What you do is start lining up the numbers in a diagonal, so that you
are going to hit lots of numbers, then what you do is pick one that's
not in the list and you list it next. In this way you list all the real numbers in an order just like the integers. Cantor was a pussy.
On 5/4/2026 6:58 AM, phoenix wrote:
What you do is start lining up the numbers in a diagonal, so that you
are going to hit lots of numbers, then what you do is pick one that's
not in the list and you list it next. In this way you list all the
real numbers in an order just like the integers. Cantor was a pussy.
Ummm... No. Think of the infinity between say .01 and .002. You will run
out of integers... big time. You cannot escape the infinite gap, and
your integers will not be enough...
On 5/4/2026 3:18 PM, Chris M. Thomasson wrote:
On 5/4/2026 6:58 AM, phoenix wrote:
What you do is start lining up the numbers in a diagonal, so that you
are going to hit lots of numbers, then what you do is pick one that's
not in the list and you list it next. In this way you list all the
real numbers in an order just like the integers. Cantor was a pussy.
Ummm... No. Think of the infinity between say .01 and .002. You will
run out of integers... big time. You cannot escape the infinite gap,
and your integers will not be enough...
It seems like you are thinking of using a countable infinity to map an uncountable infinity...
What you do is start lining up the numbers in a diagonal, so that you
are going to hit lots of numbers, then what you do is pick one that's
not in the list and you list it next. In this way you list all the real numbers in an order just like the integers. Cantor was a pussy.
On 5/4/2026 6:58 AM, phoenix wrote:
What you do is start lining up the numbers in a diagonal, so that you
are going to hit lots of numbers, then what you do is pick one that's
not in the list and you list it next. In this way you list all the
real numbers in an order just like the integers. Cantor was a pussy.
Ummm... No. Think of the infinity between say .01 and .002. You will run
out of integers...
big time. You cannot escape the infinite gap, and
your integers will not be enough...
On 5/4/2026 3:18 PM, Chris M. Thomasson wrote:
On 5/4/2026 6:58 AM, phoenix wrote:
What you do is start lining up the numbers in a diagonal, so that you
are going to hit lots of numbers, then what you do is pick one that's
not in the list and you list it next. In this way you list all the
real numbers in an order just like the integers. Cantor was a pussy.
Ummm... No. Think of the infinity between say .01 and .002. You will
run out of integers... big time. You cannot escape the infinite gap,
and your integers will not be enough...
It seems like you are thinking of using a countable infinity to map an uncountable infinity...
On 05/04/2026 03:24 PM, Chris M. Thomasson wrote:
On 5/4/2026 3:18 PM, Chris M. Thomasson wrote:
On 5/4/2026 6:58 AM, phoenix wrote:
What you do is start lining up the numbers in a diagonal, so that you
are going to hit lots of numbers, then what you do is pick one that's
not in the list and you list it next. In this way you list all the
real numbers in an order just like the integers. Cantor was a pussy.
Ummm... No. Think of the infinity between say .01 and .002. You will
run out of integers... big time. You cannot escape the infinite gap,
and your integers will not be enough...
It seems like you are thinking of using a countable infinity to map an
uncountable infinity...
Instead it's 0-iota, 1-iota, 2-iota, ..., infinity-iota = 1.
There's are about two proper examples of "countable continuous domains",
one is these line-reals or iota-values, defined exactly
between 0 and 1, and the other is for an account of the signal-reals,
that like the function that defines the line-reals, that the function
defines the signal-reals, is non-Cartesian, so that besides that it
falls out of the number-theoretic arguments otherwise for
un-countability as non-contradicted, then furthermore it's not
guilty by association as it's a bijection itself yet there aren't
any bijections from these countable continuous domains to uncountable continous domains (field-reals, complete ordered field).
I'm not sure what "Bob Dobbs" would say about it, though a usual
idea would be that Pinks don't get it.
