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In binary system, Cantor's diagonal argument lost its effect.(https://zhuanlan.zhihu.com/p/20197130)(Sorry for the Chinese needed to translating) What can we say about it? How can we improve it？

In binary system,if we construct like this: r = 0.d1d2d3… di = 1 - dii

Then we fail. For an example: r1 = 0.1 r2= 0. 001 r3 = 0.0001 r4 = 0.00001 …

The thing we diagonally inverted constructed is: r = 0.0111…

But in this way,it just is r1.

And it seems other ways of diagonal construction fails too.

(Sorry for my be utterly ignorant of things like Latex)

Noah Schweber
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    A LaTeX user guide is here. – J.G. Jan 17 '20 at 13:37
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    The failure is due to the fact that real numbers (specifically, rational numbers) do not necessarily have a unique binary expansion (or tenary, decimal, hexadecimal or whatever you want.) – Hanul Jeon Jan 17 '20 at 13:40
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    If you expand "Construction of a bijection between $T$ and $R$" here, you'll see how the subtlety @HanulJeon mentioned is handled. – J.G. Jan 17 '20 at 13:41
  • Cantors method also works for infinite bit strings (or real numbers in the range [0,1] in binary representation). To arrive at a contradiction, you assume that you can write down all possible strings (or numbers) and construct a new with the diagonal method. – Peter Jan 17 '20 at 14:31
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    @Peter: the problem is that your original list might include say $1/2 =0.100\ldots$ and the diagonal argument could come back with $0.0111\ldots = 1/2$. There are lots of ways around this problem. – Rob Arthan Jan 17 '20 at 23:48
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    @Peter Urgh; there’s no need to do an argument by contradiction. Cantor’s argument is a direct proof of the contrapositive: given any function from $\mathbb{N}$ to the set of infinite bit strings, there is at least one string not in the range; that is, no such function is surjective. See, e.g., here. – Arturo Magidin Jan 18 '20 at 01:40
  • I've removed the "diagonalization" tag (see the description of that tag). – Noah Schweber Jan 18 '20 at 02:51
  • @ArturoMagidin What else than a contradiction is this , if we find another element contradicting the assumption that we have listed all ? – Peter Jan 18 '20 at 09:48
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    @Peter: It’s not that you cannot argue by contradiction; it’s that it is unnecessary to do so. Delete your assumption that the list contains all, remove the final line saying “this contradicts our hypothesis that the list contains all”, and you get a direct proof that the list is incomplete. – Arturo Magidin Jan 18 '20 at 11:18
  • Please learn to use MathJax to properly format math expressions. If for whatever reason you don't want to fix the question, then most likely the community will vote to close it. – Lee David Chung Lin Feb 21 '20 at 03:35

2 Answers2

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There are several ways to fix this; here are two.

  • We can look at more than one digit at a time. Given our starting list $$(0.a^1_1a^2_2a^3_3, \quad 0.a^2_1a^2_2a^2_3...,\quad 0.a^3_1a^3_2a^3_3..., \quad ...)$$ of binary representations of reals in $[0,1)$, we define a sequence $b_1,b_2,b_3,...$ as follows: the two-digit block $b_{2i-1}b_{2i}$ is "$01$" if $a_{2i-1}a_{2i}$ is not "$01,$" and $b_{2i-1}b_{2i}$ is "$10$" otherwise.

  • We can just transfer between contexts. We don't have to use the binary representations; we could always switch to (say) decimal representations, apply diagonalization there (where having more than two digits gives us "room" to work without having to consider multiple digits at once), and then convert back to binary. This might feel like cheating, but it's perfectly valid.

I'd say that both approaches have their advantages; ultimately each is teaching a kind of flexibility when it comes to constructions, either at the specific implementation level or right at the formulation level.

Noah Schweber
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The argument Georg Cantor presented was in binary. And I don't mean the binary representation of real numbers. Cantor did not apply the diagonal argument to real numbers at all; he used infinite-length binary strings (quote: "there is a proof of this proposition that ... does not depend on considering the irrational numbers.") So the string "10000..." is different than the string "01111...".

There are several other discrepancies about the way it is taught, and almost all of the objections to it stem from one of them.

JeffJo
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