When the halting problem is defined as dividing up arbitrary pairs of
finite strings into those that halt on their input and those that do not:

For any program H that might determine if programs
halt, a "pathological" program D, called with some
input, can pass its own source and its input to H
and then specifically do the opposite of what H
predicts D will do. No H can exist that handles
this case. https://en.wikipedia.org/wiki/Halting_problem

Then the halting problem cannot be solved because it is a category error
comparable to the Liar Paradox and Russell's Paradox.

When we redefine the halting problem such that H correctly determines
that x(y) correctly simulated by H would never reach its final state and
terminate normally and those that would not the category error has been
eliminated making H(D,D)==0 correct.

Redefining a problem that has been defined to have no possible correct
solution has a key precedent in the handling of Russell's Paradox by
ZFC. ZFC redefined all of set theory to eliminate a key element
required by Russell's Paradox: a set being a member of itself.

When the barber says that he shaves all and only those that do not shave
themselves we know that this is not true in the same way that we know
that the Liar Paradox is not true.

*Self-contradictory expressions of language are never truth bearers*

--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 2/3/23 4:15 PM, olcott wrote:
> When the halting problem is defined as dividing up arbitrary pairs of
> finite strings into those that halt on their input and those that do not:
>
>    For any program H that might determine if programs
>    halt, a "pathological" program D, called with some
>    input, can pass its own source and its input to H
>    and then specifically do the opposite of what H
>    predicts D will do. No H can exist that handles
>    this case. https://en.wikipedia.org/wiki/Halting_problem
>
> Then the halting problem cannot be solved because it is a category error
> comparable to the Liar Paradox and Russell's Paradox.

Nope.

>
> When we redefine the halting problem such that H correctly determines
> that x(y) correctly simulated by H would never reach its final state and
> terminate normally and those that would not the category error has been
> eliminated making H(D,D)==0 correct.
>

So, Are you admitting that NOTHING you have said actually applies to the
REAL Halting problem, and thus not to any of the other theorem that
refer to the ACTAUL Halting problem?

> Redefining a problem that has been defined to have no possible correct
> solution has a key precedent in the handling of Russell's Paradox by
> ZFC. ZFC redefined all of set theory to eliminate a key element
> required by Russell's Paradox: a set being a member of itself.

No, you have things wrong. Russell's Paradox showed that Naive Set
Theory had a fundamental problem that lead to it being INCONSISTENT.
This requird a reformilation of the basics of Set Theory to keep it
conssitent.

The uncomputabilty of the Halting Problem does NOT lead to any
inconsistancy, thus there is no need to reformulate the system.

You only get an inconsistancy if you have in your system a requiremnt
that True mean Provable, and what the various proofs have shown is that
shuch a system MUST, by necessity, limit its scope to avoid being
inconsistent, and computation theory, as is most of mathematics is
beyond that scope line. You only can get as far as about to Peano
Arithmatic.

You don't seem to undestand this FUNDAMENTAL limitation of your
persumptions. But this explains a LOT of the things your don't
understand, as if your mind is stuck on the idea that things are only
true if they are provable, means you can't handle infinites well, if at
all, and definitely can't handle "categories" of infinity at all.

>
> When the barber says that he shaves all and only those that do not shave
> themselves we know that this is not true in the same way that we know
> that the Liar Paradox is not true.

So, that isn't the question here.

Yes, there are statements of Natural Language which are not Truth
Bearers, and that is a problem for people who try to make Natural
Language a basis of logic. Good thing it isn't really.

>
> *Self-contradictory expressions of language are never truth bearers*
>
>

And what is self-contradictory about the Halting Problem.

it asks if a given program when run on a given input will Halt or Run
Forver.

All inputs will do that, thus the answer IS a truth-bearer.

The fact that you can't build a computation to compute it isn't a
problem, MOST arbitrary decision problems are non-computable, since
there are an uncountable infinite number of decsion problems and only a
countable infinite number of computation to run on them.

You just can't handle that so things can be true, but unknowable.

On 2/3/2023 3:15 PM, olcott wrote:
> When the halting problem is defined as dividing up arbitrary pairs of
> finite strings into those that halt on their input and those that do not:
>
>    For any program H that might determine if programs
>    halt, a "pathological" program D, called with some
>    input, can pass its own source and its input to H
>    and then specifically do the opposite of what H
>    predicts D will do. No H can exist that handles
>    this case. https://en.wikipedia.org/wiki/Halting_problem
>
> Then the halting problem cannot be solved because it is a category error
> comparable to the Liar Paradox and Russell's Paradox.
>
> When we redefine the halting problem such that H correctly determines
> that x(y) correctly simulated by H would never reach its final state and
> terminate normally and those that would not the category error has been
> eliminated making H(D,D)==0 correct.
>
> Redefining a problem that has been defined to have no possible correct
> solution has a key precedent in the handling of Russell's Paradox by
> ZFC. ZFC redefined all of set theory to eliminate a key element
> required by Russell's Paradox: a set being a member of itself.
>
> When the barber says that he shaves all and only those that do not shave
> themselves we know that this is not true in the same way that we know
> that the Liar Paradox is not true.
>
> *Self-contradictory expressions of language are never truth bearers*

The key is that the original definition of the halting problem does not
prove that a universal halt decider cannot exist.

