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

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} .
 >>>
 >>> 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
 >
 > ⎛ 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.
 > ⎜ 2⋅(𝔊+1) < ω
Mistake.
 > ⎜ However,
 > ⎜ ω ≤ 2⋅(𝔊+1)
 > ⎝ Contradiction.
No, your mistake.
Regards, WM