Re: because g⤨(g⁻¹(x)) = g(y) [1/2] Re: how

On 5/10/2024 1:31 PM, Chris M. Thomasson wrote:

On 5/10/2024 5:18 AM, WM wrote:

Le 08/05/2024 à 23:55, Jim Burns a écrit :

On 5/8/2024 3:55 PM, WM wrote:

Le 07/05/2024 à 00:11, Jim Burns a écrit :

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All which canNOT be counted.to are not.in ℕ

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All which canNOT be counted.to are not.in ℕ_def.

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And all which CAN be counted.to are in ℕ_def.

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Yes. But every n ∈ ℕ_def has ℵ₀ successors which never vanish by counting. They can be removed only collectively such that nothing of ℕ remains.

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ℕ_def is the set of all and only
numbers which CAN be counted.to.
ℕ_def is what everyone else calls ℕ

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Because almost everyone has not yet realized, that ℕ contains also the natural numbers which cannot be counted to and which do not leave ℵ₀ successors after being removed from ℝ.

 Huh? ℕ is the natural numbers, all of them. What do you mean by ℕ _also_ contains the naturals? ℕ cannot represent say, 1.12542 because its a real number... What is wrong with you!

I need to clarify ℕ is a subset of the reals. So, ℕ and the reals have 1, 2, 3, ect... However, ℕ does not have 666.666...
Any real can be converted to ℕ by chopping off its fractional part, say, floor and ceil functions.
So, ceil(666.666...) = 667 and floor(666.666...) = 666.
;^)

 Btw, when I say ℕ is all of the naturals does not imply that there is some magical largest natural.
 

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|ℕ_def| = ℵ₀

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ℕ_def is a potentially infinite collection and as such has no fxed number of elements. We use the indefinite oo in this case.
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Regards, WM