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tech / sci.math / Re: Natural numbers and vases

SubjectAuthor
* Natural numbers and vasesWilliam
+- Re: Natural numbers and vasesRoss A. Finlayson
+* Re: Natural numbers and vasesWM
|+* Re: Natural numbers and vasesWilliam
||`* Re: Natural numbers and vasesWM
|| +- Re: Natural numbers and vasessergio
|| `* Re: Natural numbers and vasesWilliam
||  `* Re: Natural numbers and vasesWM
||   +- Re: Natural numbers and vasessergio
||   `* Re: Natural numbers and vasesWilliam
||    `* Re: Natural numbers and vasesWM
||     +- Re: Natural numbers and vasessergio
||     `* Re: Natural numbers and vasesWilliam
||      +- Re: Natural numbers and vasesFritz Feldhase
||      `* Re: Natural numbers and vasesWM
||       +* Re: Natural numbers and vasesWilliam
||       |+* Re: Natural numbers and vasesGus Gassmann
||       ||`* Re: Natural numbers and vasesFritz Feldhase
||       || `- Re: Natural numbers and vasesGus Gassmann
||       |`* Re: Natural numbers and vasesWM
||       | +- Re: Natural numbers and vasesGus Gassmann
||       | +* Re: Natural numbers and vasesWilliam
||       | |+* Re: Natural numbers and vasesWM
||       | ||+* Re: Natural numbers and vasesWilliam
||       | |||`* Re: Natural numbers and vasesWM
||       | ||| `* Re: Natural numbers and vasesWilliam
||       | |||  +* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |`- Re: Natural numbers and vasessergio
||       | |||  +* Re: Natural numbers and vasesGus Gassmann
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||+- Re: Natural numbers and vasesFromTheRafters
||       | |||  ||`- Re: Natural numbers and vasessergio
||       | |||  |+- Re: Natural numbers and vasesGus Gassmann
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||`- Re: Natural numbers and vasessergio
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesGus Gassmann
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||`- Re: Natural numbers and vasessergio
||       | |||  |+- Re: Natural numbers and vasesGus Gassmann
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||`- Re: Natural numbers and vasessergio
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesGus Gassmann
||       | |||  ||`- Re: Natural numbers and vasesChris M. Thomasson
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesGus Gassmann
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||+* Re: Natural numbers and vasesJim Burns
||       | |||  |||`* Re: Natural numbers and vasesWM
||       | |||  ||| +* Re: Natural numbers and vasesJim Burns
||       | |||  ||| |`* Re: Natural numbers and vasesWM
||       | |||  ||| | +* Re: Natural numbers and vasesJim Burns
||       | |||  ||| | |`* Re: Natural numbers and vasesWM
||       | |||  ||| | | +* Re: Natural numbers and vasesJim Burns
||       | |||  ||| | | |`* Re: Natural numbers and vasesWM
||       | |||  ||| | | | `- Re: Natural numbers and vasessergio
||       | |||  ||| | | `- Re: Natural numbers and vasessergio
||       | |||  ||| | `- Re: Natural numbers and vasessergio
||       | |||  ||| `- Re: Natural numbers and vasessergio
||       | |||  ||`* Re: Natural numbers and vasessergio
||       | |||  || `- Re: Natural numbers and vasesChet Hirasi
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||`- Re: Natural numbers and vasessergio
||       | |||  |+* Re: Natural numbers and vasesGus Gassmann
||       | |||  ||`- Re: Natural numbers and vasesChet Hirasi
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||`* Re: Natural numbers and vasessergio
||       | |||  || `* Re: Natural numbers and vasesWM
||       | |||  ||  `* Re: Natural numbers and vasessergio
||       | |||  ||   +* Re: Natural numbers and vasessergio
||       | |||  ||   |`* Re: Natural numbers and vasesFredJeffries
||       | |||  ||   | `- Re: Natural numbers and vasessergio
||       | |||  ||   `* Re: Natural numbers and vasesChris M. Thomasson
||       | |||  ||    `* Re: Natural numbers and vasessergio
||       | |||  ||     +* Re: Natural numbers and vasesChris M. Thomasson
||       | |||  ||     |`- Re: Natural numbers and vasessergio
||       | |||  ||     `* Re: Natural numbers and vasesChris M. Thomasson
||       | |||  ||      `* Re: Natural numbers and vasessergio
||       | |||  ||       `* Re: Natural numbers and vasesChris M. Thomasson
||       | |||  ||        +* Re: Natural numbers and vasesChris M. Thomasson
||       | |||  ||        |`- Re: Natural numbers and vasessergio
||       | |||  ||        `- Re: Natural numbers and vasessergio
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||`- Re: Natural numbers and vasessergio
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||`- Re: Natural numbers and vasessergio
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||+- Re: Natural numbers and vasessergio
||       | |||  ||`* Re: Natural numbers and vasesChris M. Thomasson
||       | |||  || +* Re: Natural numbers and vasesFromTheRafters
||       | |||  || `* Re: Natural numbers and vasessergio
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesGus Gassmann
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesGus Gassmann
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesGus Gassmann
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
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||       | |||  |+* Re: Natural numbers and vasesWM
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||       | |||  |+* Re: Natural numbers and vasesWM
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||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
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||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
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||       | |||  |+* Re: Natural numbers and vasesWM
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||       | |||  |+- Re: Natural numbers and vasesWM
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||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
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||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
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||       | |||  |+- Re: Natural numbers and vasesWM
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||       | |||  |+* Re: Natural numbers and vasesWM
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||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
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||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |`* Re: Natural numbers and vasesWM
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vaseszelos...@gmail.com
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vaseszelos...@gmail.com
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vaseszelos...@gmail.com
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vaseszelos...@gmail.com
||       | |||  +- Re: Natural numbers and vaseszelos...@gmail.com
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vaseszelos...@gmail.com
||       | |||  `- Re: Natural numbers and vaseszelos...@gmail.com
||       | ||`- Re: Natural numbers and vasessergio
||       | |`* Re: Natural numbers and vasesWM
||       | +- Re: Natural numbers and vasessergio
||       | `* Re: Natural numbers and vasesJim Burns
||       `- Re: Natural numbers and vasessergio
|+- Re: Natural numbers and vasesRoss A. Finlayson
|`- Re: Natural numbers and vasesFritz Feldhase
+* Re: Natural numbers and vasesChris M. Thomasson
+* Re: Natural numbers and vasesMostowski Collapse
`- Re: Natural numbers and vasesSocratis T.n.p.

