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I came across this article, by H. Vic Dannon, where he argues that the cardinality of a powerset is not necessarily strictly bigger than the cardinality of the set, and the problem arises when one starts applying properties of finite sets to infinite sets. But that's not the point I am concerned with... On pages 4-5 he describes a purported injection from reals to rationals, more precisely from (0, 1) to rationals. I am not very good at math, so as I read the description I couldn't find the mistake. Could someone please explain where his flaw is (I presume there is one)? Thank you very much.

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closed as too localized by Arturo Magidin, Grigory M, Andres Caicedo, Asaf Karagila, t.b. Jan 30 '12 at 18:41

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No, it's not right. A "lexicographic list of real numbers in binary expansion" never includes any real number whose expansion contains infinitely many 1s; e.g., $(1,0,1,0,1,0,1,0,1,0,1,0,1,0,1\ldots)$ is not on his list. So his domain is not all of $\mathbb{R}$. His "list" only includes real numbers with terminating binary expansion, and those reals are certainly bijectable with $\mathbb{Q}$. The entire article shows some deep misunderstandings and ignorance of the basics of elementary set theory. –  Arturo Magidin Jan 28 '12 at 1:30
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But everything else here is good, right? :) –  Dylan Moreland Jan 28 '12 at 1:33

1 Answer 1

up vote 15 down vote accepted

The problem with that argument is that his ‘dictionary listing of the Real numbers’ includes only those sequences that are $0$ from some point on. In other words, he gets only the so-called dyadic rationals in $(0,1)$, those that can be written as a fraction whose denominator is a power of $2$. In particular, it does not include $\sqrt2$, his assertion to the contrary notwithstanding. Since he lists only countably many of the real numbers, it’s no surprise that he is able to find an injection from them into the rationals. He could, in fact, simply have let $f(x)=x$ for each $x$ in his list, and $f$ would have been an injection from his list into the rationals!

The essay is a rather sad exhibition of someone’s reach greatly exceeding his grasp: the author clearly understands very little of the mathematics that he attempts to bring to bear on a non-existent problem.

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Fans of H. Vic Dannon might also like his proof of the Riemann Hypothesis, found here: riemann-hypothesis-journal-online.org/proof/… –  I. J. Kennedy Jun 26 '12 at 23:59

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