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NNTP-Posting-Date: Fri, 11 Oct 2024 00:54:45 +0000
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (infinitary)
Newsgroups: sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Thu, 10 Oct 2024 17:54:45 -0700
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On 10/10/2024 01:47 PM, Jim Burns wrote:
> On 10/9/2024 11:39 AM, WM wrote:
>> On 09.10.2024 17:11, Alan Mackenzie wrote:
>>> WM <wolfgang.mueckenheim@tha.de> wrote:
>
>>>> No.
>>>> When we *in actual infinity*
>>>> multiply all |ℕ|natural numbers by 2,
>>>> then we keep |ℕ| numbers
>>>> but only half of them are smaller than ω,
>>>> i.e., are natural numbers.
>>>> The other half is larger than ω.
>
> You (WM) are treating ω as though it is (our) finite.
> ω is the first (our) transfinite ordinal: not finite.
>
>>> Ha ha ha ha! This is garbage.
>>> If you think doubling some numbers gives results
>>> which are "larger than ω"
>>> you'd better be prepared to give
>>> an example of such a number.
>>> But you're surely going to tell me that
>>> these are "dark numbers
>>
>> {1, 2, 3, ..., ω}*2 = {2, 4, 6, ..., ω*2} .
>
> Each γ≠0 preceding ω is (our) finite.
>
> Each γ≠0 preceding ω is predecessored  and
> each β≠0 preceding γ is predecessored.
>
> γ < ω  ⇔
> ⎛ for each β: 0 < β ≤ γ  ⇒
> ⎝ exists α: 0 ≤ α < β  ∧  α+1 = β
>
> γ < ω  ⇔
> ∀β ∈ ⦅0,γ⟧: ∃α ∈ ⟦0,β⦆: α+1=β
>
> ----
> ⎛ If
> ⎜  γ≠0 preceding ω is predecessored and
> ⎜  each β≠0 preceding γ is predecessored,
> ⎜ then
> ⎜  γ+1≠0 is predecessored and
> ⎜  each β≠0 preceding γ+1 is predecessored, and
> ⎝  γ+1 precedes ω
>
> Therefore,
> γ < ω  ⇒  γ+1 < ω
>
> ----
> ⎛ 0+γ = γ
> ⎝ (β+1)+γ = (β+γ)+1
>
> β < ω  ∧  γ < ω  ⇒  β+γ < ω
>
> ⎛ Assume a counterexample.
> ⎜ Assume
> ⎜ β < ω  ∧  γ < ω  ∧  β+γ ≥ ω
> ⎜
> ⎜ The nonempty set
> ⎜ {β < ω: γ < ω ∧ β+γ ≥ ω)
> ⎜ holds a minimum 𝔊+1 and
> ⎜ 𝔊+1 has a predecessor 𝔊 not.in the set.
> ⎜
> ⎜ 𝔊 < 𝔊+1 < ω  ∧  γ < ω
> ⎜ 𝔊+γ < ω  ∧  (𝔊+1)+γ ≥ ω
> ⎜
> ⎜ However,
> ⎜  γ < ω  ⇒  γ+1 < ω
> ⎜ 𝔊+γ < ω  ⇒  (𝔊+γ)+1 < ω
> ⎜ (𝔊+γ)+1 = (𝔊+1)+γ
> ⎜ (𝔊+1)+γ < ω
> ⎝ Contradiction.
>
> Therefore,
> β < ω  ∧  γ < ω  ⇒  β+γ < ω
>
> ----
> ⎛ 0×γ = 0
> ⎝ (β+1)×γ = (β×γ)+γ
>
> β < ω  ∧  γ < ω  ⇒  β×γ < ω
>
> ⎛ Assume a counterexample.
> ⎜ Assume
> ⎜ β < ω  ∧  γ < ω  ∧  β×γ ≥ ω
> ⎜
> ⎜ The nonempty set
> ⎜ {β < ω: γ < ω ∧ β×γ ≥ ω)
> ⎜ holds a minimum 𝔊+1 and
> ⎜ 𝔊+1 has a predecessor 𝔊 not.in the set.
> ⎜
> ⎜ 𝔊 < 𝔊+1 < ω  ∧  γ < ω
> ⎜ 𝔊×γ < ω  ∧  (𝔊+1)×γ ≥ ω
> ⎜
> ⎜ However,
> ⎜  β < ω  ∧  γ < ω  ⇒  β+γ < ω
> ⎜ 𝔊×γ < ω  ⇒  (𝔊×γ)+γ < ω
> ⎜ (𝔊×γ)+γ = (𝔊+1)×γ
> ⎜ (𝔊+1)×γ < ω
> ⎝ Contradiction.
>
> Therefore,
> β < ω  ∧  γ < ω  ⇒  β×γ < ω
>
>> {1, 2, 3, ..., ω}*2 = {2, 4, 6, ..., ω*2} .
>
> γ < ω  ⇒  γ+1 < ω
> There is no first successor out of ⟦0,ω⦆
> There is no successor out of ⟦0,ω⦆
>
> β < ω  ∧  γ < ω  ⇒  β+γ < ω
> There is no first sum out of ⟦0,ω⦆
> There is no sum out of ⟦0,ω⦆
>
> β < ω  ∧  γ < ω  ⇒  β×γ < ω
> There is no first product out of ⟦0,ω⦆
> There is no product out of ⟦0,ω⦆
>
> {1,2,3,...} ᵉᵃᶜʰ< ω
>
> {1,2,3,...}ᵉᵃᶜʰ×2  =  {2,4,6,...} ᵉᵃᶜʰ< ω
>
>> Should all places ω+2, ω+4, ω+6, ... remain empty?
>
> Should ω be (our) finite?
>
>> Should the even numbers in spite of doubling
>> remain below ω?
>> Then they must occupy places not existing before.
>> That means
>> the original set had not contained all natural numbers.
>
> ω is the first (our) transfinite ordinal.
> ∀γ:  γ ∈ ⟦0,ω⦆  ⇔
> ∀β ∈ ⦅0,γ⟧: ∃α ∈ ⟦0,β⦆: α+1=β
>
>


Halmos has for "infinite-dimensional vector spaces",
so not only is the Archimedean contrived either "potential"
or "un-bounded", so is the matter of the count of dimensions
and the schema or quantification or comprehension of the dimensions,
where there's a space like R^N in effect, or R^w, then
for a usual sort of idea that "the first transfinite ordinal"
is only kind of after all those, ...,
like a "spiral space-filling curve".


Otherwise there's your Boolos it's a standard 20'th century
text on non-standard models of integers, starting to set up
book-keeping, then that sometimes framed in the Goedel/Kolmogorov,
or, what's called "Goedel Delta" or what it is, "sizes", of
combinatorial classes, after modularity, when simple density
seems only relevant to number theorists yet set theorists
end up "counting" the same things.


Anyways though a "spiral space-filling curve" is very much
kind of like DesCartes' or Kelvin's vortices as well it is
its own sort of continuum after line-drawing out through
infinite dimensions, in as with regards to whether an
integer lattice results multiple dimensions in the integer
lattice, or for example line-reals, in one dimension, or
for example field-reals, in one dimension, or as with
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