A UTM processes a subset of the set of finite strings such that
the first string TM description is concatenated to its input
forming the complete set of every TMD + INPUT combination.
All of the finite strings left over from the set of finite strings
are totally irrelevant because the complete set of decision problems
has already been specified.
On 1/7/23 1:19 PM, olcott wrote:
On 1/7/2023 11:23 AM, Richard Damon wrote:
On 1/7/23 12:07 PM, olcott wrote:
A UTM processes a subset of the set of finite strings such that
the first string TM description is concatenated to its input
forming the complete set of every TMD + INPUT combination.
All of the finite strings left over from the set of finite strings
are totally irrelevant because the complete set of decision problems
has already been specified.
But the subset is just the same size as the set of finite strings
since they both are of size "countable infinity".
Although it may be conventional to say it that way the set of finite
strings that are prepended with a valid TMD is a proper subset of the
set of all finite strings.
But it is still just
This is one of the "confusing" properties of countably infinite sets,
countably infinite proper subsets are still the same size as there
proper super set.
If this means that conventional set theory claims that a proper subset
of a set is also an identical set to this set then conventional set
theory is wrong.
No, not identical, just the same size.
This is part of the confusing thing with infinite numbers.
The problem is (if I am remembering right) that the number of
decision problems isn't a countably infinte number, but an
uncountable infinite, which IS larger than the number of finite
strings that could be given to a UTM, since that is, BY DEFINITION,
countable.
Yes that is the mistake of conventional wisdom.
An infinite set of unique TMD's has each element of the unique set of
finite strings appended. A UTM processes each element of these TMD+INPUT
pairs.
Which is still just a countable infinite number of inputs, with an uncountable infinte number of functions to compute.
Since you seem to have problems understanding the countable infinite,
I don't expect you to understand the uncountable infinite.
Reals are construed as uncountably infinite because there has previously
been no way to uniquely identify a pair of immediately adjacent points
on a number line. The assumption has always been that there is always a
point between two points thus no two points are immediately adjacent.
First note, I didn't say mapping to the Real, I said an uncountable
infinite set, so the same size as the reals, but is a different set.
To show they are countable, you need to show a bijection of EVERY real
to the counting numbers. (or every function over N to the counting Numbers)
Using interval notation we can see that the line segment specified by
[0,1] is exactly one geometric point longer then the line segment
specified by [0,1). Thus the point at the right end of [0,1] is
immediately adjacent to point at the right end of [0,1) with no points
in-between.
Except that there is no such thing.
What IS that point immediately adjacent to the number 1?
On 1/7/2023 11:23 AM, Richard Damon wrote:
On 1/7/23 12:07 PM, olcott wrote:
A UTM processes a subset of the set of finite strings such that
the first string TM description is concatenated to its input
forming the complete set of every TMD + INPUT combination.
All of the finite strings left over from the set of finite strings
are totally irrelevant because the complete set of decision problems
has already been specified.
But the subset is just the same size as the set of finite strings
since they both are of size "countable infinity".
Although it may be conventional to say it that way the set of finite
strings that are prepended with a valid TMD is a proper subset of the
set of all finite strings.
This is one of the "confusing" properties of countably infinite sets,
countably infinite proper subsets are still the same size as there
proper super set.
If this means that conventional set theory claims that a proper subset
of a set is also an identical set to this set then conventional set
theory is wrong.
The problem is (if I am remembering right) that the number of decision
problems isn't a countably infinte number, but an uncountable
infinite, which IS larger than the number of finite strings that could
be given to a UTM, since that is, BY DEFINITION, countable.
Yes that is the mistake of conventional wisdom.
An infinite set of unique TMD's has each element of the unique set of
finite strings appended. A UTM processes each element of these TMD+INPUT pairs.
Since you seem to have problems understanding the countable infinite,
I don't expect you to understand the uncountable infinite.
Reals are construed as uncountably infinite because there has previously
been no way to uniquely identify a pair of immediately adjacent points
on a number line. The assumption has always been that there is always a
point between two points thus no two points are immediately adjacent.
Using interval notation we can see that the line segment specified by
[0,1] is exactly one geometric point longer then the line segment
specified by [0,1). Thus the point at the right end of [0,1] is
immediately adjacent to point at the right end of [0,1) with no points in-between.
On 1/7/23 12:07 PM, olcott wrote:
A UTM processes a subset of the set of finite strings such that
the first string TM description is concatenated to its input
forming the complete set of every TMD + INPUT combination.
All of the finite strings left over from the set of finite strings
are totally irrelevant because the complete set of decision problems
has already been specified.
But the subset is just the same size as the set of finite strings since
they both are of size "countable infinity".