The same infinite set of arbitrary finite strings can be divided into
halting and non-halting when the pathological self-reference error has
been eliminated.

It really is the case that H(D,D)==0 from the relevant perspective. A
pathological input is correctly determined to be non-halting from the
halt deciders point-of-view otherwise H would get stuck in infinite
recursion. This enables compilers to be augmented with halt deciding
capabilities.

When compilers can catch bugs in code hundreds of billions of software
development costs can be saved.

--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 2/3/23 9:37 PM, olcott wrote:
> On 2/3/2023 3:15 PM, olcott wrote:
>> When the halting problem is defined as dividing up arbitrary pairs of
>> finite strings into those that halt on their input and those that do not:
>>
>>     For any program H that might determine if programs
>>     halt, a "pathological" program D, called with some
>>     input, can pass its own source and its input to H
>>     and then specifically do the opposite of what H
>>     predicts D will do. No H can exist that handles
>>     this case. https://en.wikipedia.org/wiki/Halting_problem
>>
>> Then the halting problem cannot be solved because it is a category error
>> comparable to the Liar Paradox and Russell's Paradox.
>>
>> When we redefine the halting problem such that H correctly determines
>> that x(y) correctly simulated by H would never reach its final state and
>> terminate normally and those that would not the category error has been
>> eliminated making H(D,D)==0 correct.
>>
>> Redefining a problem that has been defined to have no possible correct
>> solution has a key precedent in the handling of Russell's Paradox by
>> ZFC. ZFC redefined all of set theory to eliminate a key element
>> required by Russell's Paradox: a set being a member of itself.
>>
>> When the barber says that he shaves all and only those that do not shave
>> themselves we know that this is not true in the same way that we know
>> that the Liar Paradox is not true.
>>
>> *Self-contradictory expressions of language are never truth bearers*
>
> The key is that the original definition of the halting problem does not
> prove that a universal halt decider cannot exist.

No, the Definition of the Halting Problem doesn't say they are
impossible, that is just a cleverly proven Theory based on it and
fundamental properties in Computation Theory. One you haven't pointed
out an actual error in, but just jabber about it saying things it
doesn't actually say, because you just don't understand it, or even the
basics of logic.

>
> The same infinite set of arbitrary finite strings can be divided into
> halting and non-halting when the pathological self-reference error has
> been eliminated.

Nope, since the "pathological" case has a definite answer. Remember, you
can't make a given machine P, until you have a definite define machine
H, and that that point P(P) WILL have a defined behavior.

>
> It really is the case that H(D,D)==0 from the relevant perspective. A
> pathological input is correctly determined to be non-halting from the
> halt deciders point-of-view otherwise H would get stuck in infinite
> recursion. This enables compilers to be augmented with halt deciding
> capabilities.

Nope, because if H(D,D) returns 0, we KNOW, and you have even admitted
that D(D) will Halt when run, thus the ONLY correct answer that H should
have returned is "Halting"

IF you want to say that main calling H(D,D) can have H return 0, but
main calling D(D) calling H(D,D) can have that same pure function H not
return, you need to provide the first instruction in that execution of H
that differed between those two cases.

You have been asked for this before, and haven't answered, thus you have
admited you have no basis for your claim, and are just being an ignorant
pathological liar when you make that claim.

Note, since the DEFINITION of a Halt Decider is to answer about the
exectution of the ACTUAL MACHINE (not a "Correct Simulaiton") your Red
Herrings of trying to say its partial simulation have allowed a correct
determination of what should have happened are just proven to be
irrelevent. The fact that they make the claim that a wrong answer is
rigth, also makes them incorrect and shows the logic they are using is
flawed.

>
> When compilers can catch bugs in code hundreds of billions of software
> development costs can be saved.
>

Red Herring. We aren't talking about compilers. Note, there is nothing
in the proof that says that we can't create a machine that is right
99.999% which may be good enough for your compiler.

On 2/3/2023 3:15 PM, olcott wrote:
> When the halting problem is defined as dividing up arbitrary pairs of
> finite strings into those that halt on their input and those that do not:
>
>    For any program H that might determine if programs
>    halt, a "pathological" program D, called with some
>    input, can pass its own source and its input to H
>    and then specifically do the opposite of what H
>    predicts D will do. No H can exist that handles
>    this case. https://en.wikipedia.org/wiki/Halting_problem
>
> Then the halting problem cannot be solved because it is a category error
> comparable to the Liar Paradox and Russell's Paradox.
>
> When we redefine the halting problem such that H correctly determines
> that x(y) correctly simulated by H would never reach its final state and
> terminate normally and those that would not the category error has been
> eliminated making H(D,D)==0 correct.
>
> Redefining a problem that has been defined to have no possible correct
> solution has a key precedent in the handling of Russell's Paradox by
> ZFC. ZFC redefined all of set theory to eliminate a key element
> required by Russell's Paradox: a set being a member of itself.
>
> When the barber says that he shaves all and only those that do not shave
> themselves we know that this is not true in the same way that we know
> that the Liar Paradox is not true.
>
> *Self-contradictory expressions of language are never truth bearers*
>
>

In computability theory, the halting problem is the
problem of determining, from a description of an
arbitrary computer program and an input, whether
the program will finish running, or continue to run
forever. Alan Turing proved in 1936 that a general
algorithm to solve the halting problem for all
possible program input pairs cannot exist.

