```Date: Jan 1, 2013 1:19 PM
Author: Zaljohar@gmail.com
Subject: The Distinguishability argument of the Reals.

The distinguishability argument is a deep intuitive argument about thequestion of Countability of the reals. It is an argument of mine, itclaims that the truth is that the reals are countable. However itdoesn't claim that this truth can be put in a formal proof.The idea is that we cannot have more objects than what we candistinguish.The argument originated with discussions about the Infinite binarytree, and it is built on the following observations and generalizationand consideration:Observation 1:In any finite binary tree if we change the labeling of all nodesbeyond a specific level in such a manner that all of those receive thesame label, then the number of paths that can be distinguished by thelabels of their nodes will not increase beyond that of the lastunaltered level.Example: The infinite binary tree with Two levels below the root node.    0   /  \  0   1 / \   | \0 1 0  1Now lets alter the last level (i.e. Level 2):    0   /  \  0   1 / \   | \0  0 0  0Now the number of paths at level 1 (the unaltered paths) is 2, thoseare:0-00-1The number of paths at the altered level which is level 2 would bealso 2, those are:0-0-00-1-0Now lets add another level with fixed labeling with 0, this would be:     0    /  \   0    1  / \    | \ 0  0   0  0 /\  /\   /\  /\00 00 00 00The number of paths at level 4 would be also 2, those are:0-0-0-00-1-0-0Generalization: From the above observation we can make the followingintuitive generalization_ That in the case of ANY binary tree thetotal number of paths distinguishable by labels of their nodes of Sizen Will be equal to the total number of paths distinguishable by labelsof their nodes of Size m where m > niff distinct labeling of nodes seize to exist after nodes at the endof paths of size n.Observation 2:The complete Infinite binary tree have all its nodes labeleddistinctly occurring at end of FINITELY long paths. And accordingly Nodiscrimination by labeling of nodes occurs at the end of someinfinitely long path, so there is not discrimination by labeling thatoccurs at INFINITE level, all distinct labeling do occur at FINITElevel only.Consideration: We Consider FINITE and INFINITE to be kinds of gross(semi-quantitative) Size criteria where INFINITE size criterion isbigger than FINITE size criterion, i.e. INFINITE > FINITE.Now from Generalization, Observations 2 and Consideration we arrive atthe following:RESULT 1:The number of Infinitely long paths of the complete infinite binarytree is the same as the number of the finitely long paths of thecomplete infinite binary tree.Observation 3:The total number of FINITE paths of the complete Infinite binary treethat are distinguishable by labeling of their nodes is COUNTABLE.From Result 1 and Observation 3, we reach at:Result 2The number of all INFINITE paths and thus ALL paths of the completebinary tree is COUNTABLE.Observation 4:Each real is identified with a distinguishable path by labeling of itsnodes of the complete infinite binary tree.From Result 2 and Observation 4 we arrive finally at:FINAL CONCLUSION:The number of all reals is COUNTABLE.QEDZuhair
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