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On 7/31/24 10:27 AM, WM wrote:That is simply nonsense. Do you know what an accumalation point is? Every eps interval around 0 contains unit fractions which cannot be separated from 0 by any eps. Therefore your claim is wrong.Le 31/07/2024 à 03:28, Richard Damon a écrit :In other words, outside the Natural Nubmer, all of which are defined and definable.On 7/30/24 1:37 PM, WM wrote:In the midst, far beyond all definable numbers, far beyond ω/10^10.Le 30/07/2024 à 03:18, Richard Damon a écrit :>On 7/29/24 9:11 AM, WM wrote:>>But what number became ω when doubled?
ω/2
And where is that in {1, 2, 3, ... w} ?
They are natural numbers.They may be "dark" but they are not Natural Numbers.>ω/10^10 and ω/10 are dark natural numbers.
The input set was the Natural Numbers and w,
Natural numbers, by their definition, are reachable by a finite number of successor operations from 0.That is the opinion of Peano and his disciples. It holds only for potetial infinity, i.e., definable numbers.
What is the reason for the gap before omega? How large is it? Are these questions a blasphemy?I assume completness.I guess you definition of "completeness" is incorrect.
If I take the set of all cats, and the set of all doges, can there not be a gap between them?
That does ny formula not say. It says for all n which have successors, there is distance between 1/n and 1/(n+1).Right, for ALL n in ℕ, there exist another number in ℕ that is n+1,No. My formula says ∀n ∈ ℕ.∀n ∈ ℕ: 1/n - 1/(n+1) > 0. Note the universal quantifier.>
Right, so we can say that ∀n ∈ ℕ: 1/n > 1/(n+1), so that for every unit fraction 1/n, there exists another unit fraction smaller than itself.
It is the definition of definable numbers. Study the accumulation point. Define (separate by an eps from 0) all unit fractions. Fail.Maybe not for dark numbers, but it does for all Natural Numbers, as that is part of their DEFINITION.>Not for all dark numbers.
Remember, one property of Natural numbers that ∀n ∈ ℕ: n+1 exists.
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