Re: The non-existence of "dark numbers" [was: The existence of dark numbers proven by the thinned out harmonic series]

On 3/16/2025 6:01 AM, WM wrote:

On 16.03.2025 00:42, Jim Burns wrote:

On 3/15/2025 5:58 PM, WM wrote:

On 15.03.2025 21:14, Jim Burns wrote:

There is a very old anecdote told in which
Galileo Galilei tries to show the moons of Jupiter
to a representative of the pope.
The representative refuses to look at
anything his religion says can't exist.

>
That should not seduce anyone to
spend his time with nonsense.
Lossless exchange is lossless for every exchange,
independent of the number of exchanges.

>
Define an ovine set to be a set for which
fuller.by.one sets exist which are larger.

>
All infinitely many lossless exchanges
have finite indices.

What do 'finite' and 'infinite' mean to you (WM)?
----
Define an ovine set
to not.have same.sized proper.subsets.
For each ovine set,
the set of ovine.set.sizes has a larger subset.
No set has a larger subset.
For each ovine set,
the set of ovine.set.sizes isn't that set.
The set of ovine.set.sizes isn't any ovine set.
The set of ovine.set.sizes
doesn't not.have same.sized proper.subsets.
All size.preserving exchanges are size.preserving.
Exchanging a superovine set for
a same.sized proper.subset is size.preserving.

Which index has the first lossless exchange
where a loss happens
(by the way one of infinitely many which
are required to empty the matrix of O's)?

What do 'finite' and 'infinite' mean to you (WM)?