On 2/27/22 10:03 PM, olcott wrote:
On 2/27/2022 8:43 PM, Richard Damon wrote:
On 2/27/22 9:28 PM, olcott wrote:
On 2/27/2022 8:11 PM, Python wrote:
olcott wrote:
On 2/27/2022 7:34 PM, Python wrote:
olcott wrote:
...
An x86 machine can do many more things but they are ruled as not >>>>>>>> counting because a TM cannot do these things. A TM can only do >>>>>>>> the subset of things that are mappings from inputs to outputs.
*facepalm*
you are really that low?
https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel >>>>>> Clearly the concept of infinity is incoherently defined.
*facepalm*^2
https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
If we use a sphere instead of a ball to perform this paradox then
when we put the two spheres back together they are no longer
spherical. Instead they are comprised of line segments of
infinitesimal length.
Every other geometric point on the number line is taken up by one
sphere or the other. A line segment comprised of three immediately
adjacent geometric points is exactly one infinitesimal unit longer
than a line segment comprised of two immediately adjacent geometric
points.
Says you don't understand the properties of finite sets like the reals.
There are NOT 'immediately adjacent geometric points' in the object,
as it can be shown that between ANY two (different) real numbers (or
even just rational numbers) there exists another number between them,
so no numbers are 'adjacent'
The interval [0,1] is exactly one geometric point longer than the
interval [0,1)
https://www.mathwords.com/i/interval_notation.htm
Non-Sequitur. Doesn't show that there is anything like adjacent points.
The open interval [0,1) has no highest point, as any point you try to
name, has a point higher that is also in the interval.
On 2/27/22 11:29 PM, olcott wrote:
On 2/27/2022 10:18 PM, Richard Damon wrote:
On 2/27/22 11:08 PM, olcott wrote:
On 2/27/2022 9:20 PM, Richard Damon wrote:
On 2/27/22 10:03 PM, olcott wrote:
On 2/27/2022 8:43 PM, Richard Damon wrote:
On 2/27/22 9:28 PM, olcott wrote:
On 2/27/2022 8:11 PM, Python wrote:
olcott wrote:
On 2/27/2022 7:34 PM, Python wrote:
olcott wrote:
...
An x86 machine can do many more things but they are ruled as >>>>>>>>>>>> not counting because a TM cannot do these things. A TM can >>>>>>>>>>>> only do the subset of things that are mappings from inputs >>>>>>>>>>>> to outputs.
*facepalm*
you are really that low?
https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel >>>>>>>>>>
Clearly the concept of infinity is incoherently defined.
*facepalm*^2
https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
If we use a sphere instead of a ball to perform this paradox
then when we put the two spheres back together they are no
longer spherical. Instead they are comprised of line segments of >>>>>>>> infinitesimal length.
Every other geometric point on the number line is taken up by
one sphere or the other. A line segment comprised of three
immediately adjacent geometric points is exactly one
infinitesimal unit longer than a line segment comprised of two >>>>>>>> immediately adjacent geometric points.
Says you don't understand the properties of finite sets like the >>>>>>> reals.
There are NOT 'immediately adjacent geometric points' in the
object, as it can be shown that between ANY two (different) real >>>>>>> numbers (or even just rational numbers) there exists another
number between them, so no numbers are 'adjacent'
The interval [0,1] is exactly one geometric point longer than the
interval [0,1)
https://www.mathwords.com/i/interval_notation.htm
Non-Sequitur. Doesn't show that there is anything like adjacent
points.
The open interval [0,1) has no highest point, as any point you try
to name, has a point higher that is also in the interval.
It is exactly one geometric point (one infinitesimal unit) less than
1.0
And what is that?
The problem is geometric points are infintesimally small.
Do you disagree that the interval [0,1] is exactly one geometric point
longer than the interval [0,1) ???
Yes, because that statement doesn't actually make sense when you look at
it precisly. The problem is infinitity - 1 is the same as infinity.
