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July 18th, 2017, 05:22 AM   #41
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Two binary-digit natural numbers:
00 -> 0
01 -> 1
10 -> 2
11 -> 3

Three binary-digit natural numbers:
000 -> 0
001 -> 1
010 -> 2
011 -> 3
100 -> 4
101 -> 5
110 -> 6
111 -> 7

Infinite binary-digit natural numbers (Infinite Binary Sequences):
....00000 -> 0
....00001 -> 1
....00010 -> 2
....00011 -> 3
....00100 -> 4
....00101 -> 5
....00110 -> 6
....00111 -> 7
............
............
Preceding zeros conventionally not displayed or indicated.
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July 18th, 2017, 05:29 AM   #42
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Quote:
Originally Posted by zylo View Post
Two binary-digit natural numbers:
00 -> 0
01 -> 1
10 -> 2
11 -> 3

Three binary-digit natural numbers:
000 -> 0
001 -> 1
010 -> 2
011 -> 3
100 -> 4
101 -> 5
110 -> 6
111 -> 7

Infinite binary-digit natural numbers (Infinite Binary Sequences):
....00000 -> 0
....00001 -> 1
....00010 -> 2
....00011 -> 3
....00100 -> 4
....00101 -> 5
....00110 -> 6
....00111 -> 7
............
............
Preceding zeros conventionally not displayed or indicated.
Cool, so what is the decimal notation of ...1111111 ?
And what would be ...111111 + 1?
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July 18th, 2017, 05:30 AM   #43
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Quote:
Originally Posted by zylo View Post
Two binary-digit natural numbers:
00 -> 0
01 -> 1
10 -> 2
11 -> 3

Three binary-digit natural numbers:
000 -> 0
001 -> 1
010 -> 2
011 -> 3
100 -> 4
101 -> 5
110 -> 6
111 -> 7

Infinite binary-digit natural numbers (Infinite Binary Sequences):
....00000 -> 0
....00001 -> 1
....00010 -> 2
....00011 -> 3
....00100 -> 4
....00101 -> 5
....00110 -> 6
....00111 -> 7
............
............
Preceding zeros conventionally not displayed or indicated.
Also, are you not going to provide the definitions that I asked for?
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July 18th, 2017, 08:59 AM   #44
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Quote:
Originally Posted by zylo View Post

Infinite binary-digit natural numbers (Infinite Binary Sequences):
....00000 -> 0
....00001 -> 1
....00010 -> 2
....00011 -> 3
....00100 -> 4
....00101 -> 5
....00110 -> 6
....00111 -> 7
............
............
Preceding zeros conventionally not displayed or indicated.
All of the infinite binary sequences in your list contain a finite number of 1's. No sequences containing an infinite number of 1's are included. Therefore, you've only managed to produce a bijection between a small subset of the infinite binary sequences and the natural numbers.

I realize you think that all of the infinite binary sequences appear in your list, as that is what you assert time and time again. You are flapping your arms as though you are convinced you can fly. Move on.
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July 18th, 2017, 09:55 AM   #45
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If you write the "infinite" sequences that way round, zylo, CDA is still applicable if you start the diagonal at the top right instead of the top left.
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July 18th, 2017, 04:11 PM   #46
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Since Zylo believes that his list goes
$$\begin{gather}000\ldots000 \\ 000\ldots001 \\ 000\ldots010 \\ \vdots \\111\ldots111\end{gather}$$
This means that the first few have finite quantities of $1$s and the last few have infinite quantities of $1$s. A pertinent question is: when do we make the transition from finite $1$s to infinite $1$s?
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