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

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:

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The infinite union doesn’t.

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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.
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>
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"In-formally [naively], ...."
No, you got merely there what you
think you want, not what is.
It's called wishing yourself right,
not logic.
Perhaps cursorily research "illative"
and "univalent" with regards to what
intend to entail, w.r.t. "infinite union",
and furthermore cross-check definitions
of set theory, w.r.t. "axiom of union".
Because, otherwise what you got there
is "naive comprehension".
See, I told you, well-ordering the reals
and consequences to set theory have that
it's a very well-known and well-explored field.