Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 12/29/2024 01:48 PM, Jim Burns wrote:

On 12/29/2024 3:57 PM, Ross Finlayson wrote:

On 12/29/2024 12:34 PM, Jim Burns wrote:

On 12/29/2024 1:15 PM, Ross Finlayson wrote:

On 12/29/2024 06:43 AM, joes wrote:

>

The infinite union doesn’t.

>
There is no "infinite union" in ZF,
only "pair-wise union",
according to the axiom of union.

>
No.
https://en.wikipedia.org/wiki/Axiom_of_union
⎛ Informally, the axiom states that
⎜ for each set x there is a set y
⎜ whose elements are precisely
⎝ the elements of the elements of x.

>
"In-formally [naively], ...."

>
'Informally' isn't 'naively'.
>
⎛ In the formal language of the Zermelo–Fraenkel axioms,
⎜ the axiom reads:
⎝ ∀A∃B∀c(c∈B⟺∃D(c∈D∧D∈A))
-- ibid.
>
>

Yeah, "pair-wise".
I'm telling you, I am not wrong,
and for a long time, I am not under-informed,
about ZF and ZFC set theories.
Mistakes to the contrary
are wrong and/or under-informed.