Re: how

On 4/23/2024 2:06 PM, WM wrote:

Le 22/04/2024 à 20:02, FromTheRafters a écrit :

WM has brought this to us :

Le 21/04/2024 à 00:44, FromTheRafters a écrit :

WM wrote :

What is a gap in the ordinals?

>
It is a not existing ordinal between two ordinals.
It is a not existing natural number next to ω,
for instance.

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Sets do not have not existing elements.

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What is ω - 1?

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An absurdity.

ω-1 is that which has anything and everything
following in a finite not.first.false claim.sequence:
Rumpelstiltskin spins straw into gold,
Aladdin commands the genie of the lamp,
the check is in the mail,
anything, everything.

An absurdity.

>
Not more nor less than ω.

ω-1 requires Rumpelstiltskin and Aladdin's genie to exist.
ω requires boring.arithmetic to exist.
For i,j,k ∈ ⟦0,ω⦆:
i⁺¹≠0 ∧ ¬∃h≠i:h⁺¹=i⁺¹ ∧ ∃k=i⁺¹⁺¹ ∧ 1=0⁺¹
P(0) ∧ ∀i:P(i)⇒P(i⁺¹)  ⟹  ∀k:P(k)
i+0=i ∧ i+j⁺¹=(i+j)⁺¹
i⋅0=0 ∧ i⋅j⁺¹=(i⋅j)+i
i^0=1 ∧ i^j⁺¹=(i^j)⋅i
boringly.finite           ; excitingly.transfinite
… ⇇⃥ ⟦0,κ⦆ ⇇⃥ ⟦0,κ+1⦆ ⇇⃥ … ; ⟦0,ω⦆ ⇇ ⟦0,ω+1⦆ ⇇ …
⟦0,ω⦆ = {ordinal κ: ⟦0,κ⦆ ⇇⃥ ⟦0,κ+1⦆}
⟦0,κ⦆ ⇇⃥ ⟦0,κ+1⦆  ⟺  ⟦0,κ+1⦆ not.fits ⟦0,κ⦆  ⟺
¬∃f: ⟦0,κ+1⦆ ⇉ ⟦0,κ⦆: ¬∃λ≠λ′:f(λ)=f(λ′)  ⟺
κ is a different.sized.neighbor.haver
⟦0,ω⦆ ⇇ ⟦0,ω+1⦆  ⟺  ⟦0,ω+1⦆ fits ⟦0,ω⦆  ⟺
∃g: ⟦0,ω+1⦆ ⇉ ⟦0,ω⦆: ¬∃λ≠λ′:g(λ)=g(λ′)  ⟺
ω is a same.sized.neighbor.haver
ω stands between
boring different.sized.neighbor.havers and
exciting same.sized.neighbor.havers.