This is one of the "confusing" properties of countably infinite sets, countably infinite proper subsets are still the same size as there
proper super set.
The problem is (if I am remembering right) that the number of decision problems isn't a countably infinte number, but an uncountable infinite,
which IS larger than the number of finite strings that could be given to
a UTM, since that is, BY DEFINITION, countable.
Since you seem to have problems understanding the countable infinite, I
don't expect you to understand the uncountable infinite.
On 1/7/2023 12:41 PM, Richard Damon wrote:
On 1/7/23 1:19 PM, olcott wrote:
On 1/7/2023 11:23 AM, Richard Damon wrote:
On 1/7/23 12:07 PM, olcott wrote:
A UTM processes a subset of the set of finite strings such that
the first string TM description is concatenated to its input
forming the complete set of every TMD + INPUT combination.
All of the finite strings left over from the set of finite strings
are totally irrelevant because the complete set of decision problems >>>>> has already been specified.
But the subset is just the same size as the set of finite strings
since they both are of size "countable infinity".
Although it may be conventional to say it that way the set of finite
strings that are prepended with a valid TMD is a proper subset of the
set of all finite strings.
But it is still just
This is one of the "confusing" properties of countably infinite
sets, countably infinite proper subsets are still the same size as
there proper super set.
If this means that conventional set theory claims that a proper subset
of a set is also an identical set to this set then conventional set
theory is wrong.
No, not identical, just the same size.
Even though conventional, it is incorrect to say that a proper subset of
a set has the same size as the set.
This is part of the confusing thing with infinite numbers.
The problem is (if I am remembering right) that the number of
decision problems isn't a countably infinte number, but an
uncountable infinite, which IS larger than the number of finite
strings that could be given to a UTM, since that is, BY DEFINITION,
countable.
Yes that is the mistake of conventional wisdom.
An infinite set of unique TMD's has each element of the unique set of
finite strings appended. A UTM processes each element of these TMD+INPUT >>> pairs.
Which is still just a countable infinite number of inputs, with an
uncountable infinte number of functions to compute.
Since you seem to have problems understanding the countable
infinite, I don't expect you to understand the uncountable infinite.
Reals are construed as uncountably infinite because there has previously >>> been no way to uniquely identify a pair of immediately adjacent points
on a number line. The assumption has always been that there is always a
point between two points thus no two points are immediately adjacent.
First note, I didn't say mapping to the Real, I said an uncountable
infinite set, so the same size as the reals, but is a different set.
Reals are also construed as an uncountable infinite set.
To show they are countable, you need to show a bijection of EVERY real
to the counting numbers. (or every function over N to the counting
Numbers)
Using interval notation we can see that the line segment specified by
[0,1] is exactly one geometric point longer then the line segment
specified by [0,1). Thus the point at the right end of [0,1] is
immediately adjacent to point at the right end of [0,1) with no points
in-between.
Except that there is no such thing.
What IS that point immediately adjacent to the number 1?
The right point of the line segment [0,1) is immediately adjacent to 1.
On 1/7/23 1:49 PM, olcott wrote:It is a point that is uniquely identified as the rightmost point of the following line segment [0,1).
On 1/7/2023 12:41 PM, Richard Damon wrote:
On 1/7/23 1:19 PM, olcott wrote:
On 1/7/2023 11:23 AM, Richard Damon wrote:
On 1/7/23 12:07 PM, olcott wrote:
A UTM processes a subset of the set of finite strings such that
the first string TM description is concatenated to its input
forming the complete set of every TMD + INPUT combination.
All of the finite strings left over from the set of finite strings >>>>>> are totally irrelevant because the complete set of decision problems >>>>>> has already been specified.
But the subset is just the same size as the set of finite strings
since they both are of size "countable infinity".
Although it may be conventional to say it that way the set of finite
strings that are prepended with a valid TMD is a proper subset of the
set of all finite strings.
But it is still just
This is one of the "confusing" properties of countably infinite
sets, countably infinite proper subsets are still the same size as
there proper super set.
If this means that conventional set theory claims that a proper subset >>>> of a set is also an identical set to this set then conventional set
theory is wrong.
No, not identical, just the same size.
Even though conventional, it is incorrect to say that a proper subset
of a set has the same size as the set.
Why? You comment just shows an failure to understand how infinite
numbers work.
In fact, if you TRY to make that sort of logic work, you get
inconsistencies.
This comes form things like the set of all even numers is BOTH a subset
of all the Natural Numbers (removing all the odd numbers) or just a relabeling of the set of Natural Numbers (replacing each one with twice itself).
These are EXACTLY the same set, so the set of all even numbers must be
bigger than itself with your logic.
This is part of the confusing thing with infinite numbers.