Ross Finlayson <ross.a.finlayson@gmail.com> wrote in news:MkKdnfkAb4VIrWT0nZ2dnZfqn_WdnZ2d@giganews.com:
On 05/04/2026 03:24 PM, Chris M. Thomasson wrote:
On 5/4/2026 3:18 PM, Chris M. Thomasson wrote:
On 5/4/2026 6:58 AM, phoenix wrote:
What you do is start lining up the numbers in a diagonal, so that you >>>>> are going to hit lots of numbers, then what you do is pick one that's >>>>> not in the list and you list it next. In this way you list all the
real numbers in an order just like the integers. Cantor was a pussy. >>>>>
Ummm... No. Think of the infinity between say .01 and .002. You will
run out of integers... big time. You cannot escape the infinite gap,
and your integers will not be enough...
It seems like you are thinking of using a countable infinity to map an
uncountable infinity...
Instead it's 0-iota, 1-iota, 2-iota, ..., infinity-iota = 1.
There's are about two proper examples of "countable continuous domains",
one is these line-reals or iota-values, defined exactly
between 0 and 1, and the other is for an account of the signal-reals,
that like the function that defines the line-reals, that the function
defines the signal-reals, is non-Cartesian, so that besides that it
falls out of the number-theoretic arguments otherwise for
un-countability as non-contradicted, then furthermore it's not
guilty by association as it's a bijection itself yet there aren't
any bijections from these countable continuous domains to uncountable
continous domains (field-reals, complete ordered field).
I'm not sure what "Bob Dobbs" would say about it, though a usual
idea would be that Pinks don't get it.
You sound really dumb.
i wanted to say something about reals and integers.
reals are probably not real in the universal sense that the universe we inhabit is continuous along a real line.
jojo wrote:
i wanted to say something about reals and integers.Physicists continue to wonder whether there is a smallest unit of
reals are probably not real in the universal sense that the
universe we inhabit is continuous along a real line.
time that there cannot be smaller than but haven't had any
success recently narrowing the problem down.
Am 05.05.2026 um 00:24 schrieb Chris M. Thomasson:
On 5/4/2026 3:18 PM, Chris M. Thomasson wrote:
On 5/4/2026 6:58 AM, phoenix wrote:
What you do is start lining up the numbers in a diagonal, so that
you are going to hit lots of numbers, then what you do is pick one
that's not in the list and you list it next. In this way you list
all the real numbers in an order just like the integers. Cantor was
a pussy.
Ummm... No. Think of the infinity between say .01 and .002. You will
run out of integers... big time. You cannot escape the infinite gap,
and your integers will not be enough...
It seems like you are thinking of using a countable infinity to map an
uncountable infinity...
There is neither a countable nor an uncountable infinity.
The reason is the existence of dark numbers: https://www.academia.edu/91188101/ Proof_of_the_existence_of_dark_numbers_bilingual_version_
phoenix wrote:
jojo wrote:
i wanted to say something about reals and integers.Physicists continue to wonder whether there is a smallest unit of time
reals are probably not real in the universal sense that the universe
we inhabit is continuous along a real line.
that there cannot be smaller than but haven't had any success recently
narrowing the problem down.
speaking of time, have you heard that zeno guy and his paradox?
does it prove that the universe is discrete and not continuous?
On 05/07/2026 08:26 AM, jojo wrote:
phoenix wrote:
jojo wrote:
i wanted to say something about reals and integers.Physicists continue to wonder whether there is a smallest unit of time
reals are probably not real in the universal sense that the universe
we inhabit is continuous along a real line.
that there cannot be smaller than but haven't had any success recently
narrowing the problem down.
speaking of time, have you heard that zeno guy and his paradox?
does it prove that the universe is discrete and not continuous?
