Re: Incompleteness of Cantor's enumeration of the rational numbers (standard infinitesimals)

On 11/27/2024 10:19 AM, FromTheRafters wrote:

WM explained :

On 27.11.2024 13:32, Richard Damon wrote:

On 11/27/24 5:12 AM, WM wrote:

>

Of course. |{1, 2, 3, 4, ...}| = |ℕ| and |{2, 3, 4, ...}| = |ℕ| - 1
is consistent.

>
So you think, but that is because you brain has been exploded by the
contradiction.
>
We can get to your second set two ways, and the set itself can't know
which.
>
We could have built the set by the operation of removing 1 like your
math implies, or we can get to it by the operation of increasing each
element by its successor, which must have the same number of elements,

>
Yes, the same number of elements, but not the same number of natural
numbers.
>
Hint: Decreasing every element in the real interval (0, 1] by one
point yields the real interval [0, 1). The set of points remains the
same, the set of positive points decreases by 1.

>
If you have a successor function for the real numbers, why don't you
share it with the rest of the world?

You mean like line-reals and iota-values?
It's one of Aristotle's continuums, been around forever.
Oh, you mean stack it up again modern mathematics
and show that a sort of only-diagonal a non-Cartesian
not-a-real-function with surprising and special
real analytical character fits within the theory
otherwise our great axiomatic set theory a descriptive
set theory with a bit of stipulating LUB and measure 1.0?
I wouldn't say that usenet's "closed" as it were,
though, traffic is usually more directed to the
great maw of mammon's soup-hole, there is though
that each usenet article has a usual unique identifier
according to MLS and Chicago and other usual matters
of agreement in bibliographic reference.
The infinitesimal analysis of course has been around
for a long, long time, and these days it's called
"non-standard", which it exists at all,
now that you mention it.
Here it's "line-reals" with "iota-values", at fulfill
being a model of a continuous domain (though, only a
bounded segment, of course), that do have a least positive
iota-value, not to be confused with infinitely-divisible
members of the complete ordered field, that also fulfills
being a "continuous domain" (extent, density, completeness,
measure) after axiomatizing LUB and measure 1.0 above set theory.