Re: Contradiction of bijections as a measure for infinite sets

On 4/4/2024 5:43 AM, WM wrote:

Le 03/04/2024 à 23:48, Jim Burns a écrit :

Bijecting n and n/1 do not "destroy"
bijections between ℕ and ℚᶠʳᵃᶜ

 Mapping n/1 in the fractions
shows that no bijection is possible.

You have assumed that ℕ and ℚᶠʳᵃᶜ are Mückenheim sets.
I define a Mückenheim set such that,
for any set,  either
all injections are onto,  or
all injections are not.onto.
For a Mückenheim set,
any not.onto.injection implies no onto.injection,
which "destroys" the onto.injection I showed you.
A Mückenheim set is a finiteⁿᵒᵗᐧᵂᴹ set.
For each Mückenheim (finiteⁿᵒᵗᐧᵂᴹ) set M
there is a Mückenheim (finiteⁿᵒᵗᐧᵂᴹ) ordinal ⟦0,m⟧
such that not.exists an injection from ⟦0,m⟧ to M
such that ⟦0,m⟧ ⇉| M
such that ⟦0,m⟧ is bigger than M
Never( not.Mückenheim ⟦0,ζ⟧ < Mückenheim ⟦0,m⟧ )
Each not.Mückenheim ⟦0,ζ⟧ is
an upper.bound of {Mückenheim ⟦0,n⟧}
Never( not.Mückenheim ⟦0,ζ+1⟧ and Mückenheim ⟦0,ζ⟧ )
Never( Mückenheim ⟦0,m⟧ and not.Mückenheim ⟦0,m+1⟧ )
Each Mückenheim ⟦0,m⟧ is
not an upper.bound of {Mückenheim ⟦0,n⟧}
Define ⟦0,ω⟧ as least.upper.bound of {Mückenheim ⟦0,n⟧}
⟦0,ω⟧ := lub{Mückenheim ⟦0,n⟧}
For ⟦0,m⟧ < ⟦0,ω⟧ < ⟦0,ζ⟧
Mückenheim ⟦0,m⟧
not.Mückenheim ⟦0,ζ⟧
| Assume ⟦0,m+1⟧ = ⟦0,ω⟧
|
| For ⟦0,m⟧ < ⟦0,ω⟧ < ⟦0,m+2⟧
| Mückenheim ⟦0,m⟧
| not.Mückenheim ⟦0,m+2⟧
|
| However,
| only
| Mückenheim ⟦0,m⟧ and Mückenheim ⟦0,m+2⟧
| or
| not.Mückenheim ⟦0,m⟧ and not.Mückenheim ⟦0,m+2⟧
| Contradiction.
Therefore
always ⟦0,m+1⟧ ≠ ⟦0,ω⟧