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Natural numbers and vases

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Subject: Natural numbers and vases
From: wpihug...@gmail.com (William)
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 by: William - Sun, 1 May 2022 15:43 UTC

Let natural numbers.|N be {N_p N_d} with N_p a Peano set and N_p a set of "dark" elements. Start off with no elements in vase 1 and each element of |N in vase 2. At step n, we mover element n, from N_p in vase 2 to vase 1. There is a set n iff n is in N_p.. At every step n, vase 1 contains a finite set of elements and vase 2 contains an infinite subset of N_p and N_d. We do not need N_d to insure that vase 2 contains and infinite number of elements
Note we never touch N_d so each element of N_d remains in vase 2.

We now continue until each element of N_p is in vase 1, (This takes an infinite number of steps.) At this point vase 1 contains each element of N_p, thus as infinite number of elements and vase 2 contains the empty set (a subset of N_p) and N_d. Note that N_d just sits there. We do not need N_d for anything.

Take the axiom of infinity. The the natural numbers are a Peano set and as a Peano set cannot contain a "dark" element N_d must be empty. So |N=|N_p

William Hughes

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: ross.fin...@gmail.com (Ross A. Finlayson)
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 by: Ross A. Finlayson - Sun, 1 May 2022 16:59 UTC

On Sunday, May 1, 2022 at 8:44:05 AM UTC-7, William wrote:
> Let natural numbers.|N be {N_p N_d} with N_p a Peano set and N_p a set of "dark" elements. Start off with no elements in vase 1 and each element of |N in vase 2. At step n, we mover element n, from N_p in vase 2 to vase 1. There is a set n iff n is in N_p.. At every step n, vase 1 contains a finite set of elements and vase 2 contains an infinite subset of N_p and N_d. We do not need N_d to insure that vase 2 contains and infinite number of elements
> Note we never touch N_d so each element of N_d remains in vase 2.
>
> We now continue until each element of N_p is in vase 1, (This takes an infinite number of steps.) At this point vase 1 contains each element of N_p, thus as infinite number of elements and vase 2 contains the empty set (a subset of N_p) and N_d. Note that N_d just sits there. We do not need N_d for anything.
>
> Take the axiom of infinity. The the natural numbers are a Peano set and as a Peano set cannot contain a "dark" element N_d must be empty. So |N=|N_p
>
>
> William Hughes

So, what's the "supertask" that "takes infinite time"?

It seems that "in time" the supertask either "takes less time for each step
and in the limit sums to a finite time" or "no, never completes", or that
"there is a supertask for constant infinitesimals that results infinitely many
in 1.0s".

I.e. you seem to apply at least a model of real numbers, to time, to finish
in time (any "count" of time).

