Are you aware of the fact that in

{1}
{1, 2}
{1, 2, 3}
...
{1, 2, 3, ..., n}
...

up to every n infinitely many natural numbers of the whole set

{1, 2, 3, ...}

are missing? Infinitely many of them will never be mentioned
individually. They are dark.

Regards, WM

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On 17.05.2025 19:20, Alan Mackenzie wrote:

WM <[email protected]> wrote:

Are you aware of the fact that in

{1}
{1, 2}
{1, 2, 3}
...
{1, 2, 3, ..., n}
...

up to every n infinitely many natural numbers of the whole set

{1, 2, 3, ...}

are missing? Infinitely many of them will never be mentioned
individually. They are dark.

<Yawn>

Exciting. Many readers claim(ed) that all natural numbers could be used
as individuals. Further this would be a precondition for countability of infinite sets.

Regards, WM

Regards, WM

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WM <[email protected]> wrote:

Are you aware of the fact that in

{1}
{1, 2}
{1, 2, 3}
...
{1, 2, 3, ..., n}
...

up to every n infinitely many natural numbers of the whole set

{1, 2, 3, ...}

are missing? Infinitely many of them will never be mentioned
individually. They are dark.

<Yawn>

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).

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On 18.05.2025 08:27, Chris M. Thomasson wrote:

On 5/17/2025 11:20 PM, Chris M. Thomasson wrote:

On 5/17/2025 12:06 PM, FromTheRafters wrote:

After serious thinking Chris M. Thomasson wrote :

On 5/17/2025 10:35 AM, WM wrote:

Exciting. Many readers claim(ed) that all natural numbers could be
used as individuals. Further this would be a precondition for
countability of infinite sets.

Show me a dark natural number?

Take the greatest number that you can express. All greater numbers are
dark yet. Double your greatest number and express the result. Then you
see a hitherto dark number. Of course it is no longer dark. But
infinitely many numbers remain dark

We are building a natural number digit by digit using random rolls,
the first roll needs to be higher that zero... Fair enough? They will
all be natural numbers, right?

Of course. All will be natural numbers. It is a potentially infinite
set, {1, 2, 3, ..., n}, always finite but without a upper bound,
followed by an infinite set of dark numbers, infinitely many of which
will remain dark forever.

So, how could my process "break" when the natural numbers are infinite
any at any step of the process,

Your process will not break. One after one the dark numbers will become visible. Nevertheless almost all natural numbers will remain dark. The
stock is incredibly large. There are numbers like ω/2 or ω/10 which you
will never touch. For every defined n ∈ ℕ: ω/n is larger than you will every reach, how long ever you will increase your visible numbers.
Compared to ω the defined numbers are infinitesimal.

Regards, WM

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On 19.05.2025 01:05, Chris M. Thomasson wrote:

On 5/18/2025 8:30 AM, WM wrote:

Your process will not break. One after one the dark numbers will
become visible. Nevertheless almost all natural numbers will remain
dark. The stock is incredibly large. There are numbers like ω/2 or
ω/10 which you will never touch. For every defined n ∈ ℕ: ω/n is
larger than you will every reach, how long ever you will increase your
visible numbers. Compared to ω the defined numbers are infinitesimal.

So, you say wrt a little kid in the womb, well, perhaps all numbers are
dark?

Yes.

Regards, WM

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On 23.05.2025 01:46, Chris M. Thomasson wrote:

On 5/19/2025 12:22 PM, WM wrote:

On 19.05.2025 01:05, Chris M. Thomasson wrote:

On 5/18/2025 8:30 AM, WM wrote:

Your process will not break. One after one the dark numbers will
become visible. Nevertheless almost all natural numbers will remain
dark. The stock is incredibly large. There are numbers like ω/2 or
ω/10 which you will never touch. For every defined n ∈ ℕ: ω/n is >>>> larger than you will every reach, how long ever you will increase
your visible numbers. Compared to ω the defined numbers are
infinitesimal.

So, you say wrt a little kid in the womb, well, perhaps all numbers
are dark?

Yes.

Even if its mother says one, two, three, oh crap I have to pee? So,
perhaps, just perhaps, the entity building in her womb can hear things
via sound vibrations, and or other "mystery" things? Strange!

For the start use simpler cases like these: The pocket calculator is
limited to decimal representations below 10^100, the universe is limited
to more or less sophisticated formulas requiring less than 10^80 bit.

In every system almost all natural numbers are and remain dark - if an
actual infinity of them exists.

Regards, WM

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On 23.05.2025 22:00, Chris M. Thomasson wrote:

On 5/23/2025 1:34 AM, WM wrote:

In every system almost all natural numbers are and remain dark - if an
actual infinity of them exists.

Sounds like a conflation between real life and math?

Here is a proof, pure mathematics:
{1} has infinitely many (ℵo) successors.
If {1, 2, 3, ..., n} has infinitely many (ℵo) successors, then {1, 2, 3,
..., n, n+1} has infinitely many (ℵo) successors. For every n that can
be defined.

Regards, WM

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On 24.05.2025 17:03, Ross Finlayson wrote:

On 05/24/2025 04:44 AM, WM wrote:

Here is a proof, pure mathematics:
{1} has infinitely many (ℵo) successors.
If {1, 2, 3, ..., n} has infinitely many (ℵo) successors, then {1, 2, 3, >> ..., n, n+1} has infinitely many (ℵo) successors. For every n that can
be defined.

No, that's not relevance logic, you have just stipulated.

I have proved by induction. One of the basic proof methods for infinite
sets.

I suppose you're saying infinite sets are inexhaustible
by finite induction, ..., which is a very classical consideration.

It is obvious, but set theorists claim that every natural number can be
chosen as an individual. That is wrong.

Also you'd want to pick another word since "dark" gets
involved in ethnic discrimination,

That is not a discrimination but a description of reality. There is no
reason to avoid words like white, black, yellow, red or brown.

Regards, WM

Regards, WM

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