Re: The set of necessary FISONs

On 3/5/2025 4:28 PM, WM wrote:

Am 05.03.2025 um 20:26 schrieb Jim Burns:

On 3/5/2025 4:22 AM, WM wrote:

On 04.03.2025 23:05, Jim Burns wrote:

ω/2 ω/10 and ω/20 follow ω?

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No they follow Z₀ and UF.

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ω = ⋃{F}

No.
F = {F(1), F(2), ..., F(n), ...}
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UF = {1, 2, 3, ...}
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∀n ∈ ℕ: ω/n > n.

I think that you (WM) are using a different dictionary.
definableᵂᴹ:  finiteⁿᵒᵗᐧᵂᴹ ( #[0,x)ᵒʳᵈ < #[x+1)ᵒʳᵈ )
darkᵂᴹ:  finiteⁿᵒᵗᐧᵂᴹ ( #[0,x)ᵒʳᵈ < #[x+1)ᵒʳᵈ )
matheologicalᵂᴹ:  infiniteⁿᵒᵗᐧᵂᴹ ( #[0,x)ᵒʳᵈ = #[x+1)ᵒʳᵈ )

ω is the first non.finite ordinal.

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Yes.
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ω is the set of all and only finite ordinals.

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That is another use of ω.

It is von Neumann's use of ω