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

On 12/29/2024 8:17 PM, Ross Finlayson wrote:

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:

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], ...."

>
'Informaclly' 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".

No,
|A|.many.wise
∀A∃B ∀c(c∈B ⟺ ∃D(c∈D∧D∈A))
set A of sets D of elements c
B = ⋃A = {c: ∃D(c∈D∧D∈A)}
----
pairwise
∀D∀D′∃B ∀c(c∈B ⟺ c∈D∨c∈D′))
sets D,D′
B = D∪D′ = ⋃{D,D′} = {c: c∈D∨c∈D′}

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.

⎛ "Just the place for a Snark!" the Bellman cried,
⎜    As he landed his crew with care;
⎜ Supporting each man on the top of the tide
⎜   By a finger entwined in his hair.
⎜
⎜ "Just the place for a Snark! I have said it twice:
⎜    That alone should encourage the crew.
⎜ Just the place for a Snark! I have said it thrice:
⎝    What I tell you three times is true."
-- Lewis Carroll, "The Hunting of the Snark"