The problem is (if I am remembering right) that the number of
decision problems isn't a countably infinte number, but an
uncountable infinite, which IS larger than the number of finite
strings that could be given to a UTM, since that is, BY DEFINITION,
countable.
Yes that is the mistake of conventional wisdom.
An infinite set of unique TMD's has each element of the unique set of
finite strings appended. A UTM processes each element of these
TMD+INPUT
pairs.
Which is still just a countable infinite number of inputs, with an
uncountable infinte number of functions to compute.
Since you seem to have problems understanding the countable
infinite, I don't expect you to understand the uncountable infinite.
Reals are construed as uncountably infinite because there has
previously
been no way to uniquely identify a pair of immediately adjacent points >>>> on a number line. The assumption has always been that there is always a >>>> point between two points thus no two points are immediately adjacent.
First note, I didn't say mapping to the Real, I said an uncountable
infinite set, so the same size as the reals, but is a different set.
Reals are also construed as an uncountable infinite set.
Yep, but a DIFFERENT uncountable infiite set.
To show they are countable, you need to show a bijection of EVERY
real to the counting numbers. (or every function over N to the
counting Numbers)
Using interval notation we can see that the line segment specified
by [0,1] is exactly one geometric point longer then the line segment
specified by [0,1). Thus the point at the right end of [0,1] is
immediately adjacent to point at the right end of [0,1) with no points >>>> in-between.
Except that there is no such thing.
What IS that point immediately adjacent to the number 1?
The right point of the line segment [0,1) is immediately adjacent to 1.
Which is?
You can't name it, because it isn't a unique pooint with a value.
So it just doesn't exist.
On 1/7/23 3:25 PM, olcott wrote:
On 1/7/2023 1:49 PM, Richard Damon wrote:
On 1/7/23 1:49 PM, olcott wrote:It is a point that is uniquely identified as the rightmost point of the
On 1/7/2023 12:41 PM, Richard Damon wrote:
On 1/7/23 1:19 PM, olcott wrote:
Using interval notation we can see that the line segment specified >>>>>> by [0,1] is exactly one geometric point longer then the line segment >>>>>> specified by [0,1). Thus the point at the right end of [0,1] is
immediately adjacent to point at the right end of [0,1) with no
points
in-between.
Except that there is no such thing.
What IS that point immediately adjacent to the number 1?
The right point of the line segment [0,1) is immediately adjacent to 1. >>>>
Which is?
You can't name it, because it isn't a unique pooint with a value.
So it just doesn't exist.
following line segment [0,1).
Except what ever point x you name, has another point closer with a value
of (x+1)/2
The conventional meaning of interval notation knows that there are no
points between the rightmost point of the line segment [0,1] and the
rightmost point of the line segment [0,1).
No it means all the points 0 <= x < 1, or the points on the line
excluding that end point.
The definition NEVER talks of the "right most point" that is just less
than 1.
There is a bijection between each of these points and a real number,
thus specifying a pair of real numbers that are immediately adjacent to
each other.
Nope, you don't biject to "a real number", you biject the elements of a
set.
Bijectection also doesn't define "adjacent".
You are just showing you don't know what you are talking about.
Your brain just don't understand the concepts, so of course you are
confused.
On 1/7/2023 1:49 PM, Richard Damon wrote:
On 1/7/23 1:49 PM, olcott wrote:It is a point that is uniquely identified as the rightmost point of the following line segment [0,1).
On 1/7/2023 12:41 PM, Richard Damon wrote:
On 1/7/23 1:19 PM, olcott wrote:
Using interval notation we can see that the line segment specified
by [0,1] is exactly one geometric point longer then the line segment >>>>> specified by [0,1). Thus the point at the right end of [0,1] is
immediately adjacent to point at the right end of [0,1) with no points >>>>> in-between.
Except that there is no such thing.
What IS that point immediately adjacent to the number 1?
The right point of the line segment [0,1) is immediately adjacent to 1.
Which is?
You can't name it, because it isn't a unique pooint with a value.
So it just doesn't exist.
The conventional meaning of interval notation knows that there are no
points between the rightmost point of the line segment [0,1] and the rightmost point of the line segment [0,1).
There is a bijection between each of these points and a real number,
thus specifying a pair of real numbers that are immediately adjacent to
each other.
On 1/7/2023 2:38 PM, Richard Damon wrote:
On 1/7/23 3:25 PM, olcott wrote:
On 1/7/2023 1:49 PM, Richard Damon wrote:
On 1/7/23 1:49 PM, olcott wrote:It is a point that is uniquely identified as the rightmost point of the
On 1/7/2023 12:41 PM, Richard Damon wrote:
On 1/7/23 1:19 PM, olcott wrote:
Using interval notation we can see that the line segment
specified by [0,1] is exactly one geometric point longer then the >>>>>>> line segment
specified by [0,1). Thus the point at the right end of [0,1] is
immediately adjacent to point at the right end of [0,1) with no
points
in-between.
