Re: Incompleteness of Cantor's enumeration of the rational numbers

Am 05.11.2024 um 00:08 schrieb Chris M. Thomasson:

On 11/4/2024 2:02 PM, Jim Burns wrote:

On 11/4/2024 3:54 PM, Chris M. Thomasson wrote:

On 11/4/2024 6:37 AM, Jim Burns wrote:

>

Further there are never
two irrational numbers
without an interval between them.

>
Unless they equal each other? ;^)

>
And thus are one, not two, points?
>
>

Well, yeah in a sense.
 A = (-1, .5, 3, 7)
B = (-1, .5, 3, 7)
 A = B = true

Well, rather:
    "A = B" is true.
After all, (-1, .5, 3, 7) =/= true (I'd say). :-P
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Actually, some authors would prefer to write:
"there are never two different irrational numbers without an [nonempty] interval between them".
It's a matter of "style" or "precision", if you like.
If we talk about "two" real numbers x, y [in math], usually x = y is NOT excluded.
For example,
        for any two real numbers x, y ... bla bla
USUALLY does not exclude x = y. [Might look strange, but is just the usual math lingo.]
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