>>332
Thank you for joining my friend.
What brings you to mathchan?
How can it grow?

>gaps between two closest points are small
Hackenbush is very interesting.
https://www.youtube.com/watch?v=ZYj4NkeGPdM&t=1260
https://www.goodreads.com/book/show/1293306.Winning_Ways_for_Your_Mathematical_Plays

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>limits
\math{|\mathbb{R}| = |\mathbb{Z}|}
>muh \textbf{diagonal} argument
Invalid. Consider the following countably infinite list:
\code{.
  First Number: 0
 Second Number: 0.1
  Third Number: 0.11
 Fourth Number: 0.111
  Fifth Number: 0.1111
  Sixth Number: 0.11111
Seventh Number: 0.111111
 Eighth Number: 0.1111111
  Ninth Number: 0.11111111
  Tenth Number: 0.111111111
  (... continues ...)
}
\textbf{Question}: Is the following number in the list?
\math{9^{-1}}
\textit{i.e.} \math{\frac{1}{9}}
\textit{i.e.} \math{0.\overline{1}}
\textit{i.e.} \math{0.111111111111\ldots}

The \textbf{diagonal} argument claims \math{0.\overline{1}} isn't in the list.
\textit{(Because \math{0.\overline{1}} differs from the first number in the tenths digit, the second number in the hundredths digit, the third number in the thousandths digit, the fourth number in the ten-thousandths digit, the fifth number in the hundred-thousandths digit, and this will continue forever, then allegedly \math{0.\overline{1}} is not in the list.}

But obviously, \math{0.\overline{1}} is in the list.
The list is directly constructed so as to contain \math{0.\overline{1}}.
>but muh infinite digits
The decimal number \math{0.\overline{1}} contains \textbf{countably infinite} digits.
The list is a \textbf{countably infinite} list.

Contradiction. Therefore the \textbf{diagonal} argument is invalid.
P.S. For the curious, the countably infinite list which contains all real numbers is simply
\code{
0    , 0.1  , 0.2  , 0.3  , 0.4  , 0.5  , 0.6  , 0.7  , 0.8  , 0.9  ,
0.01 , 0.11 , 0.21 , 0.31 , 0.41 , 0.51 , 0.61 , 0.71 , 0.81 , 0.91 ,
0.02 , 0.12 , 0.22 , 0.32 , 0.42 , 0.52 , 0.62 , 0.72 , 0.82 , 0.92 ,
0.03 , 0.13 , 0.23 , 0.33 , 0.43 , 0.53 , 0.63 , 0.73 , 0.83 , 0.93 ,
0.04 , 0.14 , 0.24 , 0.34 , 0.44 , 0.54 , 0.64 , 0.74 , 0.84 , 0.94 ,
0.05 , 0.15 , 0.25 , 0.35 , 0.45 , 0.55 , 0.65 , 0.75 , 0.85 , 0.95 ,
0.06 , 0.16 , 0.26 , 0.36 , 0.46 , 0.56 , 0.66 , 0.76 , 0.86 , 0.96 ,
0.07 , 0.17 , 0.27 , 0.37 , 0.47 , 0.57 , 0.67 , 0.77 , 0.87 , 0.97 ,
0.08 , 0.18 , 0.28 , 0.38 , 0.48 , 0.58 , 0.68 , 0.78 , 0.88 , 0.98 ,
0.09 , 0.19 , 0.29 , 0.39 , 0.49 , 0.59 , 0.69 , 0.79 , 0.89 , 0.99 ,
0.001, 0.101, 0.201, 0.301, 0.401, 0.501, 0.601, 0.701, 0.801, 0.901,
0.011, 0.111, 0.211, ...
}