Except that there is no such thing.
What IS that point immediately adjacent to the number 1?
The right point of the line segment [0,1) is immediately adjacent
to 1.
Which is?
You can't name it, because it isn't a unique pooint with a value.
So it just doesn't exist.
following line segment [0,1).
Except what ever point x you name, has another point closer with a
value of (x+1)/2
The conventional meaning of interval notation knows that there are no
points between the rightmost point of the line segment [0,1] and the
rightmost point of the line segment [0,1).
No it means all the points 0 <= x < 1, or the points on the line
excluding that end point.
The definition NEVER talks of the "right most point" that is just less
than 1.
None the less it does specify a rightmost point that is immediately
adjacent to 1
There is a bijection between each of these points and a real number,
thus specifying a pair of real numbers that are immediately adjacent to
each other.
Nope, you don't biject to "a real number", you biject the elements of
a set.
Bijectection also doesn't define "adjacent".
Every point on a number line has a unique corresponding real number.
I did uniquely identify a pair of points on a number line that are immediately adjacent. Therefore these points must correspond to Real
numbers that are immediately adjacent.
You are just showing you don't know what you are talking about.
Your brain just don't understand the concepts, so of course you are
confused.
On 1/7/23 3:53 PM, olcott wrote:
On 1/7/2023 2:38 PM, Richard Damon wrote:
On 1/7/23 3:25 PM, olcott wrote:
On 1/7/2023 1:49 PM, Richard Damon wrote:
On 1/7/23 1:49 PM, olcott wrote:It is a point that is uniquely identified as the rightmost point of the >>>> following line segment [0,1).
On 1/7/2023 12:41 PM, Richard Damon wrote:
On 1/7/23 1:19 PM, olcott wrote:
Using interval notation we can see that the line segment
specified by [0,1] is exactly one geometric point longer then
the line segment
specified by [0,1). Thus the point at the right end of [0,1] is >>>>>>>> immediately adjacent to point at the right end of [0,1) with no >>>>>>>> points
in-between.
Except that there is no such thing.
What IS that point immediately adjacent to the number 1?
The right point of the line segment [0,1) is immediately adjacent
to 1.
Which is?
You can't name it, because it isn't a unique pooint with a value.
So it just doesn't exist.
Except what ever point x you name, has another point closer with a
value of (x+1)/2
The conventional meaning of interval notation knows that there are no
points between the rightmost point of the line segment [0,1] and the
rightmost point of the line segment [0,1).
No it means all the points 0 <= x < 1, or the points on the line
excluding that end point.
The definition NEVER talks of the "right most point" that is just
less than 1.
None the less it does specify a rightmost point that is immediately
adjacent to 1
Nope, because what ever point you try to chose, there is one closer.
This is called the Density property.
On 1/7/2023 3:19 PM, Richard Damon wrote:
On 1/7/23 3:53 PM, olcott wrote:You can assume that yet interval notation contradicts you.
On 1/7/2023 2:38 PM, Richard Damon wrote:
On 1/7/23 3:25 PM, olcott wrote:
On 1/7/2023 1:49 PM, Richard Damon wrote:
On 1/7/23 1:49 PM, olcott wrote:It is a point that is uniquely identified as the rightmost point of
On 1/7/2023 12:41 PM, Richard Damon wrote:
On 1/7/23 1:19 PM, olcott wrote:
Using interval notation we can see that the line segment
specified by [0,1] is exactly one geometric point longer then >>>>>>>>> the line segment
specified by [0,1). Thus the point at the right end of [0,1] is >>>>>>>>> immediately adjacent to point at the right end of [0,1) with no >>>>>>>>> points
in-between.
Except that there is no such thing.
What IS that point immediately adjacent to the number 1?
The right point of the line segment [0,1) is immediately adjacent >>>>>>> to 1.
Which is?
You can't name it, because it isn't a unique pooint with a value.
So it just doesn't exist.
the
following line segment [0,1).
Except what ever point x you name, has another point closer with a
value of (x+1)/2
The conventional meaning of interval notation knows that there are no >>>>> points between the rightmost point of the line segment [0,1] and the >>>>> rightmost point of the line segment [0,1).
No it means all the points 0 <= x < 1, or the points on the line
excluding that end point.
The definition NEVER talks of the "right most point" that is just
less than 1.
None the less it does specify a rightmost point that is immediately
adjacent to 1
Nope, because what ever point you try to chose, there is one closer.
This is called the Density property.
I will clarify the right point (not rightmost point) of the line segment specified by [0,1) is immediately adjacent to 1.
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