Re: More complex numbers than reals?

On 7/15/2024 9:26 AM, WM wrote:

Le 15/07/2024 à 00:39, Ben Bacarisse a écrit :

[...]

>
The only daredevil who tried it, Jim Burns,
has to assume that
by exchangig one of the elements can disappear.

Assume that ℕⁿᵒᵗᐧᵂᴹ is the set of finiteⁿᵒᵗᐧᵂᴹ ordinals.
If α is finiteⁿᵒᵗᐧᵂᴹ, then α⁺¹ = α∪{α} is finiteⁿᵒᵗᐧᵂᴹ.
If α ∈ ℕⁿᵒᵗᐧᵂᴹ, then α⁺¹ ∈ ℕⁿᵒᵗᐧᵂᴹ.
f(α) = α⁺¹ is 1.to.1
¬∃β≠α:  β⁺¹ = α⁺¹
f: ℕⁿᵒᵗᐧᵂᴹ → ℕⁿᵒᵗᐧᵂᴹ\{0}: 1.to.1
ℕⁿᵒᵗᐧᵂᴹ ∋ 0 ∉ f(ℕⁿᵒᵗᐧᵂᴹ)
'Bye, Bob!

No reason to pay attention.

E pur si muove.