Actually here it's established since at least about a decade
that the iota-values the range of the equivalency function
have least-upper-bound and sigma-algebras the measure 1.0,
thusly having extent density completeness measure and
thusly being a continuous domain, that there's only and
exactly this one functions between N and [ 0,1 ] that's
a bijective function (being one-to-one and onto, both
an injection and a surjection, a function), this the
"natural/unit equivalency function" or "N/U EF" or "EF",
then having also its reverse image (1 - EF = REF,
"reverse equivalency function"), so that the continuous
and discrete have a relation that's a _countable_ continuous
domain the continuous, while though the function so defining
it is a _non-Cartesian_ function, so that besides falling out
of the arguments otherwise for un-countability as un-contradicted,
that also as non-Cartesian it stays out of the way of the Cantor-Schroeder-Bernstein and can't be cancelled away algebraically.
So, these are "line-reals" not "field-reals" the usual standard complete-ordered-field after fractions and rationals the
ordered field the "completion" of those, also it makes ground for there existing instead of being "axiomatized" the completion of the complete ordered field.
There's the other account of a countable continuous domain,
the "signal-reals", about that there's a model of rationals
that since a continuous domain exists, that this sort of
f: Q <-> S \ Q is a bijection between Q and a countable
continuous domain, the signal-reals.
That the universe is continuous is always formally non-falsifiable,
so it's un-scientific to say so, though it can be scientific to
say that a discrete approximation approaches particular accuracies.
Then "running constants" in real theory paint-cans or
throws water on otherwise those claiming "grainy universe".
Ross Finlayson wrote:
On 05/07/2026 08:26 AM, jojo wrote:What is your reason for focusing on [0,1]?
phoenix wrote:
jojo wrote:
i wanted to say something about reals and integers.Physicists continue to wonder whether there is a smallest unit of time >>>> that there cannot be smaller than but haven't had any success recently >>>> narrowing the problem down.
reals are probably not real in the universal sense that the universe >>>>> we inhabit is continuous along a real line.
speaking of time, have you heard that zeno guy and his paradox?
does it prove that the universe is discrete and not continuous?
Actually here it's established since at least about a decade
that the iota-values the range of the equivalency function
have least-upper-bound and sigma-algebras the measure 1.0,
thusly having extent density completeness measure and
thusly being a continuous domain, that there's only and
exactly this one functions between N and [ 0,1 ] that's
a bijective function (being one-to-one and onto, both
an injection and a surjection, a function), this the
"natural/unit equivalency function" or "N/U EF" or "EF",
then having also its reverse image (1 - EF = REF,
"reverse equivalency function"), so that the continuous
and discrete have a relation that's a _countable_ continuous
domain the continuous, while though the function so defining
it is a _non-Cartesian_ function, so that besides falling out
of the arguments otherwise for un-countability as un-contradicted,
that also as non-Cartesian it stays out of the way of the
Cantor-Schroeder-Bernstein and can't be cancelled away algebraically.
So, these are "line-reals" not "field-reals" the usual standard
complete-ordered-field after fractions and rationals the
ordered field the "completion" of those, also it makes ground for there
existing instead of being "axiomatized" the completion of the complete
ordered field.
There's the other account of a countable continuous domain,
the "signal-reals", about that there's a model of rationals
that since a continuous domain exists, that this sort of
f: Q <-> S \ Q is a bijection between Q and a countable
continuous domain, the signal-reals.
That the universe is continuous is always formally non-falsifiable,
so it's un-scientific to say so, though it can be scientific to
say that a discrete approximation approaches particular accuracies.
Then "running constants" in real theory paint-cans or
throws water on otherwise those claiming "grainy universe".
From what you've told me, the integers map to any set [-M,M] for any M
no matter how large.
Unfortunately, this does not mean that M can be positive infinity,
however, which is where it would break down and no longer work.
