Re: Replacement of Cardinality (infinite middle)

On 8/12/2024 4:59 PM, Ross Finlayson wrote:

On 08/11/2024 09:44 PM, Jim Burns wrote:

On 8/11/2024 7:39 PM, Ross Finlayson wrote:

Starting with a theory _without_
the constant introduced named omega,
i.e., finite sets,

For P(z),
use a description 𝕆ᶠⁱⁿ(z) of a finite ordinal,
and ω := {z:𝕆ᶠⁱⁿ(z)} exists
>
For example, use
𝕆ᶠⁱⁿ(z)  ⇔
(z ∋ {} ∧ ∀y ∈ z+1: y≠{} ⇒ ∃x∈z: x+1=y)
∨ (z = {})
>
z+1 = z∪{z}

Then, omega, as you've defined it,

>
ω := {z:𝕆ᶠⁱⁿ(z)}
>

contains itself,

I'm curious, now that you have
a beginning and an end of
the finite, or 0 and omega in ZF,

>
ω is the least.upper.bound of the finites.
ω is not a finite.
ω is not the upper.end of the finites.
The upper.end of the finites doesn't exist.

>
Here though

_Where_ though?

it's beginning ... ( ... infinitely-many ...) ... end,
where the upper.end of the finites always exists.

For ω as I've defined it, no upper.end exists.
for each k ∈ ω
𝕆ᶠⁱⁿ(k)
𝕆ᶠⁱⁿ(k+1)
k+1 ∈ ω
k is not the upper end of ω
for each k ∉ ω
k is not the upper end of ω

Then you claim to have
an axiom of restriction of comprehension of the finites

To review:
What I claim is
⎛ ∃{}
⎜ ∀x∀y∃z=x∪{y}
⎝ and extensionality
⎛ ∃∃xx={z:P(z)}: ∀y: y ∈ {z:P(z)} ⇔ P(y)
⎝ and extensionality
∃∃{z:P(z)} is unrestricted comprehension.
Unless we are no longer uninterested in what words mean.

unless Russell grants you
a dispensation of Russell's retro-thesis,
and say it's always so for others, too,
congratulations,
you claim to have invented a mathematics
where you = Russell.

Ah.
I've seen this one before.
Your tacit thesis is that
it is preferable to disagree with the Old Ones
even at the cost of being wrong.
Well, it's a choice.