Re: Replacement of Cardinality (infinite middle)

On 8/30/2024 4:00 PM, Ross Finlayson wrote:

On 08/29/2024 10:24 PM, Jim Burns wrote:

[...]

>
Regularity of _difference_, and,
regularity of _dispersion_,
both _increment_, and _modularity_,
are examples of two various kinds of regularity,

There is yet another kind of regularity which is
relevant to whether ω contains ω.
⎛ Axiom of Regularity.
⎜ Every non-empty set x contains a member y such that
⎝ x and y are disjoint sets.
Consider an ordinal as the set of ordinals before it.
Each ordinal ξ except the first, 0 = {},
holds the first, {} ∈ ξ
Each ordinal ξ is disjoint from {}
Each ordinal ξ is a regular set.
A regular set is not an element of itself.
An ordinal is not an element of itself.
ω is not an element of itself.

The reals actually give a well-ordering, though,
it's their normal ordering as via a model of line-reals.

No.
The normal ordering of the reals is not a well.ordering.
In a well.ordering,
each nonempty subset holds a minimum.
In the normal ordering of ℝ,
(0,1] does not hold a minimum.
The normal ordering of ℝ is not a well.ordering.
If the non.hypocritical stance on that result is that,
there are non.standard.this and non.standard.that
_without_ that result,
consider that triangles without right angles are
not counter.examples to the Pythagorean theorem.
They are _not relevant_ to the Pythagorean theorem.
Well.ordered things which aren't ℝ
does not change that
the normal ordering of ℝ is not a well.order.

Of course
any other one you'd give would have
taking a subset of ordinals,
which of course are _always_ well-ordered,
with those being an uncountable subset's, of the reals,
_also in their normal ordering_.

I find it interesting,
though possibly not to.the.point,
that,
for any two well.ordered sets,
one set is order.isomorphic to
an initial segment of the other set.
In that sense, there is only one well.order.