You FAIL the basic principles of infinite math.
olcott wrote:
On 2/27/2022 10:18 PM, Richard Damon wrote:...
Do you disagree that the interval [0,1] is exactly one geometric point
longer than the interval [0,1) ???
Lebesgue's measures of [0,1] and [0,1) have the same value: 1.
Richard Damon <Richard@Damon-Family.org> writes:
On 2/27/22 11:29 PM, olcott wrote:
On 2/27/2022 10:18 PM, Richard Damon wrote:
On 2/27/22 11:08 PM, olcott wrote:
It is exactly one geometric point (one infinitesimal unit) less than 1.0 >>>>And what is that?
The problem is geometric points are infintesimally small.
Do you disagree that the interval [0,1] is exactly one geometric point
longer than the interval [0,1) ???
Yes, because that statement doesn't actually make sense when you look
at it precisly. The problem is infinitity - 1 is the same as infinity.
I think the problem is simpler than that. If lengths are to be useful,
one should be able to do arithmetic on them,
and they should be the sort
of quantities that appear in the intervals we are measuring
(i.e. |[0, a]| = a, and for any length l, [0, l] exists).
But then if |[0, 1)| = lx < 1 it all falls apart. For example, what
interval has length (lx+1)/2? Is the set [0, lx] equal to [0,1)? What elements are in [0, 1] \ [0, lx]? PO-lengths have to be special things
with a huge number of ad hoc "axioms" propping them up.
But PO does not care if his idea of length makes sense because its sole
purpose is to reflect the intuitions he had formed by the time he
stopped learning maths. He's never going to write a book on "PO-measure theory", so he can just keep decreeing new facts about PO-lengths to his heart's content. It does not even matter if the result is inconsistent because he rejects what everyone else thinks of as a proof anyway.
(Remember that if {A} ⊦ X, he asserts that {A,¬A} ⊬ X.)
olcott wrote:
On 2/28/2022 5:48 AM, Richard Damon wrote:
On 2/27/22 11:29 PM, olcott wrote:
On 2/27/2022 10:18 PM, Richard Damon wrote:
On 2/27/22 11:08 PM, olcott wrote:
On 2/27/2022 9:20 PM, Richard Damon wrote:
On 2/27/22 10:03 PM, olcott wrote:
On 2/27/2022 8:43 PM, Richard Damon wrote:
On 2/27/22 9:28 PM, olcott wrote:
On 2/27/2022 8:11 PM, Python wrote:
olcott wrote:
On 2/27/2022 7:34 PM, Python wrote:
olcott wrote:
...
An x86 machine can do many more things but they are ruled >>>>>>>>>>>>>> as not counting because a TM cannot do these things. A TM >>>>>>>>>>>>>> can only do the subset of things that are mappings from >>>>>>>>>>>>>> inputs to outputs.
*facepalm*
you are really that low?
https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel
Clearly the concept of infinity is incoherently defined. >>>>>>>>>>>>
*facepalm*^2
https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox >>>>>>>>>> If we use a sphere instead of a ball to perform this paradox >>>>>>>>>> then when we put the two spheres back together they are no >>>>>>>>>> longer spherical. Instead they are comprised of line segments >>>>>>>>>> of infinitesimal length.
Every other geometric point on the number line is taken up by >>>>>>>>>> one sphere or the other. A line segment comprised of three >>>>>>>>>> immediately adjacent geometric points is exactly one
infinitesimal unit longer than a line segment comprised of two >>>>>>>>>> immediately adjacent geometric points.
Says you don't understand the properties of finite sets like >>>>>>>>> the reals.
There are NOT 'immediately adjacent geometric points' in the >>>>>>>>> object, as it can be shown that between ANY two (different)
real numbers (or even just rational numbers) there exists
another number between them, so no numbers are 'adjacent'
The interval [0,1] is exactly one geometric point longer than
the interval [0,1)
https://www.mathwords.com/i/interval_notation.htm
Non-Sequitur. Doesn't show that there is anything like adjacent
points.
