On 10/20/2024 2:13 PM, Richard Damon wrote:
On 10/20/24 11:32 AM, olcott wrote:
On 10/20/2024 6:46 AM, Richard Damon wrote:
A "First Principles" approach that you refer to STARTS with an study
and understanding of the actual basic principles of the system. That
would be things like the basic definitions of things like "Program",
"Halting" "Deciding", "Turing Machine", and then from those
concepts, sees what can be done, without trying to rely on the ideas
that others have used, but see if they went down a wrong track, and
the was a different path in the same system.
The actual barest essence for formal systems and computations
is finite string transformation rules applied to finite strings.
So, show what you can do with that.
Note, WHAT the rules can be is very important, and seems to be beyond
you ability to reason about.
After all, all a Turing Machine is is a way of defining a finite
stting transformation computation.
The next minimal increment of further elaboration is that some
finite strings has an assigned or derived property of Boolean
true. At this point of elaboration Boolean true has no more
semantic meaning than FooBar.
And since you can't do the first step, you don't understand what that
actually means.
As soon as any algorithm is defined to transform any finite
string into any other finite string we have conclusively
proven that algorithms can transform finite strings.
The simplest formal system that I can think of transforms
pairs of strings of ASCII digits into their sum. This algorithm
can be easily specified in C.
Some finite strings are assigned the FooBar property and other
finite string derive the FooBar property by applying FooBar
preserving operations to the first set.
But, since we have an infinite number of finite strings to be assigned
values, we can't just enumerate that set.
The infinite set of pairs of finite strings of ASCII digits
can be easily transformed into their corresponding sum for
arbitrary elements of this infinite set.
Once finite strings have the FooBar property we can define
computations that apply Foobar preserving operations to
determine if other finite strings also have this FooBar property.
It seems you never even learned the First Principles of Logic
Systems, bcause you don't understand that Formal Systems are built
from their definitions, and those definitions can not be changed and
let you stay in the same system.
The actual First Principles are as I say they are: Finite string
transformation rules applied to finite strings. What you are
referring to are subsequent principles that have added more on
top of the actual first principles.
But it seems you never actually came up with actual "first Principles'
about what could be done at your first step, and thus you have no idea
what can be done at each of the later steps.
Also, you then want to talk about fields that HAVE defined what those
mean, but you don't understand that, so your claims about what they
can do are just baseless.
All you have done is proved that you don't really understand what you
are talking about, but try to throw around jargon that you don't
actually understand either, which makes so many of your statements
just false or meaningless.
When we establish the ultimate foundation of computation and
formal systems as transformations of finite strings having the
FooBar (or any other property) by FooBar preserving operations
into other finite strings then the membership algorithm would
seem to always be computable.
There would either be some finite sequence of FooBar preserving
operations that derives X from the set of finite strings defined
to have the FooBar property or not.
On 10/20/2024 10:26 PM, Richard Damon wrote:
On 10/20/24 5:59 PM, olcott wrote:
On 10/20/2024 2:13 PM, Richard Damon wrote:
On 10/20/24 11:32 AM, olcott wrote:
On 10/20/2024 6:46 AM, Richard Damon wrote:
A "First Principles" approach that you refer to STARTS with an study >>>>>> and understanding of the actual basic principles of the system. That >>>>>> would be things like the basic definitions of things like "Program", >>>>>> "Halting" "Deciding", "Turing Machine", and then from those concepts, >>>>>> sees what can be done, without trying to rely on the ideas that others >>>>>> have used, but see if they went down a wrong track, and the was a
different path in the same system.
The actual barest essence for formal systems and computations
is finite string transformation rules applied to finite strings.
So, show what you can do with that.
Note, WHAT the rules can be is very important, and seems to be beyond
you ability to reason about.
After all, all a Turing Machine is is a way of defining a finite stting >>>> transformation computation.