For any program H that might determine if programs halt,
a "pathological" program D, called with some input, can
pass its own source and its input to H and then
specifically do the opposite of what H predicts D will do.
No H can exist that handles this case.
https://en.wikipedia.org/wiki/Halting_problem

When we define an incorrect decision problem instance as any instance of
a decision problem such that neither Boolean return value is correct
then H(D,D) is an incorrect instance of the original halting problem.

When the halting problem is defined as dividing arbitrary finite
string input pairs: (x,y) such that x(y) correctly simulated by H would
reach the final state of x and terminate normally (or not) then the
above "impossible" input (D,D) input to H ceases to be impossible.

--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 2/5/23 4:23 PM, olcott wrote:
> On 2/3/2023 3:15 PM, olcott wrote:
>> When the halting problem is defined as dividing up arbitrary pairs of
>> finite strings into those that halt on their input and those that do not:
>>
>>     For any program H that might determine if programs
>>     halt, a "pathological" program D, called with some
>>     input, can pass its own source and its input to H
>>     and then specifically do the opposite of what H
>>     predicts D will do. No H can exist that handles
>>     this case. https://en.wikipedia.org/wiki/Halting_problem
>>
>> Then the halting problem cannot be solved because it is a category error
>> comparable to the Liar Paradox and Russell's Paradox.
>>
>> When we redefine the halting problem such that H correctly determines
>> that x(y) correctly simulated by H would never reach its final state and
>> terminate normally and those that would not the category error has been
>> eliminated making H(D,D)==0 correct.
>>
>> Redefining a problem that has been defined to have no possible correct
>> solution has a key precedent in the handling of Russell's Paradox by
>> ZFC. ZFC redefined all of set theory to eliminate a key element
>> required by Russell's Paradox: a set being a member of itself.
>>
>> When the barber says that he shaves all and only those that do not shave
>> themselves we know that this is not true in the same way that we know
>> that the Liar Paradox is not true.
>>
>> *Self-contradictory expressions of language are never truth bearers*
>>
>>
>
>    In computability theory, the halting problem is the
>    problem of determining, from a description of an
>    arbitrary computer program and an input, whether
>    the program will finish running, or continue to run
>    forever.

This is the end of the Definition!

> Alan Turing proved in 1936 that a general
>    algorithm to solve the halting problem for all
>    possible program input pairs cannot exist.
This is a point of History about it
>
>    For any program H that might determine if programs halt,
>    a "pathological" program D, called with some input, can
>    pass its own source and its input to H and then
>    specifically do the opposite of what H predicts D will do.
>    No H can exist that handles this case.
>    https://en.wikipedia.org/wiki/Halting_problem
>
> When we define an incorrect decision problem instance as any instance of
> a decision problem such that neither Boolean return value is correct
> then H(D,D) is an incorrect instance of the original halting problem.

And what is the error in the DEFINITION as I have marked.

Note, the answer to the question posed IS always Yes or No.

When YOUR arguement uses the wrong definition, then by your own words it
is INCORRECT>

>
> When the halting problem is defined as dividing arbitrary finite
> string input pairs: (x,y) such that x(y) correctly simulated by H would
> reach the final state of x and terminate normally (or not) then the
> above "impossible" input (D,D) input to H ceases to be impossible.
>

And the original problem DOES divide the input strings into two disctint
groups, those that represent a Halting Machine+Input, and those that don't.

By YOUR definition, a decider could call ALL inputs non-halting by just
aborting its simulation before it started. Since your definition is just
that every step simulated is correct, if you do 0 steps, all of them
were correct. This makes your definition WORTHLESS.

Note, also there becomes not a single correcet answer to a given problem
as not Halting is defined by not a pair of Machine + Input, but a triple
of Machine + Input + Decider, as different deciders can come up with
different answers.

If you try to add correctly simulating until it can correctly determine
that its input would not halt, Your H isn't allowed to ever abort, as it
can never ACTUALLY prove that result, as if it ever DOES abort and
return 0, so does the copy of H, and thus whatever proof you used is
proved to be INCORRECT. Your rule of H(P,P) seeing it simulation of P(P)
reaching a call to H(P,P) has been proven incorrext.

You have failed to provide the point in the execution of the two paths
main -> H(P,P) and main -> P(P) -> H(P,P) where H diverges to provide
the two claimed different behaviors, so you are just admitting that you
are just making up stuff and lying, and your "proof" is invalid because
it uses KNOWN false premises.

Since your dfinition is based on an impossible condition, it is
mal-formed, and invalid.