On 11/06/2024 06:22 AM, Jim Burns wrote:
On 11/6/2024 5:35 AM, WM wrote:
On 05.11.2024 18:25, Jim Burns wrote:
On 11/4/2024 12:32 PM, WM wrote:
>
The intervals together cover a length of less than 3.
The whole length is infinite.
Therefore there is plenty of space for
a point not in contact with any interval.
>
⎛ Assuming the covering intervals are translated
⎜ to where they are end.to.end.to.end,
⎜ there is plenty of space for
⎝ not.in.contact exterior points.
>
This plentiness does not change
when the intervals are translated.
>
⎛ When the intervals are end.to.end.to.end,
⎜ there are exterior points
⎝ a distance 10¹⁰⁰⁰⁰⁰ from any interval.
>
Are there points 10¹⁰⁰⁰⁰⁰ from any interval
when midpoints of intervals include
each of {...,-3,-2,-1,0,1,2,3,...} ?
>
Isn't that a plentiness which changes?
>
I mean 'exterior' in the topological sense.
>
For a point x in the boundary ∂A of set A
each open set Oₓ which holds x
holds points in A and points not.in A
>
The intervals are closed with irrational endpoints.
>
'Exterior' seems like a good way to say
'not in contact'.
>
It seems to me that you have a better argument
with open intervals instead of closed,
but let them be closed, if you like.
>
Either way,
there are no points 10¹⁰⁰⁰⁰⁰ from any interval.
>
Each of {...,-3,-2,-1,0,1,2,3,...} is
the midpoint of an interval.
There can't be any exterior point
a distance 1 from any interval.
>
There can't be any exterior point
a distance ⅟2 from any interval.
Nor ⅟3. Nor ⅟4. Nor any positive distance.
>
Nice try.
But there are points outside of intervals,
>
Are any of these points.outside
⅟2 from any interval? ⅟3? ⅟4?
>
and they are closer to interval ends
than to the interior, independent of
the configuration of the intervals.
>
Shouldn't I be pointing that out
to you?
>
If there is no point with more.than.⅟2
between it and any midpoint,
shouldn't there be fewer.than.no points
with more.than.⅟2 between it and
any closer endpoint?
>
Note that
only 3/oo of the points are inside.
>
Yes, less than 2³ᐟ²⋅ε
>
If the intervals were open,
all of that would be "inside"
in the interior of their union.
>
Of the rest,
none of it is more.than.⅟2 from any interval.
>
An exterior point which is not
a positive distance from any interval
is not an exterior point.
>
Positive is what you can define,
>
Positive ℕ⁺ holds countable.to from.1
Positive ℚ⁺ holds ratios of elements of ℕ⁺
Positive ℝ⁺ holds points.between.splits of Q⁺
>
but there is much more in smaller distance.
>
Distances are positive or zero.
Two distinct points are a positive distance apart.
>
Remember the infinitely many unit fractions
within every eps > 0 that you can define.
>
For each of the infinitely.many unit fractions
there is no point a distance of that unit fraction
or more from any interval.
>
Therefore,
in what is _almost_ your conclusion,
there are no exterior points.
>
There are 3/oo of all points exterior.
>
Did you intend to write "interior"?
>
An exterior point is in
an open interval holding no rational.
>
There are no
open intervals holding no rational.
>
There are no exterior points.
>
However,
there are boundary points.
All but 2³ᐟ²⋅ε are boundary points.
>
Instead, there are boundary points.
For each x not.in the intervals,
each open set Oₓ which holds x
holds points in the intervals and
points not.in the intervals.
x is a boundary point.
>
The intervals are closed
>
We are only told
that Oₓ is an open set holding x
not that Oₓ is one of the ε.cover of ℚ
The question is whether x is a boundary point.
>
The rationals are dense
>
Yes.
Each multi.point interval [x,x′] holds
rationals.
>
but the intervals are not.
>
No.
Each multi.point interval [x,x′] holds
ε.cover intervals.
>
Therefore not all rationals are enumerated.
>
Explain why.
>
⎛ i/j ↦ kᵢⱼ = (i+j-1)(i+j-2)/2+i
⎜ k ↦ iₖ+jₖ = ⌈(2⋅k+¼)¹ᐟ²+½⌉
⎜ iₖ = k-(iₖ+jₖ-1)(iₖ+jₖ-2)/2
⎜ jₖ = (iₖ+jₖ)-iₖ
⎝ (iₖ+jₖ-1)(iₖ+jₖ-2)/2+iₖ = k
proves that
the rationals are countable.
>
Contradiction.
>
It contradicts a non.empty exterior.
It doesn't contradict an almost.all boundary.
>
Something of your theory is inconsistent.
>
Your intuition is disturbed by
an almost.all boundary.
>
Disturbed intuitions and inconsistencies
are different.
>
>
The definitions of "functions", and "topologies",
vary, and while each may be considered as their
own sort of theory, for relations and continua or wholes,
as well, in various theories like "ZF" or "when
the topology is defined as the open topology",
with regards to metric spaces and so on, it can help
to understand that in the usual account of descriptive
set theory, they are _opinions_ and there are _others_.
So, WM's "disturbed intuitions" here reflect variously
on unstated assumptions, or un-necessarily opinionated
or simply oppositely opinionated, with regards to the
theories, each their own, then as with regards to most
common aspects of:
geometry
number theory
arithmetic
algebra
function theory
topology
operator calculus
that it comes back around to geometry, that each these
has their theory where they're primary, i.e. elementary
in the theory and all else derived from them or defined
after them: a sort of heno-theory, with regards to
foundations from the many perspectives and the many
perspectives of foundations.
Then, when fields make their own restrictions of comprehension,
definitions being stipulations and from them are derivable
contradictions, then those are no longer foundations, instead
"practical theories", of sorts, where for example "functions
aren't necessarily ZF's Cartesian functions" and "topologies
aren't necessarily the standard open topology", with respect
to otherwise the central and primary objects of the universe
of mathematics, which according to theories of types, for
example, according to model theory, are either equi-interpretable,
or, aren't, AND, making a deconstructive account into assumptions
or stipulations, DOES make a wider and fuller dialectic, which
MAY thusly point out that thusly non-logical assumptions or
stipulations may be negated, logically.
Or, "ZF set theory with respect to number theory or geometry,
is, at best: incomplete, and in some theories, there are
counterexamples to otherwise its results, according to each
non-logical stipulation, or restriction of comprehension".
Re: Incompleteness of Cantor's enumeration of the rational numbers
Re: Incompleteness of Cantor's enumeration of the rational numbers (re-Vitali-ized)