The distinguishability argument is a deep intuitive argument about the question of Countability of the reals. It is an argument of mine, it claims that the truth is that the reals are countable. However it doesn't claim that this truth can be put in a formal proof.
The idea is that we cannot have more objects than what we can distinguish.
The argument originated with discussions about the Infinite binary tree, and it is built on the following observations and generalization and consideration:
In any finite binary tree if we change the labeling of all nodes beyond a specific level in such a manner that all of those receive the same label, then the number of paths that can be distinguished by the labels of their nodes will not increase beyond that of the last unaltered level.
Example: The infinite binary tree with Two levels below the root node.
0 / \ 0 1 / \ | \ 0 1 0 1
Now lets alter the last level (i.e. Level 2):
0 / \ 0 1 / \ | \ 0 0 0 0
Now the number of paths at level 1 (the unaltered paths) is 2, those are:
The number of paths at the altered level which is level 2 would be also 2, those are:
Now lets add another level with fixed labeling with 0, this would be:
0 / \ 0 1 / \ | \ 0 0 0 0 /\ /\ /\ /\ 00 00 00 00
The number of paths at level 4 would be also 2, those are:
Generalization: From the above observation we can make the following intuitive generalization_ That in the case of ANY binary tree the total number of paths distinguishable by labels of their nodes of Size n Will be equal to the total number of paths distinguishable by labels of their nodes of Size m where m > n iff distinct labeling of nodes seize to exist after nodes at the end of paths of size n.
The complete Infinite binary tree have all its nodes labeled distinctly occurring at end of FINITELY long paths. And accordingly No discrimination by labeling of nodes occurs at the end of some infinitely long path, so there is not discrimination by labeling that occurs at INFINITE level, all distinct labeling do occur at FINITE level only.
Consideration: We Consider FINITE and INFINITE to be kinds of gross (semi-quantitative) Size criteria where INFINITE size criterion is bigger than FINITE size criterion, i.e. INFINITE > FINITE.
Now from Generalization, Observations 2 and Consideration we arrive at the following:
The number of Infinitely long paths of the complete infinite binary tree is the same as the number of the finitely long paths of the complete infinite binary tree.
Observation 3: The total number of FINITE paths of the complete Infinite binary tree that are distinguishable by labeling of their nodes is COUNTABLE.
From Result 1 and Observation 3, we reach at:
Result 2 The number of all INFINITE paths and thus ALL paths of the complete binary tree is COUNTABLE.
Observation 4: Each real is identified with a distinguishable path by labeling of its nodes of the complete infinite binary tree.
From Result 2 and Observation 4 we arrive finally at: