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

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

On 03.09.2024 19:50, Jim Burns wrote:

If the smallest unit.fraction existed,
you could see it

>
No, that is impossible by your argument:

My argument is that
an existing smallest unit.fraction requires impossibles.
Therefore, an existing smallest unit.fraction is not.

positive and undercut by
a visibleᵂᴹ smaller.than.smallest unit.fraction.
>
But you can't see that.
The smallest unit fraction doesn't exist.

>
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.

Peano has been generalized from
the small natural numbers.

Peano describes the finite natural numbers.
'Finite' doesn't need to be 'small'.
⎛ The natural numbers are well.ordered.
⎜ Each non.0 natural number and each non.0 before it
⎜ has a predecessor.natural,
⎝ Each natural number has a successor.natural.
However large 𝔊 is,
if
⎛ [0…𝔊]ᵒʳᵈ is well.ordered
⎜ ∀k ∈ (0…𝔊]ᵒʳᵈ: [0…𝔊)ᵒʳᵈ ∋ k-1
⎝ ∀j ∈ [0…𝔊]ᵒʳᵈ: (0…𝔊+1]ᵒʳᵈ ∋ j+1
then
𝔊 is one of what Peano describes.
Moreover,
if
[0…ω)ᵒʳᵈ ∋ ω-1
then
⎛ [0…ω]ᵒʳᵈ is well.ordered
⎜ ∀k ∈ (0…ω]ᵒʳᵈ: [0…ω)ᵒʳᵈ ∋ k-1
⎝ ∀j ∈ [0…ω]ᵒʳᵈ: (0…ω+1]ᵒʳᵈ ∋ j+1
and
ω is one of what Peano describes,
which is incorrect --
ω is _first.after_ what Peano describes.
ω is infinite.
𝔊 is finite.
'Infinite and 'finite' don't mean 'large' and 'small'.

"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.