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

On 20.10.2024 21:42, Jim Burns wrote:

On 10/20/2024 3:24 PM, WM wrote:

On 20.10.2024 20:20, Jim Burns wrote:

On 10/20/2024 3:48 AM, WM wrote:

On 20.10.2024 00:54, Jim Burns wrote:

On 10/19/2024 2:19 PM, WM wrote:

 

A doubled finite is finite.

>
If all finites are doubled,
then not all results can be in that set.

>
If all finites are doubled,
then all results are in the set of finites.

>
But not in the mapped or multiplied range.

 If n is countable.to from 0
then ⟨0,1,...,n-1,n⟩ exists
 If ⟨0,1,...,n-1,n⟩ exists
then ⟨n,n+1,...,2⋅n-1,2⋅n⟩ exists
 If ⟨0,1,...,n-1,n⟩ and ⟨n,n+1,...,2⋅n-1,2⋅n⟩ exist
then ⟨0,1,...,n-1,n,n+1,...,2⋅n-1,2⋅n⟩ exists
 If ⟨0,1,...,n-1,n,n+1,...,2⋅n-1,2⋅n⟩ exists
then 2⋅n is countable.to from 0
 ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
If n is countable.to from 0
then 2⋅n is countable.to from 0
 

If all finites are doubled,
then all results are in the set of finites.

>
But not in the mapped or multiplied range.

 If n is countable.to from 0
then 2⋅n is countable.to from 0

>
Nevertheless 2n is not in the set {1, ..., n}.