The next minimal increment of further elaboration is that some
finite strings has an assigned or derived property of Boolean
true. At this point of elaboration Boolean true has no more
semantic meaning than FooBar.
And since you can't do the first step, you don't understand what that
actually means.
As soon as any algorithm is defined to transform any finite
string into any other finite string we have conclusively
proven that algorithms can transform finite strings.
So?
The simplest formal system that I can think of transforms
pairs of strings of ASCII digits into their sum. This algorithm
can be easily specified in C.
So?
Some finite strings are assigned the FooBar property and other
finite string derive the FooBar property by applying FooBar
preserving operations to the first set.
But, since we have an infinite number of finite strings to be assigned >>>> values, we can't just enumerate that set.
The infinite set of pairs of finite strings of ASCII digits
can be easily transformed into their corresponding sum for
arbitrary elements of this infinite set.
So?
Once finite strings have the FooBar property we can define
computations that apply Foobar preserving operations to
determine if other finite strings also have this FooBar property.
It seems you never even learned the First Principles of Logic Systems, >>>>>> bcause you don't understand that Formal Systems are built from their >>>>>> definitions, and those definitions can not be changed and let you stay >>>>>> in the same system.
The actual First Principles are as I say they are: Finite string
transformation rules applied to finite strings. What you are
referring to are subsequent principles that have added more on
top of the actual first principles.
But it seems you never actually came up with actual "first Principles' >>>> about what could be done at your first step, and thus you have no idea >>>> what can be done at each of the later steps.
Also, you then want to talk about fields that HAVE defined what those
mean, but you don't understand that, so your claims about what they can >>>> do are just baseless.
All you have done is proved that you don't really understand what you
are talking about, but try to throw around jargon that you don't
actually understand either, which makes so many of your statements just >>>> false or meaningless.
When we establish the ultimate foundation of computation and
formal systems as transformations of finite strings having the
FooBar (or any other property) by FooBar preserving operations
into other finite strings then the membership algorithm would
seem to always be computable.
There would either be some finite sequence of FooBar preserving
operations that derives X from the set of finite strings defined
to have the FooBar property or not.
But you don't understand that if you need to answer a question that
isn;t based on a computable function, you get a question that you can
not compute.
Remember, a problem statement is effectively asking for a machine to
compute a mapping from EVERY POSSIBLE finite string input to the
corresponding answer.
By simple counting, there are Aleph_0 possible deciders (since we can
express the algorithm of the system as a finite string, so we must have
only a countable infinite number of possible computations.
When we count the possible problems to ask, even for a binary question,
we have Aleph_0 possible inputs too, and thus 2 ^ Aleph_0 possible
mappings (as each mapping can have a unique combinations of output for
every possible input).
It turns out that 2 ^ Aleph_0 is Aleph_1, and that is greater than Aleph_0. >>
This means we have more problems than deciders, and thus there MUST be
problems that can not be solved.
The problem is always:
Can this finite string be derived in L by applying FooBar
preserving operations to a set of strings in L having the
FooBar property?
With finite strings that express all human knowledge that
can be expressed in language we can always reduce what could
otherwise be infinities into a finite set of categories.
When we look at the problem of proof finding, the problem is that from
the finite number of statements, we can build an arbitrary length
finite string that establishes the theorem. Trying to find an arbitrary
length finite s
Human knowledge expressed in language just doesn't seem
to work that way. When you ask someone a question as long
as they are not brain damaged they give you a succinct answer.
On 10/20/2024 10:26 PM, Richard Damon wrote:
On 10/20/24 5:59 PM, olcott wrote:
On 10/20/2024 2:13 PM, Richard Damon wrote:
On 10/20/24 11:32 AM, olcott wrote:
On 10/20/2024 6:46 AM, Richard Damon wrote:
A "First Principles" approach that you refer to STARTS with an
study and understanding of the actual basic principles of the
system. That would be things like the basic definitions of things
like "Program", "Halting" "Deciding", "Turing Machine", and then
from those concepts, sees what can be done, without trying to rely >>>>>> on the ideas that others have used, but see if they went down a
wrong track, and the was a different path in the same system.
