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

On 02.11.2024 19:45, Jim Burns wrote:

On 11/2/2024 6:15 AM, WM wrote:

On 02.11.2024 01:55, Richard Damon wrote:

 

Infinite sets do not need to have both ends in them.

>
Between 0 and 1 there are unit fractions.
Hence there must be a first one.

 A.
⎛ S can be given a total ordering which is
⎜ well-ordered both forwards and backwards.
⎜ That is, every non-empty subset of S has
⎝ both a least and a greatest element in the subset.
[1]
 A. isn't true of all sets.

It is true for all sets of separated points on the real line.

If you augment ⅟ℕ with more elements
that is a different set.
A. is still not true of ⅟ℕ
 If you, nonetheless, assert A.
A. is true of _all the sets in the discourse_
Because A. is not true of ⅟ℕ,
⅟ℕ is not in the discourse.
After A., claims about "sets" are not
claims about ⅟ℕ

If NUF(x) has grown to ℵ₀ at x₀, then ℵ₀ unit fractions must be between 0 and x₀. Hence at least ℵ₀ points with ℵ₀ intervals of uncountably many points must be between 0 and x₀. That cannot happen at x₀ = 0, and it cannot happen at points in smaller distance than uncountable many points from zero.
Regards, WM