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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 by: WM - Sun, 1 May 2022 20:43 UTC

William schrieb am Sonntag, 1. Mai 2022 um 17:44:05 UTC+2:
> Let natural numbers.|N be {N_p N_d} with N_p a Peano set and N_d a set of "dark" elements. Start off with no elements in vase 1 and each element of |N in vase 2. At step n, we mover element n, from N_p in vase 2 to vase 1. There is a set n iff n is in N_p.. At every step n, vase 1 contains a finite set of elements and vase 2 contains an infinite subset of N_p and N_d. We do not need N_d to insure that vase 2 contains and infinite number of elements

Vase 2 contains an infinite number of elements as long as you remove only definable elements n. You will never remove more:
∀n ∈ ℕ_P: |ℕ \ {1, 2, 3, ..., n}| = ℵo (*)
But if you remove all elements collectively (which you fail to remove individually) then vase 2 is empty:
|ℕ \ {1, 2, 3, ...}| = 0 .

> Note we never touch N_d so each element of N_d remains in vase 2.

They cannot be touched individually.
>
> We now continue until each element of N_p is in vase 1, (This takes an infinite number of steps.)

Not even that would be sufficient because by definition (*). To remove all individually would require a last individual, and a next to the last and so on.

> At this point vase 1 contains each element of N_p, thus as infinite number of elements and vase 2 contains the empty set (a subset of N_p) and N_d. Note that N_d just sits there. We do not need N_d for anything.

We do need it for extending N_p as far as we like without losing the infinite buffer.
>
> Take the axiom of infinity. The the natural numbers are a Peano set and as a Peano set cannot contain a "dark" element N_d must be empty. So |N=|N_p
>
Every element of N_p can be removed individually. But there is no last one. And there is the condition (*) valid for every element of N_p.

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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 by: William - Mon, 2 May 2022 00:20 UTC

... a last individual, and a next to the last and so on.

"Let's start at the very beginning. A very good place to start"
There us a subset of N_p with no "last individual" (Follows trivially from the fact that N_p is a Peano set)

--
William Hughes

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: ross.fin...@gmail.com (Ross A. Finlayson)
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 by: Ross A. Finlayson - Mon, 2 May 2022 01:19 UTC

On Sunday, May 1, 2022 at 1:43:47 PM UTC-7, WM wrote:
> William schrieb am Sonntag, 1. Mai 2022 um 17:44:05 UTC+2:
> > Let natural numbers.|N be {N_p N_d} with N_p a Peano set and N_d a set of "dark" elements. Start off with no elements in vase 1 and each element of |N in vase 2. At step n, we mover element n, from N_p in vase 2 to vase 1.. There is a set n iff n is in N_p.. At every step n, vase 1 contains a finite set of elements and vase 2 contains an infinite subset of N_p and N_d. We do not need N_d to insure that vase 2 contains and infinite number of elements
>
> Vase 2 contains an infinite number of elements as long as you remove only definable elements n. You will never remove more:
> ∀n ∈ ℕ_P: |ℕ \ {1, 2, 3, ..., n}| = ℵo (*)
> But if you remove all elements collectively (which you fail to remove individually) then vase 2 is empty:
> |ℕ \ {1, 2, 3, ...}| = 0 .
> > Note we never touch N_d so each element of N_d remains in vase 2.
> They cannot be touched individually.
> >
> > We now continue until each element of N_p is in vase 1, (This takes an infinite number of steps.)
> Not even that would be sufficient because by definition (*). To remove all individually would require a last individual, and a next to the last and so on.
> > At this point vase 1 contains each element of N_p, thus as infinite number of elements and vase 2 contains the empty set (a subset of N_p) and N_d.. Note that N_d just sits there. We do not need N_d for anything.
> We do need it for extending N_p as far as we like without losing the infinite buffer.
> >
> > Take the axiom of infinity. The the natural numbers are a Peano set and as a Peano set cannot contain a "dark" element N_d must be empty. So |N=|N_p
> >
> Every element of N_p can be removed individually. But there is no last one. And there is the condition (*) valid for every element of N_p.
>
> Regards, WM

I try and keep it for the vase and balls, that,
"there is only picking out a random ball, or,
pouring out balls at random", that any "balls
put in the vase, with a label", don't necessarily
come out in the same order, out, the labels, in.

Then algorithms in random selection result
"to retrieve a ball from the vase, retrieve a
random ball until it is the given ball", not even
saying whether samples are _replaced_ in
terms of them being random samples, under access:
this is for example a usual data structure with "most
first" when it works out that membership test, is, for
"components: principal, major, minor, and the tail",
why random selection results as enforcing why algorithms
have their terms, the asymptotic, or exhaustive.

One might aver "the balls besides the vase,
are marbles in a bounds, that can be seen at once".

