Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity)

On 1/2/2025 2:04 PM, Ross Finlayson wrote:

On 01/02/2025 08:47 AM, Jim Burns wrote:

[...]

>
"Well-Order the Universe" is
what "ubiquitous ordinals" is.

I read that as
"Each set can be well.ordered".
( It'd be great if you (RF) were to indicate
how close what I read comes to what you write.

Which, you snipped?

You're welcome.
I snip a lot of what you send down the thread,
in a possibly.forlorn attempt to
make what you write comprehensible.
I do it for love of comprehensibility, at no extra charge.
If I snip context needed for understanding,
it would be great if you (RF) took that as
as sign for you to explain more fully,
upon which sign, you (RF) explained more fully.

It makes for a model of set theory where
"powerset is successor is order type",

You (RF) probably don't mean
powerset x = {y: y⊆x}
successor x = x∪{x}
because, in general, {y: y⊆x} = x∪{x} is wrong.
It would be great if you (RF) said what
the other thing you mean by 'powerset' and 'successor'.

thusly where the Powerset Theorem
holds different, as it were.

⎛ ¬∃F onto: A → 𝒫(A)
⎜
⎜ because
⎝ ¬∃a′ ∈ A: F(a′) = {a∈A: a∉F(a)} ∈ 𝒫(A)
Is that what you (RF) mean by the Powerset Theorem?
Where does it hold different?
Why does it hold different there?