The actual barest essence for formal systems and computations
is finite string transformation rules applied to finite strings.
So, show what you can do with that.
Note, WHAT the rules can be is very important, and seems to be
beyond you ability to reason about.
After all, all a Turing Machine is is a way of defining a finite
stting transformation computation.
The next minimal increment of further elaboration is that some
finite strings has an assigned or derived property of Boolean
true. At this point of elaboration Boolean true has no more
semantic meaning than FooBar.
And since you can't do the first step, you don't understand what
that actually means.
As soon as any algorithm is defined to transform any finite
string into any other finite string we have conclusively
proven that algorithms can transform finite strings.
So?
The simplest formal system that I can think of transforms
pairs of strings of ASCII digits into their sum. This algorithm
can be easily specified in C.
So?
Some finite strings are assigned the FooBar property and other
finite string derive the FooBar property by applying FooBar
preserving operations to the first set.
But, since we have an infinite number of finite strings to be
assigned values, we can't just enumerate that set.
The infinite set of pairs of finite strings of ASCII digits
can be easily transformed into their corresponding sum for
arbitrary elements of this infinite set.
So?
Once finite strings have the FooBar property we can define
computations that apply Foobar preserving operations to
determine if other finite strings also have this FooBar property.
It seems you never even learned the First Principles of Logic
Systems, bcause you don't understand that Formal Systems are built >>>>>> from their definitions, and those definitions can not be changed
and let you stay in the same system.
The actual First Principles are as I say they are: Finite string
transformation rules applied to finite strings. What you are
referring to are subsequent principles that have added more on
top of the actual first principles.
But it seems you never actually came up with actual "first
Principles' about what could be done at your first step, and thus
you have no idea what can be done at each of the later steps.
Also, you then want to talk about fields that HAVE defined what
those mean, but you don't understand that, so your claims about what
they can do are just baseless.
All you have done is proved that you don't really understand what
you are talking about, but try to throw around jargon that you don't
actually understand either, which makes so many of your statements
just false or meaningless.
When we establish the ultimate foundation of computation and
formal systems as transformations of finite strings having the
FooBar (or any other property) by FooBar preserving operations
into other finite strings then the membership algorithm would
seem to always be computable.
There would either be some finite sequence of FooBar preserving
operations that derives X from the set of finite strings defined
to have the FooBar property or not.
But you don't understand that if you need to answer a question that
isn;t based on a computable function, you get a question that you can
not compute.
Remember, a problem statement is effectively asking for a machine to
compute a mapping from EVERY POSSIBLE finite string input to the
corresponding answer.
By simple counting, there are Aleph_0 possible deciders (since we can
express the algorithm of the system as a finite string, so we must
have only a countable infinite number of possible computations.
When we count the possible problems to ask, even for a binary
question, we have Aleph_0 possible inputs too, and thus 2 ^ Aleph_0
possible mappings (as each mapping can have a unique combinations of
output for every possible input).
It turns out that 2 ^ Aleph_0 is Aleph_1, and that is greater than
Aleph_0.
This means we have more problems than deciders, and thus there MUST be
problems that can not be solved.
The problem is always:
Can this finite string be derived in L by applying FooBar
preserving operations to a set of strings in L having the
FooBar property?
With finite strings that express all human knowledge that
can be expressed in language we can always reduce what could
otherwise be infinities into a finite set of categories.
When we look at the problem of proof finding, the problem is that from
the finite number of statements, we can build an arbitrary length
finite string that establishes the theorem. Trying to find an
arbitrary length finite s
Human knowledge expressed in language just doesn't seem
to work that way. When you ask someone a question as long
as they are not brain damaged they give you a succinct answer.
