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:

>> [...]
>
> If a well-ordering exists, then,
> consider it as a bijective function from ordinal O,
> and thus its "elements" or ordinals O,
> to domain D.

We are well.ordering the reals, so...
let O = ℶ₁ = |ℝ| = |𝒫(ℕ)|
let D = ℝ
let #: ℝ → ℶ₁: bijective

https://en.wikipedia.org/wiki/Beth_number

> As a Cartesian function the usual way, that's thusly
> a set of ordered pairs (o, d) which then
> via usual axioms and schema of comprehension and
> the existence of choice,
> read out in order the element (o_alpha, d).
>
> So, a well-ordering of the reals, this function, takes
> any subset of uncountably many elements (o_alpha, d, alpha).
> Now, what's so is that
> only countably many of the d can be in their normal order,
> that alpha < beta -> d_alpha < d_beta.

...or x,y such that #x < #y  ⇒  x < y

> This is because
> there are rational numbers between any of those,
> and only countably many of those.

I don't see what you're getting at.

In the usual order,
there are rational numbers between any two real numbers,
and only countably many rational numbers,
and uncountably.many real numbers in their usual order.
One doesn't prevent the other.

Maybe there is a different argument available.