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

Le 04/09/2024 à 22:52, Jim Burns a écrit :

On 9/4/2024 3:10 PM, WM wrote:

Either
∀n ∈ ℕ: 1/n - 1/(n+1) > 0
is wrong or Peano is wrong.

 ...or natural numbers aren't what you think they are.

That is possible. My arguments hold only under the premise of actual infinity showing that Hilbert's hotel is nonsense because the set of natural numbers cannot be extended. If all rooms are occupied than no guest can leave his room for a not occupied room. (When I was in USA or the first time, I asked in a Hilton whether they had free rooms. They laughed.)

 

Peano has been generalized from
the small natural numbers.

 Peano describes the finite natural numbers.
'Finite' doesn't need to be 'small'.

Finite is much larger than Peano or you could/can imagine.

"All different unit fractions are different"
however is a basic truth. Therefore I accept the latter.

 You also accept quantifier shifts,
which breaks your logicᵂᴹ.
Quantifier shifts are unreliable.

Do you believe that it needs a shift to state:
All different unit fractions are different.
∀n ∈ ℕ: 1/n - 1/(n+1) > 0 I can see no shift.
Regards, WM