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File: distracted-math.jpg ( 39.64 KB , 854x480 , 1667702420031.jpg )

Image
  No. 329
I've tried writing longform posts but don't get much traction. Let's try a shortform post.
>>
  No. 332 >>333
I had a thought that continuous sets are not real, but they are limits of discrete sets such as that gaps between two closest points are small, but we don't really know (or don't want to know) how small they are.
>>
  No. 333 >>336 >>340

File: hackenbush-1.png ( 6.28 KB , 229x220 , 1668744680246.png )

Image
>>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

-

>limits
∣R∣=∣Z∣|\mathbb{R}| = |\mathbb{Z}|

>muh diagonal argument
Invalid. Consider the following countably infinite list: 
.
  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 ...)

Question: Is the following number in the list?
9−19^{-1}

i.e.
19\frac{1}{9}

i.e.
0.1‾0.\overline{1}

i.e.
0.111111111111…0.111111111111\ldots


The diagonal argument claims
0.1‾0.\overline{1}
isn't in the list.
(Because
0.1‾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
0.1‾0.\overline{1}
is not in the list.

But obviously,
0.1‾0.\overline{1}
is in the list.
The list is directly constructed so as to contain
0.1‾0.\overline{1}
.
>but muh infinite digits
The decimal number
0.1‾0.\overline{1}
contains countably infinite digits.
The list is a countably infinite list.

Contradiction. Therefore the diagonal argument is invalid.
P.S. For the curious, the countably infinite list which contains all real numbers is simply 
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, ...
>>
  No. 336
>>333
>
∣R∣=∣Z∣|\mathbb{R}| = |\mathbb{Z}|

I didn't really mean that real numbers are badly contructed, by mathematicians, but the world is more likely based on discrete numbers than on "real" numbers and the latter is a formal contruction where distance between numbers is "really infinitely" small.

You seem to use axiom or principle that you can count from 0.1 to 0.(1). I won't probably disprove or confirm your theorem because it is not my focus in my journey through the land of mathematics. xp I only had thought that you can add from 1 to
+∞+\infty
with it, but it's not limit of adding smaller values so they en up in a fixed value.
>>
  No. 340
>>333
>contains countably infinite digits
You're confusing the cardinality of sets with natural numbers.
Let your set of 0.1* numbers be S.
S≅N S \cong \mathbb{N}
by the counting the ones in each decimal number.
0.1‾ 0.\overline{1}
has
ℵ0 \aleph_0
digits, and
ℵ0∉N \aleph_0 \notin \mathbb{N}
, therefore
0.1‾∉S 0.\overline{1} \notin S
.
Just because something is countably infinite, doesn't mean you can reach it by counting.
A more simple refutation would be so say that by nature of the construction of S, every number in it that is not the first has a single unique predecessor that you can follow back to 0.
The predecessor of
0.1‾ 0.\overline{1}
would be
0.1‾ 0.\overline{1}
, which clearly puts it ouside S.
This is a lot more intuitive if you imagine a graph of S with each number connected to the next.
This forms a straight line stretching from 0 outwards.
0.1‾ 0.\overline{1}
is in a single node graph who's only edge is to itself, it clearly can not connect to the line.
>>
  No. 341
test

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