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

On 10/12/2024 2:39 PM, WM wrote:

On 11.10.2024 20:02, Jim Burns wrote:

We have a description of ℕ and its elements,
free of mentions of 'potential' and 'actual'.
From its description,
we can reason about it and its elements.
"Potential ℕ" vs. "actual ℕ" leaves unchanged
which claims we reason to.

>
Potential infinity:
All natural numbers when double yield
natural numbers
but larger numbers are among the result.

Finites have larger finites.
⎛ Assume k is a finite ordinal.
⎜
⎜ For each ordinal j: 0 < j ≤ k
⎜ ordinal j-1 exists
⎜
⎜ For each ordinal j: 0 < j ≤ 2⋅k
⎜ ordinal j-1 exists
⎜
⎝ 2⋅k is a finite ordinal.

In actual infinity
no new naturals can be created

Finites have larger finites.

but since doubling doubles the value of each number,
numbers of the second number class are created.

ω is the first transfinite ordinal.
⎛ Assume k < ω
⎜
⎜⎛ k is a finite ordinal.
⎜⎜
⎜⎜ For each ordinal j: 0 < j ≤ k
⎜⎜ ordinal j-1 exists
⎜⎜
⎜⎜ For each ordinal j: 0 < j ≤ 2⋅k
⎜⎜ ordinal j-1 exists
⎜⎜
⎜⎝ 2⋅k is a finite ordinal.
⎜
⎝ 2⋅k < ω
⎛
⎜ Big fleas have little fleas,
⎜ Upon their backs to bite ’em,
⎜ And little fleas have lesser fleas,
⎜ And so, ad infinitum.
⎝
-- Ogden Nash