On 10/21/2024 4:40 AM, Mikko wrote:
On 2024-10-21 03:58:05 +0000, olcott said:
On 10/20/2024 10:26 PM, Richard Damon wrote:
On 10/20/24 5:59 PM, olcott wrote:
On 10/20/2024 2:13 PM, Richard Damon wrote:
On 10/20/24 11:32 AM, olcott wrote:
On 10/20/2024 6:46 AM, Richard Damon wrote:
A "First Principles" approach that you refer to STARTS with an >>>>>>>> study and understanding of the actual basic principles of the
system. That would be things like the basic definitions of
things like "Program", "Halting" "Deciding", "Turing Machine", >>>>>>>> and then from those concepts, sees what can be done, without
trying to rely on the ideas that others have used, but see if
they went down a wrong track, and the was a different path in
the same system.
The actual barest essence for formal systems and computations
is finite string transformation rules applied to finite strings.
So, show what you can do with that.
Note, WHAT the rules can be is very important, and seems to be
beyond you ability to reason about.
After all, all a Turing Machine is is a way of defining a finite
stting transformation computation.
The next minimal increment of further elaboration is that some
finite strings has an assigned or derived property of Boolean
true. At this point of elaboration Boolean true has no more
semantic meaning than FooBar.
And since you can't do the first step, you don't understand what
that actually means.
As soon as any algorithm is defined to transform any finite
string into any other finite string we have conclusively
proven that algorithms can transform finite strings.
So?
The simplest formal system that I can think of transforms
pairs of strings of ASCII digits into their sum. This algorithm
can be easily specified in C.
So?
Some finite strings are assigned the FooBar property and other
finite string derive the FooBar property by applying FooBar
preserving operations to the first set.
But, since we have an infinite number of finite strings to be
assigned values, we can't just enumerate that set.
The infinite set of pairs of finite strings of ASCII digits
can be easily transformed into their corresponding sum for
arbitrary elements of this infinite set.
So?
Once finite strings have the FooBar property we can define
computations that apply Foobar preserving operations to
determine if other finite strings also have this FooBar property. >>>>>>>
It seems you never even learned the First Principles of Logic
Systems, bcause you don't understand that Formal Systems are
built from their definitions, and those definitions can not be >>>>>>>> changed and let you stay in the same system.
The actual First Principles are as I say they are: Finite string >>>>>>> transformation rules applied to finite strings. What you are
referring to are subsequent principles that have added more on
top of the actual first principles.
But it seems you never actually came up with actual "first
Principles' about what could be done at your first step, and thus
you have no idea what can be done at each of the later steps.
Also, you then want to talk about fields that HAVE defined what
those mean, but you don't understand that, so your claims about
what they can do are just baseless.
All you have done is proved that you don't really understand what
you are talking about, but try to throw around jargon that you
don't actually understand either, which makes so many of your
statements just false or meaningless.
When we establish the ultimate foundation of computation and
formal systems as transformations of finite strings having the
FooBar (or any other property) by FooBar preserving operations
into other finite strings then the membership algorithm would
seem to always be computable.
There would either be some finite sequence of FooBar preserving
operations that derives X from the set of finite strings defined
to have the FooBar property or not.
But you don't understand that if you need to answer a question that
isn;t based on a computable function, you get a question that you
can not compute.
Remember, a problem statement is effectively asking for a machine to
compute a mapping from EVERY POSSIBLE finite string input to the
corresponding answer.
By simple counting, there are Aleph_0 possible deciders (since we
can express the algorithm of the system as a finite string, so we
must have only a countable infinite number of possible computations.
When we count the possible problems to ask, even for a binary
question, we have Aleph_0 possible inputs too, and thus 2 ^ Aleph_0
possible mappings (as each mapping can have a unique combinations of
output for every possible input).
It turns out that 2 ^ Aleph_0 is Aleph_1, and that is greater than
Aleph_0.
This means we have more problems than deciders, and thus there MUST
be problems that can not be solved.