Ross Finlayson wrote:
On 05/07/2026 08:26 AM, jojo wrote:What is your reason for focusing on [0,1]?
phoenix wrote:
jojo wrote:
i wanted to say something about reals and integers.Physicists continue to wonder whether there is a smallest unit of time >>>> that there cannot be smaller than but haven't had any success recently >>>> narrowing the problem down.
reals are probably not real in the universal sense that the universe >>>>> we inhabit is continuous along a real line.
speaking of time, have you heard that zeno guy and his paradox?
does it prove that the universe is discrete and not continuous?
Actually here it's established since at least about a decade
that the iota-values the range of the equivalency function
have least-upper-bound and sigma-algebras the measure 1.0,
thusly having extent density completeness measure and
thusly being a continuous domain, that there's only and
exactly this one functions between N and [ 0,1 ] that's
a bijective function (being one-to-one and onto, both
an injection and a surjection, a function), this the
"natural/unit equivalency function" or "N/U EF" or "EF",
then having also its reverse image (1 - EF = REF,
"reverse equivalency function"), so that the continuous
and discrete have a relation that's a _countable_ continuous
domain the continuous, while though the function so defining
it is a _non-Cartesian_ function, so that besides falling out
of the arguments otherwise for un-countability as un-contradicted,
that also as non-Cartesian it stays out of the way of the
Cantor-Schroeder-Bernstein and can't be cancelled away algebraically.
So, these are "line-reals" not "field-reals" the usual standard
complete-ordered-field after fractions and rationals the
ordered field the "completion" of those, also it makes ground for there
existing instead of being "axiomatized" the completion of the complete
ordered field.
There's the other account of a countable continuous domain,
the "signal-reals", about that there's a model of rationals
that since a continuous domain exists, that this sort of
f: Q <-> S \ Q is a bijection between Q and a countable
continuous domain, the signal-reals.
That the universe is continuous is always formally non-falsifiable,
so it's un-scientific to say so, though it can be scientific to
say that a discrete approximation approaches particular accuracies.
Then "running constants" in real theory paint-cans or
throws water on otherwise those claiming "grainy universe".
From what you've told me, the integers map to any set [-M,M] for any M
no matter how large.
Unfortunately, this does not mean that M can be positive infinity,
however, which is where it would break down and no longer work.
On 05/09/2026 11:25 AM, phoenix wrote:
Ross Finlayson wrote:
On 05/07/2026 08:26 AM, jojo wrote:What is your reason for focusing on [0,1]?
phoenix wrote:
jojo wrote:
i wanted to say something about reals and integers.Physicists continue to wonder whether there is a smallest unit of time >>>>> that there cannot be smaller than but haven't had any success recently >>>>> narrowing the problem down.
reals are probably not real in the universal sense that the universe >>>>>> we inhabit is continuous along a real line.
speaking of time, have you heard that zeno guy and his paradox?
does it prove that the universe is discrete and not continuous?
Actually here it's established since at least about a decade
that the iota-values the range of the equivalency function
have least-upper-bound and sigma-algebras the measure 1.0,
thusly having extent density completeness measure and
thusly being a continuous domain, that there's only and
exactly this one functions between N and [ 0,1 ] that's
a bijective function (being one-to-one and onto, both
an injection and a surjection, a function), this the
"natural/unit equivalency function" or "N/U EF" or "EF",
then having also its reverse image (1 - EF = REF,
"reverse equivalency function"), so that the continuous
and discrete have a relation that's a _countable_ continuous
domain the continuous, while though the function so defining
it is a _non-Cartesian_ function, so that besides falling out
of the arguments otherwise for un-countability as un-contradicted,
that also as non-Cartesian it stays out of the way of the
Cantor-Schroeder-Bernstein and can't be cancelled away algebraically.
So, these are "line-reals" not "field-reals" the usual standard
complete-ordered-field after fractions and rationals the
ordered field the "completion" of those, also it makes ground for there
existing instead of being "axiomatized" the completion of the complete
ordered field.