The open interval [0,1) has no highest point, as any point you
try to name, has a point higher that is also in the interval.
It is exactly one geometric point (one infinitesimal unit) less
than 1.0
And what is that?
The problem is geometric points are infintesimally small.
Do you disagree that the interval [0,1] is exactly one geometric
point longer than the interval [0,1) ???
Yes, because that statement doesn't actually make sense when you look
at it precisly. The problem is infinitity - 1 is the same as infinity.
You FAIL the basic principles of infinite math.
It is the case based on the conventional meanings of intervals that
[0,1] is exactly one geometric point longer than [0,1).
No with any sensible definition of "longer".
On 2/28/22 10:17 AM, olcott wrote:
On 2/28/2022 5:48 AM, Richard Damon wrote:
On 2/27/22 11:29 PM, olcott wrote:
On 2/27/2022 10:18 PM, Richard Damon wrote:
On 2/27/22 11:08 PM, olcott wrote:
On 2/27/2022 9:20 PM, Richard Damon wrote:
On 2/27/22 10:03 PM, olcott wrote:
On 2/27/2022 8:43 PM, Richard Damon wrote:
On 2/27/22 9:28 PM, olcott wrote:
On 2/27/2022 8:11 PM, Python wrote:
olcott wrote:
On 2/27/2022 7:34 PM, Python wrote:
olcott wrote:
...
An x86 machine can do many more things but they are ruled >>>>>>>>>>>>>> as not counting because a TM cannot do these things. A TM >>>>>>>>>>>>>> can only do the subset of things that are mappings from >>>>>>>>>>>>>> inputs to outputs.
*facepalm*
you are really that low?
https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel
Clearly the concept of infinity is incoherently defined. >>>>>>>>>>>>
*facepalm*^2
https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox >>>>>>>>>> If we use a sphere instead of a ball to perform this paradox >>>>>>>>>> then when we put the two spheres back together they are no >>>>>>>>>> longer spherical. Instead they are comprised of line segments >>>>>>>>>> of infinitesimal length.
Every other geometric point on the number line is taken up by >>>>>>>>>> one sphere or the other. A line segment comprised of three >>>>>>>>>> immediately adjacent geometric points is exactly one
infinitesimal unit longer than a line segment comprised of two >>>>>>>>>> immediately adjacent geometric points.
Says you don't understand the properties of finite sets like >>>>>>>>> the reals.
There are NOT 'immediately adjacent geometric points' in the >>>>>>>>> object, as it can be shown that between ANY two (different)
real numbers (or even just rational numbers) there exists
another number between them, so no numbers are 'adjacent'
The interval [0,1] is exactly one geometric point longer than
the interval [0,1)
https://www.mathwords.com/i/interval_notation.htm
Non-Sequitur. Doesn't show that there is anything like adjacent
points.
The open interval [0,1) has no highest point, as any point you
try to name, has a point higher that is also in the interval.
It is exactly one geometric point (one infinitesimal unit) less
than 1.0
And what is that?
The problem is geometric points are infintesimally small.
Do you disagree that the interval [0,1] is exactly one geometric
point longer than the interval [0,1) ???
Yes, because that statement doesn't actually make sense when you look
at it precisly. The problem is infinitity - 1 is the same as infinity.
You FAIL the basic principles of infinite math.
It is the case based on the conventional meanings of intervals that
[0,1] is exactly one geometric point longer than [0,1).
Nope, because 'longer' doesn't apply to an infinitesimal difference to a finite value.
FAIL.