The problem is always:
Can this finite string be derived in L by applying FooBar
preserving operations to a set of strings in L having the
FooBar property?
With finite strings that express all human knowledge that
can be expressed in language we can always reduce what could
otherwise be infinities into a finite set of categories.
When we look at the problem of proof finding, the problem is that
from the finite number of statements, we can build an arbitrary
length finite string that establishes the theorem. Trying to find an
arbitrary length finite s
Human knowledge expressed in language just doesn't seem
to work that way. When you ask someone a question as long
as they are not brain damaged they give you a succinct answer.
Answers like "I don't know" and "What are you talking about" are
fairly common.
For the Golbach conjecture IDK is the only correct answer.
On 10/21/2024 5:46 PM, Richard Damon wrote:
On 10/21/24 9:31 AM, olcott wrote:
On 10/21/2024 4:40 AM, Mikko wrote:
On 2024-10-21 03:58:05 +0000, olcott said:
On 10/20/2024 10:26 PM, Richard Damon wrote:
On 10/20/24 5:59 PM, olcott wrote:
On 10/20/2024 2:13 PM, Richard Damon wrote:
On 10/20/24 11:32 AM, olcott wrote:
On 10/20/2024 6:46 AM, Richard Damon wrote:So, show what you can do with that.
A "First Principles" approach that you refer to STARTS with an >>>>>>>>>> study and understanding of the actual basic principles of the >>>>>>>>>> system. That would be things like the basic definitions of >>>>>>>>>> things like "Program", "Halting" "Deciding", "Turing Machine", >>>>>>>>>> and then from those concepts, sees what can be done, without >>>>>>>>>> trying to rely on the ideas that others have used, but see if >>>>>>>>>> they went down a wrong track, and the was a different path in >>>>>>>>>> the same system.
The actual barest essence for formal systems and computations >>>>>>>>> is finite string transformation rules applied to finite strings. >>>>>>>>
Note, WHAT the rules can be is very important, and seems to be >>>>>>>> beyond you ability to reason about.
After all, all a Turing Machine is is a way of defining a finite >>>>>>>> stting transformation computation.
The next minimal increment of further elaboration is that some >>>>>>>>> finite strings has an assigned or derived property of Boolean >>>>>>>>> true. At this point of elaboration Boolean true has no more
semantic meaning than FooBar.
And since you can't do the first step, you don't understand what >>>>>>>> that actually means.
As soon as any algorithm is defined to transform any finite
string into any other finite string we have conclusively
proven that algorithms can transform finite strings.
So?
The simplest formal system that I can think of transforms
pairs of strings of ASCII digits into their sum. This algorithm
can be easily specified in C.
So?
Some finite strings are assigned the FooBar property and other >>>>>>>>> finite string derive the FooBar property by applying FooBar
preserving operations to the first set.
But, since we have an infinite number of finite strings to be
assigned values, we can't just enumerate that set.
The infinite set of pairs of finite strings of ASCII digits
can be easily transformed into their corresponding sum for
arbitrary elements of this infinite set.
So?
Once finite strings have the FooBar property we can define
computations that apply Foobar preserving operations to
determine if other finite strings also have this FooBar property. >>>>>>>>>
It seems you never even learned the First Principles of Logic >>>>>>>>>> Systems, bcause you don't understand that Formal Systems are >>>>>>>>>> built from their definitions, and those definitions can not be >>>>>>>>>> changed and let you stay in the same system.
The actual First Principles are as I say they are: Finite string >>>>>>>>> transformation rules applied to finite strings. What you are >>>>>>>>> referring to are subsequent principles that have added more on >>>>>>>>> top of the actual first principles.
But it seems you never actually came up with actual "first
Principles' about what could be done at your first step, and
thus you have no idea what can be done at each of the later steps. >>>>>>>>
Also, you then want to talk about fields that HAVE defined what >>>>>>>> those mean, but you don't understand that, so your claims about >>>>>>>> what they can do are just baseless.
