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

On 1/1/2025 1:10 PM, Ross Finlayson wrote:

On 07/29/2024 12:46 PM, Ross Finlayson wrote:

On 07/29/2024 12:44 PM, Ross Finlayson wrote:

On 07/29/2024 05:32 AM, Jim Burns wrote:

On 7/28/2024 7:42 PM, Ross Finlayson wrote:

about ubiquitous ordinals

>
What are ubiquitous ordinal?

The "ubiquitous ordinals", sort of recalls Kronecker's
"G-d made the integers, the rest is the work of Man",

I think Kronecker's integers are the finite ordinals.
Are you saying that 'ubiquitous' means 'finite'?

So, "ubiquitous ordinals" is exactly what it says.

It's surprising how unhelpful some answers can be.

Or, you know, "infinity plus one".

Consider the definition of a finite.cardinal as
the cardinal #A of a set A
smaller.by.one than sets fuller.by.one
#A ∈ ⟦0,ℵ₀⦆  :⇔  (#A < #(A∪{a}) ⇐ A ≠ A∪{a})
If,
as might be expected,
infinity.plus.one is different from simple.infinity,
then,
under that definition,
infinity is finite.