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NNTP-Posting-Date: Fri, 10 Jan 2025 02:15:12 +0000
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Newsgroups: sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Thu, 9 Jan 2025 18:15:18 -0800
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On 01/06/2025 09:46 PM, Ross Finlayson wrote:
> On 01/06/2025 05:36 PM, Ross Finlayson wrote:
>> On 01/06/2025 02:43 PM, Jim Burns wrote:
>>> On 1/5/2025 1:14 PM, WM wrote:
>>>> On 05.01.2025 19:03, joes wrote:
>>>>> Am Sun, 05 Jan 2025 12:14:47 +0100 schrieb WM:
>>>>>> On 04.01.2025 21:38, Chris M. Thomasson wrote:
>>>
>>>>>>> For me,
>>>>>>> there are infinitely many natural numbers, period...
>>>>>>> Do you totally disagree?
>>>>>>
>>>>>> No.
>>>>>> There are actually infinitely many natural numbers.
>>>>>> All can be removed from ℕ, but only collectively
>>>>>> ℕ \ {1, 2, 3, ...} = { }.
>>>>>> It is impossible to remove the numbers individually
>>>>>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
>>>>>
>>>>> Well yes,
>>>>> the size of N is itself
>>>>> not a natural number.
>>>>> Big surprise.
>>>>
>>>> ℕ cannot be covered by FISONs,
>>>> neither by many nor by their union.
>>>> If ℕ could be covered by FISONs
>>>> then one would be sufficient.
>>>
>>> ℕ is the set of finite.ordinals.
>>> ℕ holds each finite ordinal.
>>> ℕ holds only finite.ordinals.
>>>
>>> ⎛ A FISON is a set of finite.ordinals
>>> ⎝ up to that FISON's maximum (finite.ordinal) element.
>>>
>>> A finite.ordinal is an ordinal
>>> smaller.than fuller.by.one sets.
>>>
>>> Lemma 1.
>>> ⎛ For sets A∪{a} ≠ A and B∪{b} ≠ B
>>> ⎜⎛ if A is smaller.than B
>>> ⎜⎝ then A∪{a} is smaller.than B∪{b}
>>> ⎝ #A < #B  ⇒  #(A∪{a}) < #(B∪{b})
>>>
>>> Lemma 1
>>> is true for both the darkᵂᴹ and the visibleᵂᴹ.
>>>
>>> Consider finite.ordinal k.
>>> Finite: ⟦0,k⦆ is smaller.than ⟦0,k⦆∪⦃k⦄
>>>
>>> A = ⟦0,k⦆
>>> A∪{a} = ⟦0,k⦆∪⦃k⦄
>>> B = ⟦0,k⦆∪⦃k⦄ = ⟦0,k+1⦆
>>> B∪{b} = (⟦0,k⦆∪⦃k⦄)∪⦃k+1⦄ = ⟦0,k+1⦆∪⦃k+1⦄
>>>
>>> ⎛ By lemma 1
>>> ⎜ if ⟦0,k⦆ is smaller.than ⟦0,k+1⦆
>>> ⎜ then ⟦0,k⦆∪⦃k⦄ is smaller.than ⟦0,k+1⦆∪⦃k+1⦄
>>> ⎜
>>> ⎜ If
>>> ⎜ k is in ℕ and
>>> ⎜ k is finite and
>>> ⎜ ⟦0,k⦆ is smaller.than ⟦0,k⦆∪⦃k⦄
>>> ⎜ then
>>> ⎜ ⟦0,k+1⦆ is smaller.than ⟦0,k+1⦆∪⦃k+1⦄ and
>>> ⎜ k+1 is finite and
>>> ⎝ k+1 is in ℕ.
>>>
>>> k ∈ ℕ  ⇒  k+1 ∈ ℕ
>>> is true for both the darkᵂᴹ and the visibleᵂᴹ.
>>>
>>>> If ℕ could be covered by FISONs
>>>> then one would be sufficient.
>>>
>>> ℕ is the set of finite.ordinals.
>>>
>>> A FISON is a set of finite.ordinals
>>> up to that FISON's maximum (finite.ordinal) element.
>>>
>>> If one FISON covered ℕ,
>>> that FISON.cover would equal ℕ,
>>> and the maximum of that FISON.cover
>>> would be the maximum.of.all finite.ordinal.
>>>
>>> However,
>>> no finite.ordinal k is the maximum.of.all.
>>> k ∈ ℕ  ⇒  k+1 ∈ ℕ
>>> That is true for both the darkᵂᴹ and the visibleᵂᴹ.
>>>
>>> Contradiction.
>>> No one FISON covers ℕ.
>>>
>>>> ℕ cannot be covered by FISONs,
>>>> neither by many nor by their union.
>>>
>>> No.
>>>
>>> ℕ is the set of finite ordinals.
>>>
>>> Each finite.ordinal k is in
>>> at least one FISON: ⟦0,k⟧
>>>
>>> Each finite.ordinal is in
>>> the union of FISONs
>>>
>>> The union of FISONs covers
>>> the set ℕ of finite.ordinals
>>>
>>>> But for all we have:
>>>> Extension by 100 is insufficient.
>>>
>>> Correct.
>>> Which is weird, but accurate.
>>>
>>> The source of that weird result is lemma 1.
>>> ⎛ For sets A∪{a} ≠ A and B∪{b} ≠ B
>>> ⎜⎛ if A is smaller.than B
>>> ⎜⎝ then A∪{a} is smaller.than B∪{b}
>>> ⎝ #A < #B  ⇒  #(A∪{a}) < #(B∪{b})
>>>
>>> It would be great if you (WM) did NOT
>>> find lemma 1 weird,
>>> but it is what it is.
>>>
>>>
>>
>> But, if I said it was a waste of time,
>> wouldn't that be a waste of time?
>>
>>
>> The inductive set being covered by
>> initial segments is an _axiom_ of ZF.
>>
>> There are lesser theories where it's not
>> so, of course, why they added something
>> like "Infinity" as an _axiom_, vis-a-vis
>> the illative or univalent or infinite-union
>> which is _not_ an axiom, and furthermore
>> not by itself a theorem.
>>
>> So, ..., I suppose that's part of the
>> idea of the "Reverse Mathematics" program,
>> which is about theories with less axioms,
>> about what's so, and what's not so.
>>
>> Then, of course one can show that according
>> to pair-wise union is the _un-bounded_, then
>> as with regards to whether comprehension
>> brings the Russell Paradox on, on the way
>> from going from _fragments_ to _extensions_,
>> that is a simple result in, "set theory".
>>
>> ... That it's either not infinite or,
>> you know, not finite.
>>
>>
>>
>
> It's pretty simple,
> you've invoked Russell as your ruler,
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