All you have done is proved that you don't really understand
what you are talking about, but try to throw around jargon that >>>>>>>> you don't actually understand either, which makes so many of
your statements just false or meaningless.
When we establish the ultimate foundation of computation and
formal systems as transformations of finite strings having the
FooBar (or any other property) by FooBar preserving operations
into other finite strings then the membership algorithm would
seem to always be computable.
There would either be some finite sequence of FooBar preserving
operations that derives X from the set of finite strings defined >>>>>>> to have the FooBar property or not.
But you don't understand that if you need to answer a question
that isn;t based on a computable function, you get a question that >>>>>> you can not compute.
Remember, a problem statement is effectively asking for a machine
to compute a mapping from EVERY POSSIBLE finite string input to
the corresponding answer.
By simple counting, there are Aleph_0 possible deciders (since we
can express the algorithm of the system as a finite string, so we
must have only a countable infinite number of possible computations. >>>>>>
When we count the possible problems to ask, even for a binary
question, we have Aleph_0 possible inputs too, and thus 2 ^
Aleph_0 possible mappings (as each mapping can have a unique
combinations of output for every possible input).
It turns out that 2 ^ Aleph_0 is Aleph_1, and that is greater than >>>>>> Aleph_0.
This means we have more problems than deciders, and thus there
MUST be problems that can not be solved.
The problem is always:
Can this finite string be derived in L by applying FooBar
preserving operations to a set of strings in L having the
FooBar property?
With finite strings that express all human knowledge that
can be expressed in language we can always reduce what could
otherwise be infinities into a finite set of categories.
When we look at the problem of proof finding, the problem is that
from the finite number of statements, we can build an arbitrary
length finite string that establishes the theorem. Trying to find
an arbitrary length finite s
Human knowledge expressed in language just doesn't seem
to work that way. When you ask someone a question as long
as they are not brain damaged they give you a succinct answer.
Answers like "I don't know" and "What are you talking about" are
fairly common.
For the Golbach conjecture IDK is the only correct answer.
So, you admit that the statment might be true and unprovable?
There are some expressions of language that seem to
have a truth value of UNKNOWABLE.
All other expressions of language have a truth value
of True, False, Not a truth bearer.
Most undecidability is the mistake of trying to
determine the truth value of an expression that has none.
On 10/21/2024 5:46 PM, Richard Damon wrote:
On 10/21/24 9:31 AM, olcott wrote:
On 10/21/2024 4:40 AM, Mikko wrote:
On 2024-10-21 03:58:05 +0000, olcott said:
On 10/20/2024 10:26 PM, Richard Damon wrote:
On 10/20/24 5:59 PM, olcott wrote:
On 10/20/2024 2:13 PM, Richard Damon wrote:
On 10/20/24 11:32 AM, olcott wrote:
On 10/20/2024 6:46 AM, Richard Damon wrote:So, show what you can do with that.
A "First Principles" approach that you refer to STARTS with an study >>>>>>>>>> and understanding of the actual basic principles of the system. That >>>>>>>>>> would be things like the basic definitions of things like "Program", >>>>>>>>>> "Halting" "Deciding", "Turing Machine", and then from those concepts,
sees what can be done, without trying to rely on the ideas that others
have used, but see if they went down a wrong track, and the was a >>>>>>>>>> different path in the same system.
The actual barest essence for formal systems and computations >>>>>>>>> is finite string transformation rules applied to finite strings. >>>>>>>>
Note, WHAT the rules can be is very important, and seems to be beyond >>>>>>>> you ability to reason about.
After all, all a Turing Machine is is a way of defining a finite stting
transformation computation.
The next minimal increment of further elaboration is that some >>>>>>>>> finite strings has an assigned or derived property of Boolean >>>>>>>>> true. At this point of elaboration Boolean true has no more
semantic meaning than FooBar.