It's not necessary "real numbers" only limits or
exhaustion - what results why supertasks complete
as explaining "summed up the infinite parts, sum".

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: franz.fr...@gmail.com (Fritz Feldhase)
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 by: Fritz Feldhase - Mon, 2 May 2022 02:20 UTC

On Sunday, May 1, 2022 at 10:43:47 PM UTC+2, WM wrote:

> Vase 2 contains an infinite number of elements as long as you remove only [finitely many] elements[:]
>
> ∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo

Indeed!

Though the formalisation of your claim is not quite correct. You might adopt the following one:

∀n ∈ ℕ: |ℕ \ {k_1, k_2, k_3, ..., k_n}| = ℵo ,

where (k_i) is defined with k_i e IN for all i e IN.

Hint: infinite - finite = infinite .

> But if you remove all elements [...] then vase 2 is empty:
>
> |ℕ \ ℕ| = 0 .

Indeed!

Hint: For integers x: x - x = 0, for sets X: X \ X = {}.

Well done, Mückenheim!

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
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 by: WM - Mon, 2 May 2022 12:11 UTC

William schrieb am Montag, 2. Mai 2022 um 02:20:31 UTC+2:
> ... a last individual, and a next to the last and so on.
>
> "Let's start at the very beginning. A very good place to start"
> There us a subset of N_p with no "last individual" (Follows trivially from the fact that N_p is a Peano set)

How can that be? What would it imply?

No last number exists if beyond every number, which hitherto appears to be the last one, we can create new numbers.

But if we have a complete set of natnumbers n all of which are smaller than the limit omega, then we can calculate the difference omega - n. If this is infinite for every n, then we have an infinite gap between all n and omega. This gap cannot be bridged because every n has the same infinite distance.

On the other hand, this gap can be bridged. We know it from

ℕ \ {1, 2, 3, ...} = { } .

Regards, WM

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 by: sergio - Mon, 2 May 2022 13:55 UTC

On 5/2/2022 7:11 AM, WM wrote:
> William schrieb am Montag, 2. Mai 2022 um 02:20:31 UTC+2:
>> ... a last individual, and a next to the last and so on.
>>
>> "Let's start at the very beginning. A very good place to start"
>> There us a subset of N_p with no "last individual" (Follows trivially from the fact that N_p is a Peano set)
>
> How can that be? What would it imply?
>
> No last number exists if beyond every number, which hitherto appears to be the last one, we can create new numbers.
>
> But if we have a complete set of natnumbers n all of which are smaller than the limit omega, then we can calculate the difference omega - n. If this is infinite for every n, then we have an infinite gap between all n and omega. This gap cannot be bridged because every n has the same infinite distance.
>
> On the other hand, this gap can be bridged. We know it from
>
> ℕ \ {1, 2, 3, ...} = { } .
>
> Regards, WM

Whoa! "hitherto appears to be the lastist one"

WM has a real fear of being the Last One in line.

Perhaps he cuts in line to feel safer, to the detriment of n+1,n+2,... , which he turns his back on...

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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 by: William - Mon, 2 May 2022 17:29 UTC

On Monday, May 2, 2022 at 9:11:30 AM UTC-3, WM wrote:

> gap

Nope, There is no "gap". The fact that every element of N_p is followed by a subset of N_p with cardinality aleph_0 does not mean there is a "gap" between N_p and omega. Cardinality is not a metric.

--
William Hughes

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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 by: WM - Mon, 2 May 2022 20:14 UTC

William schrieb am Montag, 2. Mai 2022 um 19:29:53 UTC+2:
> On Monday, May 2, 2022 at 9:11:30 AM UTC-3, WM wrote:
>
> > gap
>
> Nope, There is no "gap".

If we have a complete set of natnumbers n all of which are smaller than the limit omega, then we can calculate the difference omega - n.

What do you not understand?

If this is infinite for every n, then we have an infinite gap between all n and omega.
This gap cannot be bridged because every n has the same infinite distance.

> The fact that every element of N_p is followed by a subset of N_p with cardinality aleph_0 does not mean there is a "gap" between N_p and omega.

No, but the fact that omega - n infinite proves that there is a gap.

> Cardinality is not a metric.

since cardinality is nonsense. But if omega exists and all n satisfy omega - n > 100, then there is a gap before omega.

