Re: Replacement of Cardinality (infinite middle)

On 9/2/2024 8:25 PM, Ross Finlayson wrote:

On 09/02/2024 02:46 PM, Jim Burns wrote:

On 9/1/2024 2:44 PM, Ross Finlayson wrote:

least-upper-bound, has that
that's been given as an axiom above or "in" ZFC,

>
No, least.upper.bound isn't an axiom above or in ZFC.

Then,
about the least-upper-bound actually being an axiom,

...in ZFC.
Least.upper.bound is an axiom of the complete.ordered.field.
ZFC:
Extensionality, Regularity, Restricted Comprehension schema,
Pairing, Union, Replacement schema, Infinity, Power Set,
Well.ordering (Choice)

it sort of is,
that Dedekind-Eudoxus-Cauchy or
"there are all the infinite sequences",
as that there are "enough" elements in Cantor space
to fulfill least-upper-bound, it's an axiom.

I'm curious what you think it is to be an axiom.
I would ask, but you (RF) don't answer questions.
So, I'll just muddle along, with my curiosity unsatisfied.
----
A construction of ℝ in ZFC does not use bulldozers.
It is a proof that something, for example
{S⊆ℚ:∅≠Sᵉᵃᶜʰ<ᵉˣⁱˢᵗˢSᵉᵃᶜʰ<ᵉᵃᶜʰℚ\S≠∅}
satisfies all the axioms of the complete ordered field,
among which is the least.upper.bound property.
An axiom in one context can be a theorem in another,
and vice versa.
Axioms set the topic of discussion.
A different discussion will have a different topic
described by different axioms.
Changing axioms doesn't make one _wrong_
It makes one _outside the discussion_
or, as has been said before, _not even wrong_