Re: How many different unit fractions are lessorequal than all unit fractions? (infinitary)

On 10/14/2024 10:43 AM, WM wrote:

On 13.10.2024 18:28, Jim Burns wrote:

On 10/12/2024 2:06 PM, WM wrote:

On 10.10.2024 22:47, Jim Burns wrote:

On 10/9/2024 11:39 AM, WM wrote:

 

{1, 2, 3, ..., ω}*2 = {2, 4, 6, ..., ω*2} .

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Each γ≠0 preceding ω is (our) finite.

>
There are two alternatives:
Either doubling creates natnumbers,
  then they are not among those doubled,
  then we have not doubled all.
Or we have doubled all
  but then larger numbers have been created.

>
There are two alternatives:
Either 𝔊 exists such that 2⋅𝔊 < ω ≤ 2⋅(𝔊+1)
Or {2,4,6,...} ᵉᵃᶜʰ< ω
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⎛ Assume 2⋅𝔊 < ω ≤ 2⋅(𝔊+1)
⎜
⎜ 2⋅𝔊 < ω
⎜ For each j such that 0 < j ≤ 2⋅𝔊
⎜  j-1 exists.
⎜ 2⋅𝔊 = (2⋅𝔊+1)-1 exists
⎜ 2⋅𝔊+1 = (2⋅𝔊+2)-1 exists
⎜ 2⋅𝔊+2 = 2⋅(𝔊+1)

>
Correct.
>

⎜ For each j such that 0 < j ≤ 2⋅(𝔊+1)
⎜  j-1 exists. ⎜ 2⋅(𝔊+1) < ω

>
Mistake.

 ω is the first not.finite ordinal.
Each finite ordinal is before ω

 If 2⋅𝔊 is countable.to from 0
then 2⋅(𝔊+1) is countable.to from 0

 If 2⋅𝔊 is finite
then 2⋅(𝔊+1) is finite

 If 2⋅𝔊 < ω
then 2⋅(𝔊+1) < ω