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
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 by: sergio - Mon, 2 May 2022 20:46 UTC

On 5/2/2022 3:14 PM, WM wrote:
> William schrieb am Montag, 2. Mai 2022 um 19:29:53 UTC+2:
>> On Monday, May 2, 2022 at 9:11:30 AM UTC-3, WM wrote:
>>
>>> gap
>>
>> Nope, There is no "gap".
>
> If we have a complete set of natnumbers n all of which are smaller than the limit omega, then we can calculate the difference omega - n.

yes, like this omega - n = omega

>
> What do you not understand?

there is no gap

>
> If this is infinite for every n, then we have an infinite gap between all n and omega.

no, there is no gap. That is called an Endsegment

> This gap cannot be bridged because every n has the same infinite distance.

wrong. there is no gap, it is filled with natural numbers.

>
>> The fact that every element of N_p is followed by a subset of N_p with cardinality aleph_0 does not mean there is a "gap" between N_p and omega.
>
> No, but the fact that omega - n infinite proves that there is a gap.

Wrong, you misunderstand infinity.

>
>> Cardinality is not a metric.
>
> since cardinality is nonsense. But if omega exists and all n satisfy omega - n > 100, then there is a gap before omega.

still wrong, a lost little sheep that wants to be counted

>
> Regards, WM

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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 by: William - Mon, 2 May 2022 21:38 UTC

On Monday, May 2, 2022 at 5:15:01 PM UTC-3, WM wrote:
> William schrieb am Montag, 2. Mai 2022 um 19:29:53 UTC+2:
> > On Monday, May 2, 2022 at 9:11:30 AM UTC-3, WM wrote:
> >
> > > gap
> >
> > Nope, There is no "gap".
> If we have a complete set of natnumbers n all of which are smaller than the limit omega, then we can calculate the difference omega - n.
>

Piffle. The "difference" is a set difference not a distance (metric).

--
William Hughes

Re: Natural numbers and vases

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From: chris.m....@gmail.com (Chris M. Thomasson)
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Subject: Re: Natural numbers and vases
Date: Mon, 2 May 2022 15:14:46 -0700
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 by: Chris M. Thomasson - Mon, 2 May 2022 22:14 UTC

On 5/1/2022 8:43 AM, William wrote:
>
> Let natural numbers.|N be {N_p N_d} with N_p a Peano set and N_p a set of "dark" elements. Start off with no elements in vase 1 and each element of |N in vase 2. At step n, we mover element n, from N_p in vase 2 to vase 1. There is a set n iff n is in N_p.. At every step n, vase 1 contains a finite set of elements and vase 2 contains an infinite subset of N_p and N_d. We do not need N_d to insure that vase 2 contains and infinite number of elements
> Note we never touch N_d so each element of N_d remains in vase 2.
>
> We now continue until each element of N_p is in vase 1, (This takes an infinite number of steps.)

> At this point

Humm... If it takes an infinite number of steps to fill vase 1, then at
what point can you say its full?

> vase 1 contains each element of N_p, thus as infinite number of elements and vase 2 contains the empty set (a subset of N_p) and N_d. Note that N_d just sits there. We do not need N_d for > anything.
>
> Take the axiom of infinity. The the natural numbers are a Peano set and as a Peano set cannot contain a "dark" element N_d must be empty. So |N=|N_p
>
>
> William Hughes
>

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Subject: Re: Natural numbers and vases
From: franz.fr...@gmail.com (Fritz Feldhase)
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 by: Fritz Feldhase - Mon, 2 May 2022 23:43 UTC

On Tuesday, May 3, 2022 at 12:14:56 AM UTC+2, Chris M. Thomasson wrote:
> On 5/1/2022 8:43 AM, William wrote:
> >
> > Let natural numbers.|N be {N_p N_d} with N_p a Peano set and N_p a set of "dark" elements. Start off with no elements in vase 1 and each element of |N in vase 2. At step n, we mover element n, from N_p in vase 2 to vase 1.. There is a set n iff n is in N_p.. At every step n, vase 1 contains a finite set of elements and vase 2 contains an infinite subset of N_p and N_d. We do not need N_d to insure that vase 2 contains and infinite number of elements
> > Note we never touch N_d so each element of N_d remains in vase 2.
> >
> > We now continue until each element of N_p is in vase 1, (This takes an infinite number of steps.)
> >
> > At this point
> Humm... If it takes an infinite number of steps to fill vase 1, then at
> what point can you say its full?

I guess, he's considering a "super task". We start at t = 0. At t = 1 - 1/2 the first number, 1, is moved to vase 1. At t = 1 - 1/3 the second number, 2, is moved, at t = 1 - 1/4 the third number, 3, is moved and so on. Actually, it's easy to show that (in this case) for each and every number n there's a certain time t < 1, namely 1 - 1/(n + 1) , where n is moved to vase 1. Hence at t = 1 each and every number (in N_p) has been moved to vase 1.

So I'd say at t = 1 it's reasonable to claim that /now/ vase 1 is filled with all elements in N_p.

