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NNTP-Posting-Date: Tue, 07 Jan 2025 05:46:26 +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: Mon, 6 Jan 2025 21:46:26 -0800
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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,
others don't, and, there's a land
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