Re: 2N=E

On 11/2/2024 1:24 PM, WM wrote:

On 02.11.2024 13:20, joes wrote:

Why "absorbed"?
Do you think
some multiple of a power of 2 is not natural?

>
If all multiples of 2 smaller than ω are doubled,
then this doubling results in larger numbers than doubled.

If n is finite  ∧  ω ≤ n+n
then ω is finite.
⎛ Assume n is finite
⎜ ⟦0,n⟧ is finite
⎜ ⟦n,n+n⟧ = n+ᵉᵃᶜʰ ⟦0,n⟧
⎜ ⟦0,n+n⟧ = ⟦0,n⟧∪⟦n,n+n⟧
⎜
⎜ Assume ω ≤ n+n
⎜ ω ∈ ⟦0,n+n⟧
⎜
⎜ ⟦0,n⟧ is finite, which means that
⎜ each subset of ⟦0,n⟧ is two.ended.or.{}
⎜
⎜ Each subset of ⟦n,n+n⟧ is two.ended.or.{}
⎜ Each subset of ⟦0,n+n⟧ is two.ended.or.{}
⎜ ⟦0,n+n⟧ is finite.
⎜
⎜ ω ∈ ⟦0,n+n⟧
⎜ Each subset of ⟦0,ω⟧ is
⎜ a subset of ⟦0,n+n⟧
⎜ and, thus, that subset is two.ended.or.{}
⎜
⎜ ⟦0,ω⟧ is finite.
⎝ ω is finite.
Therefore,
if n is finite  ∧  ω ≤ n+n
then ω is finite.
'Finite' means
each subset is two.ended.or.{}
'Infinite' means not finite.