Yes, the process of "moving the numbers" does not have a last step. On the other hand, which number has n o t been moved to vase 1 at t = 1? :-P

For each and every t < 1 there are still infinitely many numbers which have to be moved to vase 1, but at t = 1 the job has been done. Such is the power of an "unlimited" speed up of a process.

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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 by: William - Tue, 3 May 2022 01:04 UTC

On Monday, May 2, 2022 at 7:14:56 PM UTC-3, Chris M. Thomasson wrote:
> On 5/1/2022 8:43 AM, William wrote:
> >
> > Let natural numbers.|N be {N_p N_d} with N_p a Peano set and N_p a set of "dark" elements. Start off with no elements in vase 1 and each element of |N in vase 2. At step n, we mover element n, from N_p in vase 2 to vase 1.. There is a set n iff n is in N_p.. At every step n, vase 1 contains a finite set of elements and vase 2 contains an infinite subset of N_p and N_d. We do not need N_d to insure that vase 2 contains and infinite number of elements
> > Note we never touch N_d so each element of N_d remains in vase 2.
> >
> > We now continue until each element of N_p is in vase 1, (This takes an infinite number of steps.)
>
>
> > At this point
> Humm... If it takes an infinite number of steps to fill vase 1, then at
> what point can you say its full?

There is no step at which the task is done (there is no step omega). We do not need to define a state for the vases after all steps are done. However, whether or not we define such a state it is clear that N_d remains in vase 2 (i,e, we do not need N_d). It remains true that N_p cannot contain a "dark" element so we never move a "dark" element from vase 2 to vase 1.

--
William Hughes

Re: Natural numbers and vases

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 by: Chris M. Thomasson - Tue, 3 May 2022 02:40 UTC

On 5/2/2022 4:43 PM, Fritz Feldhase wrote:
> On Tuesday, May 3, 2022 at 12:14:56 AM UTC+2, Chris M. Thomasson wrote:
>> On 5/1/2022 8:43 AM, William wrote:
>>>
>>> Let natural numbers.|N be {N_p N_d} with N_p a Peano set and N_p a set of "dark" elements. Start off with no elements in vase 1 and each element of |N in vase 2. At step n, we mover element n, from N_p in vase 2 to vase 1. There is a set n iff n is in N_p.. At every step n, vase 1 contains a finite set of elements and vase 2 contains an infinite subset of N_p and N_d. We do not need N_d to insure that vase 2 contains and infinite number of elements
>>> Note we never touch N_d so each element of N_d remains in vase 2.
>>>
>>> We now continue until each element of N_p is in vase 1, (This takes an infinite number of steps.)
>>>
>>> At this point
>> Humm... If it takes an infinite number of steps to fill vase 1, then at
>> what point can you say its full?
>
> I guess, he's considering a "super task". We start at t = 0. At t = 1 - 1/2 the first number, 1, is moved to vase 1. At t = 1 - 1/3 the second number, 2, is moved, at t = 1 - 1/4 the third number, 3, is moved and so on. Actually, it's easy to show that (in this case) for each and every number n there's a certain time t < 1, namely 1 - 1/(n + 1) , where n is moved to vase 1. Hence at t = 1 each and every number (in N_p) has been moved to vase 1.
>
> So I'd say at t = 1 it's reasonable to claim that /now/ vase 1 is filled with all elements in N_p.
>
> Yes, the process of "moving the numbers" does not have a last step. On the other hand, which number has n o t been moved to vase 1 at t = 1? :-P
>
> For each and every t < 1 there are still infinitely many numbers which have to be moved to vase 1, but at t = 1 the job has been done. Such is the power of an "unlimited" speed up of a process.

I see. It seems to be akin to me writing the word "pi". I mean pi in
_all_ of it's infinite glory; t = 1, 100% complete. Not, t = .001, or t
= .5... I mean pi as in t = 1. ;^)

pi means pi with infinite precision in a sense. Does that sound kosher
at all?

I suppose I can write something like pi(5) which means 5 digits of pi
where pi(5) = 3.1415, and p(1) = 3, ect.. ;^)

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 by: FromTheRafters - Tue, 3 May 2022 11:08 UTC

Chris M. Thomasson brought next idea :
> On 5/1/2022 8:43 AM, William wrote:
>>
>> Let natural numbers.|N be {N_p N_d} with N_p a Peano set and N_p a set of
>> "dark" elements. Start off with no elements in vase 1 and each element of
>> |N in vase 2. At step n, we mover element n, from N_p in vase 2 to vase 1.
>> There is a set n iff n is in N_p.. At every step n, vase 1 contains
>> a finite set of elements and vase 2 contains an infinite subset of N_p and
>> N_d. We do not need N_d to insure that vase 2 contains and infinite number
>> of elements
>> Note we never touch N_d so each element of N_d remains in vase 2.
>>
>> We now continue until each element of N_p is in vase 1, (This takes an
>> infinite number of steps.)
>
>
>> At this point
>
> Humm... If it takes an infinite number of steps to fill vase 1, then at what
> point can you say its full?