On 2/28/22 7:07 PM, olcott wrote:
On 2/28/2022 5:55 PM, Richard Damon wrote:
On 2/28/22 10:17 AM, olcott wrote:
On 2/28/2022 5:48 AM, Richard Damon wrote:
On 2/27/22 11:29 PM, olcott wrote:
On 2/27/2022 10:18 PM, Richard Damon wrote:
On 2/27/22 11:08 PM, olcott wrote:
On 2/27/2022 9:20 PM, Richard Damon wrote:
On 2/27/22 10:03 PM, olcott wrote:It is exactly one geometric point (one infinitesimal unit) less >>>>>>>> than 1.0
On 2/27/2022 8:43 PM, Richard Damon wrote:
On 2/27/22 9:28 PM, olcott wrote:The interval [0,1] is exactly one geometric point longer than >>>>>>>>>> the interval [0,1)
On 2/27/2022 8:11 PM, Python wrote:
olcott wrote:
On 2/27/2022 7:34 PM, Python wrote:
olcott wrote:
...
An x86 machine can do many more things but they are >>>>>>>>>>>>>>>> ruled as not counting because a TM cannot do these >>>>>>>>>>>>>>>> things. A TM can only do the subset of things that are >>>>>>>>>>>>>>>> mappings from inputs to outputs.
*facepalm*
you are really that low?
https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel
Clearly the concept of infinity is incoherently defined. >>>>>>>>>>>>>>
*facepalm*^2
https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox >>>>>>>>>>>> If we use a sphere instead of a ball to perform this paradox >>>>>>>>>>>> then when we put the two spheres back together they are no >>>>>>>>>>>> longer spherical. Instead they are comprised of line
segments of infinitesimal length.
Every other geometric point on the number line is taken up >>>>>>>>>>>> by one sphere or the other. A line segment comprised of >>>>>>>>>>>> three immediately adjacent geometric points is exactly one >>>>>>>>>>>> infinitesimal unit longer than a line segment comprised of >>>>>>>>>>>> two immediately adjacent geometric points.
Says you don't understand the properties of finite sets like >>>>>>>>>>> the reals.
There are NOT 'immediately adjacent geometric points' in the >>>>>>>>>>> object, as it can be shown that between ANY two (different) >>>>>>>>>>> real numbers (or even just rational numbers) there exists >>>>>>>>>>> another number between them, so no numbers are 'adjacent' >>>>>>>>>>
https://www.mathwords.com/i/interval_notation.htm
Non-Sequitur. Doesn't show that there is anything like adjacent >>>>>>>>> points.
The open interval [0,1) has no highest point, as any point you >>>>>>>>> try to name, has a point higher that is also in the interval. >>>>>>>>
And what is that?
The problem is geometric points are infintesimally small.
Do you disagree that the interval [0,1] is exactly one geometric
point longer than the interval [0,1) ???
Yes, because that statement doesn't actually make sense when you
look at it precisly. The problem is infinitity - 1 is the same as
infinity.
You FAIL the basic principles of infinite math.
It is the case based on the conventional meanings of intervals that
[0,1] is exactly one geometric point longer than [0,1).
Nope, because 'longer' doesn't apply to an infinitesimal difference
to a finite value.
FAIL.
It is self evident that the first interval maps to a line segment that
includes all of the points of the second interval and one additional
point that is immediately adjacent to the right end point that
corresponds to the second interval.
Maybe to someone who doesn't understand the effects of infinite and infinitesimals,
it makes sense, but you make the mistake of assuming the
existance of the point 'adjacent' to 1. There is no such point, or that
the omission of a finite number of points in an uncountably infinite
line affects its length.
On 2/28/22 7:59 PM, olcott wrote:
On 2/28/2022 6:34 PM, Richard Damon wrote:
On 2/28/22 7:07 PM, olcott wrote:
On 2/28/2022 5:55 PM, Richard Damon wrote:
On 2/28/22 10:17 AM, olcott wrote:
On 2/28/2022 5:48 AM, Richard Damon wrote:
On 2/27/22 11:29 PM, olcott wrote:
On 2/27/2022 10:18 PM, Richard Damon wrote:
On 2/27/22 11:08 PM, olcott wrote:
On 2/27/2022 9:20 PM, Richard Damon wrote:
On 2/27/22 10:03 PM, olcott wrote:
On 2/27/2022 8:43 PM, Richard Damon wrote:
On 2/27/22 9:28 PM, olcott wrote:
On 2/27/2022 8:11 PM, Python wrote:
olcott wrote:
On 2/27/2022 7:34 PM, Python wrote:
olcott wrote:
...
