Countability of the continuum

Question

I googled the word countability of continuum and the first result (Ok, second to this thread!) was from Arxiv.

I was wondering how valid this argument is.

I would also appreciate any additional comments on this paper.

Thank You.

Answer 1

Not to put too fine a point on it, the paper is rubbish.

In Proposition 2.2 the author confuses the set of nodes of his tree, which has cardinality $\aleph_0$, with the set of branches through the tree, which has cardinality $10^{\aleph_0}$; because he mistakenly thinks that these are the same set, he mistakenly concludes that $\aleph_0=10^{\aleph_0}$. Proposition 2.3 is merely a restatement of the same error in a very slightly different guise.

In his argument for Proposition 3.1 he (mostly) establishes an injection from the set of real numbers to the set of branches of his tree, but (as he himself acknowledges in the argument for Lemma 3.1) this is not a bijection, since a real number may correspond to more than one branch of the tree. (The assertion on page $4$ that $3.1415926$ ‘is the number $\pi$’ is of course false, since $\pi$ is irrational and therefore does not have a finite decimal expansion.)

Proposition 3.2 is yet another instance of his inability to distinguish the nodes of a tree from its branches. The rest of the paper is just more of the same.

Answer 2

The arguments given in the paper for the claims supposedly proved are nonsense. Moreover, as the author says in the beginning, Cantor proved that $\mathbb N$ and $\mathbb R$ have different cardinalities. So, in fact, if the author did prove what he claims (which he did not, but never mind) then unless he can demonstrate that all of the proofs of the uncountability of the reals (and there are several, essentially different ones) then the author establishes the inconsistency of mathematics by proving a contradiction. Needless to say, he did not do that.

Answer 3

This is one of the common false proofs that $\mathcal P(\Bbb N)$ is countable, we forget that there are subset of $\Bbb N$ which are infinite, and that their complement is infinite as well.

If we think of the full binary tree of height $\omega$ (i.e. every finite height is realized, and no point has infinitely points below it), then $\mathcal P(\Bbb N)$ corresponds to the branches of the tree, and not the vertices of the tree.

The paper constructs a rooted tree where every point has $10$ successors, and the tree has height $\omega$. Then he claims that the real numbers correspond to the vertices, rather than the branches (do note that some numbers can be represented by two branches, but that's not a big deal for now).

Of course this is false, as Brian points out, $\pi$ cannot be represented by a vertex and only by a branch.