When it is *all* done. :)

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 by: sergio - Tue, 3 May 2022 13:43 UTC

On 5/2/2022 6:43 PM, Fritz Feldhase wrote:
> On Tuesday, May 3, 2022 at 12:14:56 AM UTC+2, Chris M. Thomasson wrote:
>> On 5/1/2022 8:43 AM, William wrote:
>>>
>>> Let natural numbers.|N be {N_p N_d} with N_p a Peano set and N_p a set of "dark" elements. Start off with no elements in vase 1 and each element of |N in vase 2. At step n, we mover element n, from N_p in vase 2 to vase 1. There is a set n iff n is in N_p.. At every step n, vase 1 contains a finite set of elements and vase 2 contains an infinite subset of N_p and N_d. We do not need N_d to insure that vase 2 contains and infinite number of elements
>>> Note we never touch N_d so each element of N_d remains in vase 2.
>>>
>>> We now continue until each element of N_p is in vase 1, (This takes an infinite number of steps.)
>>>
>>> At this point
>> Humm... If it takes an infinite number of steps to fill vase 1, then at
>> what point can you say its full?
>
> I guess, he's considering a "super task". We start at t = 0. At t = 1 - 1/2 the first number, 1, is moved to vase 1. At t = 1 - 1/3 the second number, 2, is moved, at t = 1 - 1/4 the third number, 3, is moved and so on. Actually, it's easy to show that (in this case) for each and every number n there's a certain time t < 1, namely 1 - 1/(n + 1) , where n is moved to vase 1. Hence at t = 1 each and every number (in N_p) has been moved to vase 1.
>
> So I'd say at t = 1 it's reasonable to claim that /now/ vase 1 is filled with all elements in N_p.
>
> Yes, the process of "moving the numbers" does not have a last step. On the other hand, which number has n o t been moved to vase 1 at t = 1? :-P

this points out some of the limitations of language (which is poor at describing science), there is a good chapter in "the nature of physical science"
on drawbacks of languages.

t<1 it's "moving"
t=>1 it's "moved"

if you stop at k, when t<1, like WM does, then it is "truncated" move.

>
> For each and every t < 1 there are still infinitely many numbers which have to be moved to vase 1, but at t = 1 the job has been done. Such is the power of an "unlimited" speed up of a process.

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Subject: Re: Natural numbers and vases
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 by: WM - Tue, 3 May 2022 13:47 UTC

William schrieb am Montag, 2. Mai 2022 um 23:38:48 UTC+2:
> On Monday, May 2, 2022 at 5:15:01 PM UTC-3, WM wrote:
> > William schrieb am Montag, 2. Mai 2022 um 19:29:53 UTC+2:
> > > On Monday, May 2, 2022 at 9:11:30 AM UTC-3, WM wrote:
> > >
> > > > gap
> > >
> > > Nope, There is no "gap".
> > If we have a complete set of natnumbers n all of which are smaller than the limit omega, then we can calculate the difference omega - n.
> >
> Piffle. The "difference" is a set difference not a distance (metric).
>
The difference either consists of natural numbers or it is a gap. Cantor: ω - n = ω.

Regards, WM

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Subject: Re: Natural numbers and vases
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 by: WM - Tue, 3 May 2022 13:49 UTC

William schrieb am Dienstag, 3. Mai 2022 um 03:04:55 UTC+2:
> On Monday, May 2, 2022 at 7:14:56 PM UTC-3, Chris M. Thomasson wrote:
> > On 5/1/2022 8:43 AM, William wrote:
> > >
> > > Let natural numbers.|N be {N_p N_d} with N_p a Peano set and N_p a set of "dark" elements. Start off with no elements in vase 1 and each element of |N in vase 2. At step n, we mover element n, from N_p in vase 2 to vase 1. There is a set n iff n is in N_p.. At every step n, vase 1 contains a finite set of elements and vase 2 contains an infinite subset of N_p and N_d.. We do not need N_d to insure that vase 2 contains and infinite number of elements
> > > Note we never touch N_d so each element of N_d remains in vase 2.
> > >
> > > We now continue until each element of N_p is in vase 1, (This takes an infinite number of steps.)
> >
> >
> > > At this point
> > Humm... If it takes an infinite number of steps to fill vase 1, then at
> > what point can you say its full?
> There is no step at which the task is done (there is no step omega). We do not need to define a state for the vases after all steps are done. However, whether or not we define such a state it is clear that N_d remains in vase 2 (i,e, we do not need N_d).