There's the other account of a countable continuous domain,
the "signal-reals", about that there's a model of rationals
that since a continuous domain exists, that this sort of
f: Q <-> S \ Q is a bijection between Q and a countable
continuous domain, the signal-reals.
That the universe is continuous is always formally non-falsifiable,
so it's un-scientific to say so, though it can be scientific to
say that a discrete approximation approaches particular accuracies.
Then "running constants" in real theory paint-cans or
throws water on otherwise those claiming "grainy universe".
-aFrom what you've told me, the integers map to any set [-M,M] for any M
no matter how large.
Unfortunately, this does not mean that M can be positive infinity,
however, which is where it would break down and no longer work.
The reason why is because the function is n/d, just like numerator/denominator, for only non-negative numbers,
those being the simplest, say, for n between 0 and d,
and "in the limit" as d goes to infinity, that the
most usual way to visualize that is drawing a line-segment
with a pencil the line-drawing, about the usual idea that
a "clock-arithmetic" is monotone and constant in time, time, time, ....
Then, about whether the infinitely-many numbers contain
at least one infinite-grand number, if you only have a model
of infinitely-many each-finite, then the idea that anything
quantifies over those (collects) can be called "Russell's paradox"
then instead of being a paradox, it's just that Archimedes
(infinitely-many and no-infinitely-grand) is just a reductionism
already, the original sort of restriction-of-comprehension,
since expansion-of-comprehension makes for models of integers
the infinitely-many with at least one infinitely-grand, or
for example half infinitely-grand or mostly infinitely-grand.
So, the reason for focusing on [0,1] as a magnitude like a
classical finite magnitude, is that if if were [0, oo) then
the definition of the function would fall flat and disappear,
that it's "non-Cartesian among otherwise Cartesian functions"
yet "a function" has that it's non-reorderable and basically
establishes itself only on [0,1] (or by a finite scale),
thusly mathematics has it as an object of mathematics.
Ross Finlayson wrote:
On 05/09/2026 11:25 AM, phoenix wrote:
Ross Finlayson wrote:
On 05/07/2026 08:26 AM, jojo wrote:What is your reason for focusing on [0,1]?
phoenix wrote:
jojo wrote:
i wanted to say something about reals and integers.Physicists continue to wonder whether there is a smallest unit of
reals are probably not real in the universal sense that the universe >>>>>>> we inhabit is continuous along a real line.
time
that there cannot be smaller than but haven't had any success
recently
narrowing the problem down.
speaking of time, have you heard that zeno guy and his paradox?
does it prove that the universe is discrete and not continuous?
Actually here it's established since at least about a decade
that the iota-values the range of the equivalency function
have least-upper-bound and sigma-algebras the measure 1.0,
thusly having extent density completeness measure and
thusly being a continuous domain, that there's only and
exactly this one functions between N and [ 0,1 ] that's
a bijective function (being one-to-one and onto, both
an injection and a surjection, a function), this the
"natural/unit equivalency function" or "N/U EF" or "EF",
then having also its reverse image (1 - EF = REF,
"reverse equivalency function"), so that the continuous
and discrete have a relation that's a _countable_ continuous
domain the continuous, while though the function so defining
it is a _non-Cartesian_ function, so that besides falling out
of the arguments otherwise for un-countability as un-contradicted,
that also as non-Cartesian it stays out of the way of the
Cantor-Schroeder-Bernstein and can't be cancelled away algebraically.
So, these are "line-reals" not "field-reals" the usual standard
complete-ordered-field after fractions and rationals the
ordered field the "completion" of those, also it makes ground for there >>>> existing instead of being "axiomatized" the completion of the complete >>>> ordered field.
There's the other account of a countable continuous domain,
the "signal-reals", about that there's a model of rationals
that since a continuous domain exists, that this sort of
f: Q <-> S \ Q is a bijection between Q and a countable
continuous domain, the signal-reals.
