Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

Ross Finlayson used his keyboard to write :

On 01/07/2025 01:40 AM, FromTheRafters wrote:

WM has brought this to us :

On 06.01.2025 23:43, Jim Burns wrote:

On 1/5/2025 1:14 PM, WM wrote:

>

ℕ cannot be covered by FISONs,
neither by many nor by their union.
If ℕ could be covered by FISONs
then one would be sufficient.

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ℕ is the set of finite.ordinals.
ℕ holds each finite ordinal.
ℕ holds only finite.ordinals.

>

ℕ| is by definition the smallest transfinite number, larger than every

natural number by every visible factor.

>
k ∈ ℕ  ⇒  k+1 ∈ ℕ
is true for both the darkᵂᴹ and the visibleᵂᴹ.

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One exception exists: ω-1.

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Which remains undefined.

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That's _under_-defined, with regards to
its term, omega, 1, and subtraction,
or just omega and decrement,
or just constant and increment.
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Well, now it's sort of defined, ....
>
>
The "counting backward from infinity"
is one of Cantor's great ideas,
you saying he can't just define it?

No, I'm saying that omega is the first infinite ordinal and there is no predecessor function for it just as there isn't for zero.