Re: Contradiction of bijections as a measure for infinite sets

On 3/25/2024 7:10 AM, WM wrote:

Le 24/03/2024 à 22:02, Dieter Heidorn a écrit :

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Important is only this:
You cannot add a natural number to the set ℕ.

not.exists finiteⁿᵒᵗᐧᵂᴹ ordinal n not.in ℕ
not.exists finiteⁿᵒᵗᐧᵂᴹ ordinal n such that
⟦0,n⟧ not.fits ℕ
== not.exists 1.to.1.map ⟦0,n⟧ ⇉ ℕ
for each finiteⁿᵒᵗᐧᵂᴹ set S
exists finiteⁿᵒᵗᐧᵂᴹ ordinal n such that
⟦0,n⟧ not.fits S
== not.exists 1.to.1.map ⟦0,n⟧ ⇉ S
ℕ is not finiteⁿᵒᵗᐧᵂᴹ.
for each finiteⁿᵒᵗᐧᵂᴹ set S
not.exists 1.to.1.map S∪{Bob} > S
ℕ is not finiteⁿᵒᵗᐧᵂᴹ
for not.finiteⁿᵒᵗᐧᵂᴹ ℕ
exists 1.to.1.map ℕ∪{Bob} ⇉ ℕ
You can add Bob to ℕ