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

On 12/12/2024 02:18 PM, Chris M. Thomasson wrote:

On 12/12/2024 2:11 PM, WM wrote:

On 12.12.2024 17:15, Jim Burns wrote:

On 12/12/2024 4:07 AM, WM wrote:

>

The existence of a bridge implies that,
somewhere we can't see,
a size which cannot change by 1
changes by 1 to
a size which can change by 1

>
Every set can change by one element. No size is required and no size
is possible if this is forbidden.

>

There must be
a continuous sequence of steps of height 1
from many elements to none.

>
What you describe is many, not infinity.
>

Can you confirm this?

>
For many, yes,
For infinity, never.
Having a continuous sequence of steps of height 1
from many elements to none
is what makes it finite.

>
It is needed by Cantors mappings.

>
>
Again, Cantor Pairing works with any natural number. Not just many of
them... ALL of them. :^)
>

What you got there is "the space of Cartesian functions",
the Cartesian Product, D X R, the set of all tuples (d, r),
any subset of those, making a function if non-empty from
D to R, and under various conditions, a 1-1 and onto function,
a bijection.
Not all functions are Cartesian - some have no way to re-order
the elements of the domain and range, for something like the
only-diagonal of the natural/unit equivalency function, what
makes for a bijection between D a discrete domain and R
a continuous range.
Then, also, the idea that infinite sets are countable
usually is called "Galileo's", among the various authors'
various things that Cantor pulled together and called "Mengenlehre",
"reading sets", "set theory".
For example, the anti-diagonal argument was discovered decades
before, at least, and probably was known and recorded since
antiquity, and the whole "Infinitarcarcul" of duBois-Reymond
has even more going on about the "long-line" of all the
real-valued expressions and whether and where they intersect
the real line, and being of the set of all functions, even
a higher cardinality than the complete ordered field's!
Yeah, it's not gratifying when trolls never learn.