The difference ω - n = ω for every definablePeano n. Therefore the dark numbers are required.

Regards, WM

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 by: WM - Tue, 3 May 2022 13:51 UTC

FromTheRafters schrieb am Dienstag, 3. Mai 2022 um 13:08:28 UTC+2:

> >> At this point
> >
> > Humm... If it takes an infinite number of steps to fill vase 1, then at what
> > point can you say its full?
> When it is *all* done. :)

Then there remain ω - n = ω dark numbers nevertheless.

Regards, WM

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 by: sergio - Tue, 3 May 2022 14:00 UTC

On 5/3/2022 8:47 AM, WM wrote:
> William schrieb am Montag, 2. Mai 2022 um 23:38:48 UTC+2:
>> On Monday, May 2, 2022 at 5:15:01 PM UTC-3, WM wrote:
>>> William schrieb am Montag, 2. Mai 2022 um 19:29:53 UTC+2:
>>>> On Monday, May 2, 2022 at 9:11:30 AM UTC-3, WM wrote:
>>>>
>>>>> gap
>>>>
>>>> Nope, There is no "gap".
>>> If we have a complete set of natnumbers n all of which are smaller than the limit omega, then we can calculate the difference omega - n.
>>>
>> Piffle. The "difference" is a set difference not a distance (metric).
>>
> The difference either consists of natural numbers or it is a gap. Cantor: ω - n = ω.

no, the difference is natural numbers, because you stopped at k.

and no, you cannot "calculate" the difference, it is ω

>
> Regards, WM

WM, The Diversionator

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 by: sergio - Tue, 3 May 2022 14:29 UTC

On 5/3/2022 8:51 AM, WM wrote:
> FromTheRafters schrieb am Dienstag, 3. Mai 2022 um 13:08:28 UTC+2:
>
>>>> At this point
>>>
>>> Humm... If it takes an infinite number of steps to fill vase 1, then at what
>>> point can you say its full?
>> When it is *all* done. :)
>
> Then there remain ω - n = ω dark numbers nevertheless.
>
> Regards, WM

not at all, they are the numbers in vase 2, and they are not dark.

Just pour them into vase 1 and be done with it.

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 by: sergio - Tue, 3 May 2022 14:31 UTC

On 5/3/2022 8:49 AM, WM wrote:
> William schrieb am Dienstag, 3. Mai 2022 um 03:04:55 UTC+2:
>> On Monday, May 2, 2022 at 7:14:56 PM UTC-3, Chris M. Thomasson wrote:
>>> On 5/1/2022 8:43 AM, William wrote:
>>>>
>>>> Let natural numbers.|N be {N_p N_d} with N_p a Peano set and N_p a set of "dark" elements. Start off with no elements in vase 1 and each element of |N in vase 2. At step n, we mover element n, from N_p in vase 2 to vase 1. There is a set n iff n is in N_p.. At every step n, vase 1 contains a finite set of elements and vase 2 contains an infinite subset of N_p and N_d. We do not need N_d to insure that vase 2 contains and infinite number of elements
>>>> Note we never touch N_d so each element of N_d remains in vase 2.
>>>>
>>>> We now continue until each element of N_p is in vase 1, (This takes an infinite number of steps.)
>>>
>>>
>>>> At this point
>>> Humm... If it takes an infinite number of steps to fill vase 1, then at
>>> what point can you say its full?
>> There is no step at which the task is done (there is no step omega). We do not need to define a state for the vases after all steps are done. However, whether or not we define such a state it is clear that N_d remains in vase 2 (i,e, we do not need N_d).
>
> The difference ω - n = ω for every definablePeano n. Therefore the dark numbers are required.
>
> Regards, WM

so where are your dark numbers ? now you state there are Peano numbers ?

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
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 by: Mostowski Collapse - Tue, 3 May 2022 14:59 UTC

Do you mean an unordered pair {N_p N_d} or the
union N_p u N_d. These are two different things.
Otherwise I can recommend Dan Christensen,

he is quite a prolific dark matter researcher with
a fist full of logic, he first went after radonized water,
then snake oil, he is now into Half-functions

A Borjomi mineral water ad from 1929, advertising the water as "radioactive"
https://en.wikipedia.org/wiki/Radioactive_quackery

Do these Half-functions of Dan Christensen have
some Half-life time. Do they still exist after 10 years?

William schrieb am Sonntag, 1. Mai 2022 um 17:44:05 UTC+2:
> Let natural numbers.|N be {N_p N_d} with N_p a Peano set and N_p a set of "dark" elements.

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