And since you can't do the first step, you don't understand what that >>>>>>>> actually means.
As soon as any algorithm is defined to transform any finite
string into any other finite string we have conclusively
proven that algorithms can transform finite strings.
So?
The simplest formal system that I can think of transforms
pairs of strings of ASCII digits into their sum. This algorithm
can be easily specified in C.
So?
Some finite strings are assigned the FooBar property and other >>>>>>>>> finite string derive the FooBar property by applying FooBar
preserving operations to the first set.
But, since we have an infinite number of finite strings to be assigned >>>>>>>> values, we can't just enumerate that set.
The infinite set of pairs of finite strings of ASCII digits
can be easily transformed into their corresponding sum for
arbitrary elements of this infinite set.
So?
Once finite strings have the FooBar property we can define
computations that apply Foobar preserving operations to
determine if other finite strings also have this FooBar property. >>>>>>>>>
It seems you never even learned the First Principles of Logic Systems,
bcause you don't understand that Formal Systems are built from their >>>>>>>>>> definitions, and those definitions can not be changed and let you stay
in the same system.
The actual First Principles are as I say they are: Finite string >>>>>>>>> transformation rules applied to finite strings. What you are >>>>>>>>> referring to are subsequent principles that have added more on >>>>>>>>> top of the actual first principles.
But it seems you never actually came up with actual "first Principles' >>>>>>>> about what could be done at your first step, and thus you have no idea >>>>>>>> what can be done at each of the later steps.
Also, you then want to talk about fields that HAVE defined what those >>>>>>>> mean, but you don't understand that, so your claims about what they can
do are just baseless.
All you have done is proved that you don't really understand what you >>>>>>>> are talking about, but try to throw around jargon that you don't >>>>>>>> actually understand either, which makes so many of your statements just
false or meaningless.
When we establish the ultimate foundation of computation and
formal systems as transformations of finite strings having the
FooBar (or any other property) by FooBar preserving operations
into other finite strings then the membership algorithm would
seem to always be computable.
There would either be some finite sequence of FooBar preserving
operations that derives X from the set of finite strings defined >>>>>>> to have the FooBar property or not.
But you don't understand that if you need to answer a question that >>>>>> isn;t based on a computable function, you get a question that you can >>>>>> not compute.
Remember, a problem statement is effectively asking for a machine to >>>>>> compute a mapping from EVERY POSSIBLE finite string input to the
corresponding answer.
By simple counting, there are Aleph_0 possible deciders (since we can >>>>>> express the algorithm of the system as a finite string, so we must have >>>>>> only a countable infinite number of possible computations.
When we count the possible problems to ask, even for a binary question, >>>>>> we have Aleph_0 possible inputs too, and thus 2 ^ Aleph_0 possible >>>>>> mappings (as each mapping can have a unique combinations of output for >>>>>> every possible input).
It turns out that 2 ^ Aleph_0 is Aleph_1, and that is greater than Aleph_0.
This means we have more problems than deciders, and thus there MUST be >>>>>> problems that can not be solved.
The problem is always:
Can this finite string be derived in L by applying FooBar
preserving operations to a set of strings in L having the
FooBar property?
With finite strings that express all human knowledge that
can be expressed in language we can always reduce what could
otherwise be infinities into a finite set of categories.
When we look at the problem of proof finding, the problem is that from >>>>>> the finite number of statements, we can build an arbitrary length
finite string that establishes the theorem. Trying to find an arbitrary >>>>>> length finite s
Human knowledge expressed in language just doesn't seem
to work that way. When you ask someone a question as long
as they are not brain damaged they give you a succinct answer.
Answers like "I don't know" and "What are you talking about" are
fairly common.
For the Golbach conjecture IDK is the only correct answer.
So, you admit that the statment might be true and unprovable?
There are some expressions of language that seem to
have a truth value of UNKNOWABLE.
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