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

On 10/22/2024 4:13 AM, WM wrote:

On 21.10.2024 22:34, Jim Burns wrote:

On 10/21/2024 2:25 PM, WM wrote:

On 21.10.2024 14:30, Jim Burns wrote:

On 10/21/2024 6:01 AM, WM wrote:

The range.

>
A complete set of natural numbers.

ℕ is defined such that
n ∈ ℕ  ⇔  ∃{0,1,...,n-1,n}

It is a complete set by assumption.
All its numbers can be mapped,
some of them not to the preimage.

>
Not to the mapped set.

The map we refer to is doubling.
max.{0,1,...,n-1,n} ↦
max.{0,1,...,n-1,n,n+1,...,n+n-1,n+n}
Name the map 2×
2×n = n+n
∀n ∈ ℕ:
  ∃{0,1,...,n-1,n}
  ∃{0,1,...,n-1,n,n+1,...,n+n-1,n+n}
  n+n = 2×n
  2×n ∈ ℕ
∀n ∈ ℕ:  2×n ∈ ℕ

All its numbers can be mapped,

Yes.
∃{0,1,...,n-1,n,n+1,...,n+n-1,n+n}

some of them not to the preimage.

No.
2×n ∈ ℕ
----
image 2×ℕ = {2×n: n∈ℕ}
n′ ∈ ℕ\2×ℕ  ⇔
∃{0,1,...,n-1,n,n+1,...,n+n-1,n+n,n+n+1}
  ∧ n′ = n+n+1
'Bye, Bob.