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

On 9/9/2024 6:39 AM, WM wrote:

On 08.09.2024 22:21, Jim Burns wrote:

There aren't two.
There isn't one.
There is no smallest unit fraction.

>
Then there is no unit fraction.

⎛ Each real number > 0 is a rational > 0 or
⎜ between sides of a split of rationals > 0
⎜
⎜ Each rational > 0 is the ratio of
⎜ two integers > 0
⎜
⎜ Each nonempty set S of integers > 0 holds
⎜  minimum.S
⎜ Each integer > 0 has a successor.integer > 0 (+1)
⎜ Each integer > 0 has a predecessor.integer > 0 (-1)
⎝  except the minimum integer > 0 (1).
Those are the points we are talking about.
If you aren't talking about those,
what you say is irrelevant _to those_
If you are talking about those,
then, no,
-- there are unit fractions,
-- each real number > 0 is undercut by ⅟k
  where ⅟k is countable.down.to from ⅟1
-- for each x > 0, for each j countable.up.to from 1
  there are more.than.j unit.fractions in (0,x)
-- there is no first (smallest) unit fraction,  and
-- there aren't fewer.than.ℵ₀ unit.fractions in (0,x)

ℵo unit fractions cannot exist without
1, 2, 3, ... unit fractions before.

Each unit.fraction has 1,2,3,... unit.fraction before.
It is incorrect (it is a quantifier shift)
to conclude from that correct claim
that
🛇 there are 1,2,3,... unit.fractions before each