /1 1 / \0 1 /\ /1 / 0 / \0 1 |\ /1 | \ 1 | \/ \0 / 0 | \ /1 | 0 | \0 ..0 | /1 | 1 | / \0 \ 1 | /\ /1 | / 0 |/ \0 0 \ /1 \ 1 \/ \0 0 \ /1 0 \0 ...
 Recognitions: Gold Member Science Advisor Staff Emeritus How exactly are you using it for output?
 Recognitions: Gold Member Science Advisor Staff Emeritus A list of properties that uniquely specify it would be nice. Just to be entirely clear, let me ask this question: Can I label each leaf with a unique natural number?
 Hurkyl, The property of my Binary tree is based on this invariant structure: Code:  1 = child / / Father = ? \ \ 0 = child The number of the Childs depends on any existing Z* member = {0,1,2,3,...} used as the power_value of each level in the tree. Because |{0,1,2,3,...}| = aleph0, and these members are used as power_values for each level in the tree, the result can't be but a tree width with aleph0 magnitude and a tree length with 2^aleph0 magnitude. Each child is the beginning of infinitely long sequence of 01 unique combinations. We can label each child with a unique natural number but this is only an illusion of a bijection that can clearly shown here: http://www.geocities.com/complementa.../Countable.pdf
 Dont you see that we can always find any given 01 sequence and its opposite in the tree? Code:  {..4,3,2,1,0}=Z* 2 2 2 2 2 ^ ^ ^ ^ ^ | | | | | v v v v v /1 1 / \0 1 /\ /1 / 0 / \0 1 |\ /1 | \ 1 | \/ \0 / 0 | \ /1 | 0 | \0 ..1 | /1 | 1 | / \0 \ 1 | /\ /1 | / 0 |/ \0 0 \ /1 \ 1 \/ \0 0 \ /1 0 \0 /1 1 / \0 1 /\ /1 / 0 / \0 1 |\ /1 | \ 1 | \/ \0 / 0 | \ /1 | 0 | \0 ..0 | /1 | 1 | / \0 \ 1 | /\ /1 | / 0 |/ \0 0 \ /1 \ 1 \/ \0 0 \ /1 0 \0 ...
 In standard Math The opposite of any given diagonal has to be added to the list, therefore no list of magnitude aleph0 can be in a bijection with R members which means that |R| is uncountable. But look at this: ...0101 and ...1010 are in the list, for example: Let us take again our set: Code:  {...,3,2,1,0}=Z* 2 2 2 2 ^ ^ ^ ^ | | | | v v v v {...,1,1,1,1}<--> 1 ...,1,1,1,0 <--> 2 ...,1,1,0,1 <--> 3 ...,1,1,0,0 <--> 4 ...,1,0,1,1 <--> 5 ...,1,0,1,0 <--> 6 ...,1,0,0,1 <--> 7 ...,1,0,0,0 <--> 8 ...,0,1,1,1 <--> 9 ...,0,1,1,0 <--> 10 ...,0,1,0,1 <--> 11 ...,0,1,0,0 <--> 12 ...,0,0,1,1 <--> 13 ...,0,0,1,0 <--> 14 ...,0,0,0,1 <--> 15 ...,0,0,0,0 <--> 16 ... Now let us make a little redundancy diet: Code:  {...,3,2,1,0}=Z* 2 2 2 2 ^ ^ ^ ^ | | | | v v v v ... 1-1-1-1 <--> 1 \ \ \0 <--> 2 \ 0-1 <--> 3 \ \0 <--> 4 0-1-1 <--> 5 \ \0 <--> 6 0-1 <--> 7 \0 <--> 8 ... 0-1-1-1 <--> 9 \ \ \0 <--> 10 \ 0-1 <--> 11 \ \0 <--> 12 0-1-1 <--> 13 \ \0 <--> 14 0-1 <--> 15 \0 <--> 16 ... and we get: Code:  {...,3,2,1,0}=Z* 2 2 2 2 ^ ^ ^ ^ | | | | v v v v /1 <--> 1 1 / \0 <--> 2 1 /\ /1 <--> 3 / 0 / \0 <--> 4 ... 1 \ /1 <--> 5 \ 1 \/ \0 <--> 6 0 \ /1 <--> 7 0 \0 <--> 8 /1 <--> 9 1 / \0 <--> 10 1 /\ /1 <--> 11 / 0 / \0 <--> 12 ... 0 \ /1 <--> 13 \ 1 \/ \0 <--> 14 0 \ /1 <--> 15 0 \0 <--> 16 ...