I was studying Cantor's diagonal argument etc. I was testing the ideas and I thought of the following mapping between the naturals and the reals and I need some help to find the flaw in it. For convenience, we use binary.
We start with the root node (the dot). Recursively, every node has two children, 0 and 1. Every n-th layer of the tree are possible values for the n-th decimal. In a BFS fashion, we set the first two children as 0 and 1, the next layer 2,3,4,5 etc.
Can't this be considered a mapping from the naturals to the reals? What assumption am I missing? Thank you.
We establish the existence of infinite sets by the Axiom of Infinity:
Some obvious but subtle points here:
What you are trying to do, essentially, is use a nested version of the process in the Axiom of Infinity to define real numbers. The problem is that every number you define this way has a finite length, just like every n in N has a finite value. Every number you define this way is the binary representation of a rational number whose denominator is 2^n.
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BTW, Cantor did not apply his Diagonal Argument to real numbers. He even said "there is a proof of this proposition ... which does not depend on considering the irrational numbers." He did apply it to infinite-length strings like you are trying to define. But they cannot be not defined by a tree like you attempt. They are defined by a mapping from N to {0,1}.