That the universe is continuous is always formally non-falsifiable,
so it's un-scientific to say so, though it can be scientific to
say that a discrete approximation approaches particular accuracies.
Then "running constants" in real theory paint-cans or
throws water on otherwise those claiming "grainy universe".
From what you've told me, the integers map to any set [-M,M] for any M
no matter how large.
Unfortunately, this does not mean that M can be positive infinity,
however, which is where it would break down and no longer work.
The reason why is because the function is n/d, just like
numerator/denominator, for only non-negative numbers,
those being the simplest, say, for n between 0 and d,
and "in the limit" as d goes to infinity, that the
most usual way to visualize that is drawing a line-segment
with a pencil the line-drawing, about the usual idea that
a "clock-arithmetic" is monotone and constant in time, time, time, ....
Then, about whether the infinitely-many numbers contain
at least one infinite-grand number, if you only have a model
of infinitely-many each-finite, then the idea that anything
quantifies over those (collects) can be called "Russell's paradox"
then instead of being a paradox, it's just that Archimedes
(infinitely-many and no-infinitely-grand) is just a reductionism
already, the original sort of restriction-of-comprehension,
since expansion-of-comprehension makes for models of integers
the infinitely-many with at least one infinitely-grand, or
for example half infinitely-grand or mostly infinitely-grand.
I suppose this is a good reason. I'm not certain why Brennus called you
dumb. Maybe he was trying to make me feel good because you were refuting
what I wrote... He often gives me compliments but I don't understand
them because on the other hand he is often insulting me too.
So, the reason for focusing on [0,1] as a magnitude like a
classical finite magnitude, is that if if were [0, oo) then
the definition of the function would fall flat and disappear,
that it's "non-Cartesian among otherwise Cartesian functions"
yet "a function" has that it's non-reorderable and basically
establishes itself only on [0,1] (or by a finite scale),
thusly mathematics has it as an object of mathematics.
On 05/07/2026 08:26 AM, jojo wrote:
phoenix wrote:
jojo wrote:
i wanted to say something about reals and integers.Physicists continue to wonder whether there is a smallest unit
reals are probably not real in the universal sense that the
universe
we inhabit is continuous along a real line.
of time
that there cannot be smaller than but haven't had any success
recently
narrowing the problem down.
speaking of time, have you heard that zeno guy and his paradox?
does it prove that the universe is discrete and not continuous?
Actually here it's established since at least about a decade
that the iota-values the range of the equivalency function
have least-upper-bound and sigma-algebras the measure 1.0,
thusly having extent density completeness measure and
thusly being a continuous domain, that there's only and
exactly this one functions between N and [ 0,1 ] that's
a bijective function (being one-to-one and onto, both
an injection and a surjection, a function), this the
"natural/unit equivalency function" or "N/U EF" or "EF",
then having also its reverse image (1 - EF = REF,
"reverse equivalency function"), so that the continuous
and discrete have a relation that's a _countable_ continuous
domain the continuous, while though the function so defining
it is a _non-Cartesian_ function, so that besides falling out
of the arguments otherwise for un-countability as un-contradicted,
that also as non-Cartesian it stays out of the way of the Cantor-Schroeder-Bernstein and can't be cancelled away
algebraically.
So, these are "line-reals" not "field-reals" the usual standard complete-ordered-field after fractions and rationals the
ordered field the "completion" of those, also it makes ground for
there
existing instead of being "axiomatized" the completion of the
complete
ordered field.
There's the other account of a countable continuous domain,
the "signal-reals", about that there's a model of rationals
that since a continuous domain exists, that this sort of
f: Q <-> S \ Q is a bijection between Q and a countable
continuous domain, the signal-reals.
That the universe is continuous is always formally non-falsifiable,
so it's un-scientific to say so, though it can be scientific to
say that a discrete approximation approaches particular accuracies.
Then "running constants" in real theory paint-cans or
throws water on otherwise those claiming "grainy universe".
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