An x86 machine can do many more things but they are >>>>>>>>>>>>>>>>>> ruled as not counting because a TM cannot do these >>>>>>>>>>>>>>>>>> things. A TM can only do the subset of things that are >>>>>>>>>>>>>>>>>> mappings from inputs to outputs.
*facepalm*
you are really that low?
https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel
Clearly the concept of infinity is incoherently defined. >>>>>>>>>>>>>>>>
*facepalm*^2
https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox >>>>>>>>>>>>>> If we use a sphere instead of a ball to perform this >>>>>>>>>>>>>> paradox then when we put the two spheres back together >>>>>>>>>>>>>> they are no longer spherical. Instead they are comprised >>>>>>>>>>>>>> of line segments of infinitesimal length.
Every other geometric point on the number line is taken up >>>>>>>>>>>>>> by one sphere or the other. A line segment comprised of >>>>>>>>>>>>>> three immediately adjacent geometric points is exactly one >>>>>>>>>>>>>> infinitesimal unit longer than a line segment comprised of >>>>>>>>>>>>>> two immediately adjacent geometric points.
Says you don't understand the properties of finite sets >>>>>>>>>>>>> like the reals.
There are NOT 'immediately adjacent geometric points' in >>>>>>>>>>>>> the object, as it can be shown that between ANY two
(different) real numbers (or even just rational numbers) >>>>>>>>>>>>> there exists another number between them, so no numbers are >>>>>>>>>>>>> 'adjacent'
The interval [0,1] is exactly one geometric point longer >>>>>>>>>>>> than the interval [0,1)
https://www.mathwords.com/i/interval_notation.htm
Non-Sequitur. Doesn't show that there is anything like
adjacent points.
The open interval [0,1) has no highest point, as any point >>>>>>>>>>> you try to name, has a point higher that is also in the
interval.
It is exactly one geometric point (one infinitesimal unit) >>>>>>>>>> less than 1.0
And what is that?
The problem is geometric points are infintesimally small.
Do you disagree that the interval [0,1] is exactly one geometric >>>>>>>> point longer than the interval [0,1) ???
Yes, because that statement doesn't actually make sense when you >>>>>>> look at it precisly. The problem is infinitity - 1 is the same as >>>>>>> infinity.
You FAIL the basic principles of infinite math.
It is the case based on the conventional meanings of intervals
that [0,1] is exactly one geometric point longer than [0,1).
Nope, because 'longer' doesn't apply to an infinitesimal difference
to a finite value.
FAIL.
It is self evident that the first interval maps to a line segment
that includes all of the points of the second interval and one
additional point that is immediately adjacent to the right end point
that corresponds to the second interval.
Maybe to someone who doesn't understand the effects of infinite and
infinitesimals,
It is the same as Cantors proof, every point in the open interval maps
to a point in the closed interval and there is one point left over in
the closed interval.
I think you misunderstand the proof (unless you are thinking of some
other proof of his). Cantor proved that the Reals were uncountable, and
thus the matching arguement doesn't hold for comparing sets.
Note also, even on the countable infinte sets, that doesn't show that
the closed interval is larger.
Take for example the Natural Numbers and the even Natural Numbers.
One mapping shows that we can map half the natural numbers to the evens
and thus seem to show that there are more natural numbers than even
number, but since we can built a map that matchs every natural number
1:1 with the evens, we can show that they must be the same size.
So FAIL again.
it makes sense, but you make the mistake of assuming the existance of
the point 'adjacent' to 1. There is no such point, or that the
omission of a finite number of points in an uncountably infinite line
affects its length.
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