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

On 10/22/2024 2:34 PM, WM wrote:

On 22.10.2024 19:38, Jim Burns wrote:

The description of each element in ℕ
requires its double to also be in ℕ

>
All that is in ℕ, according to your opinion,
is accepted.

In the ℕ which is our ℕ,
n ∈ ℕ  ⇔  ∃⟨0,1,...,n-1,n⟩
In the ℕ which is our ℕ,
for each j ∈ ℕ, ∃k ∈ ℕ: k = j+1
for each j ∈ ℕ, j=0  ∨  ∃i ∈ ℕ: i+1=j
for each S s ℕ, S = {}  ∨  ∃m ∈ S: m=min.S
Bob is in room 0 of our ℕ.Hotel
Swap guests in 0⇄1, 1⇄2, 2⇄3, 3⇄4, ...
After all swaps,
there is no first room Bob is in.
There is no ▒▒▒▒▒ room Bob is in.
'Bye, Bob.

If you find that the set ic complete,
then it is doubled.

In a WM.complete ℕ.Hotel,
dark rooms are added for Bob to disappear to
when he isn't in the visible rooms,
repairing 'bye.Bob,
  leaving can't.see.Bob.
However,
none of these swaps 0⇄1, 1⇄2, 2⇄3, 3⇄4, ...
move Bob to a dark room.
Bob disappeared
without going to a dark room.
Adding dark rooms without Bob in them
does not repair 'bye.Bob.
There is no
'bye.Bob.repairing WM.complete Hotel.

...not so different from the way in which
the description of a right triangle
requires
the square of its longest side to equal
the sum of the squares of the two other sides.

>
Your above expression is in fact
a very closely related one.

The most mathematical action one can perform is
to accept a result one sees is correct but
does not want to be correct.