Philosophy Now Forum

PeteOlcott wrote: ↑Fri May 13, 2022 5:16 am All that I have to do to refute all of the conventional Halting problem proofs is defeat this one case:
For any program H that might determine if programs halt, a "pathological" program P, called with some input, can pass its own source and its input to H and then specifically do the opposite of what H predicts P will do. No H can exist that handles this case.
https://en.wikipedia.org/wiki/Halting_problem

All I have to do to defeat your refutation is to defeat your halting decider.

What part of this is confusing you? Every H has a degenerate case! Which can be crafted specifically to defeat H!

Skepdick wrote: ↑Fri May 13, 2022 5:19 am

PeteOlcott wrote: ↑Fri May 13, 2022 5:16 am All that I have to do to refute all of the conventional Halting problem proofs is defeat this one case:
For any program H that might determine if programs halt, a "pathological" program P, called with some input, can pass its own source and its input to H and then specifically do the opposite of what H predicts P will do. No H can exist that handles this case.
https://en.wikipedia.org/wiki/Halting_problem

All I have to do to defeat your refutation is to defeat your halting decider.

What part of this is confusing you? Every H has a degenerate case! Which can be crafted specifically to defeat H!

Your reasoning is incorrect! As long as I defeat that one case
I HAVE defeated all the conventional proofs (not the same as solving the halting problem)

PeteOlcott wrote: ↑Fri May 13, 2022 5:21 am Your reasoning is incorrect! As long as I defeat that one case
I HAVE defeated all the conventional proofs (not the same as solving the halting problem)

YOUR reasoning is incorrect. Defeating one, or 10, or million or Aleph0 cases is not sufficient. You have to defeat ALL cases!

You produce an H.
I produce a P which defeats H.
You produce a H2 which defeats P
I produce a P2 which defeats H2
You produce a H3 which defeats P2
I produce a P3 which defeats H3
...
TO INFINITY AND BEYOND

You can't produce an H which defeats ALL of my Ps!
I can't produce a P which defeats ALL of your Hs!

Which part of this is confusing you? This IS mutual recursion in practice!

PeteOlcott wrote: ↑Fri May 13, 2022 5:21 am I HAVE defeated all the conventional proofs (not the same as solving the halting problem)[/color][/b][/size]

Also, what you are observing is nothing new!

ALL proofs of the halting problem are classical proofs - they use proof by contradiction.
Proofs by contradiction are NOT constructively valid.

Constructive proofs are stronger proofs than proofs by contradiction!
Constructive proofs require the presence of positive evidence, whereas proofs by contradiction require only the absence of negative evidence.

https://en.wikipedia.org/wiki/Constructive_proof

WE KNOW THIS!

Skepdick wrote: ↑Fri May 13, 2022 5:28 am

PeteOlcott wrote: ↑Fri May 13, 2022 5:21 am I HAVE defeated all the conventional proofs (not the same as solving the halting problem)[/color][/b][/size]

Also, what you are observing is nothing new!

ALL proofs of the halting problem are classical proofs - they use proof by contradiction.
Proofs by contradiction are NOT constructively valid.

Constructive proofs are stronger proofs than proofs by contradiction!
Constructive proofs require positive evidence, whereas proofs by contradiction require the absence of negative evidence.

https://en.wikipedia.org/wiki/Constructive_proof

WE KNOW THIS!

What you are saying is a different answer to a different question.
I am not solving the halting problem that would take 10,000 man-years.
I HAVE defeated all the conventional proofs that the halting problem cannot be solved.

PeteOlcott wrote: ↑Fri May 13, 2022 5:38 am What you are saying is a different answer to a different question.
I am not solving the halting problem that would take 10,000 man-years.
I HAVE defeated all the conventional proofs.

You don't even understand what I am saying. Clearly. I am not trying to solve the halting problem.

I am pointing out that ALL proofs of the halting problem being unsolvable are NOT CONSTRUCTIVELY VALID.

Proofs of the halting problem done by classical mathematicians are not valid in the eyes of constructive mathematicians.

This is not a technical point. This is a sociological point.

Skepdick wrote: ↑Fri May 13, 2022 5:39 am

PeteOlcott wrote: ↑Fri May 13, 2022 5:38 am What you are saying is a different answer to a different question.
I am not solving the halting problem that would take 10,000 man-years.
I HAVE defeated all the conventional proofs.

You don't even understand what I am saying. Clearly. I am not trying to solve the halting problem.

I am pointing out that ALL proofs of the halting problem being unsolvable are NOT CONSTRUCTIVELY VALID.

Proofs of the halting problem done by classical mathematicians are not valid in the eyes of constructive mathematicians.

This is not a technical point. This is a sociological point.

NONE-THE-LESS
I HAVE defeated all the conventional proofs that the Halting Problem cannot be solved!

PeteOlcott wrote: ↑Fri May 13, 2022 5:44 am I HAVE defeated all the conventional proofs that the Halting Problem cannot be solved![/b][/size][/color]

No, you haven't.

You have rejected the classical proofs because they are not constructively valid.
Rejecting a paradigm is not the same thing as defeating a paradigm.

The people who subscribe to the paradigm are still here to object to your claim of defeat.

Classical mathematician say: The proof is valid.
Constructive mathematician say: The proof is invalid.

Rinse, repeat. Instead of a halting problem, now you have yourself a proof-validity problem.

Skepdick wrote: ↑Fri May 13, 2022 5:46 am

PeteOlcott wrote: ↑Fri May 13, 2022 5:44 am I HAVE defeated all the conventional proofs that the Halting Problem cannot be solved![/b][/size][/color]

No, you haven't.

You have rejected the classical proofs because they are not constructively valid.

Classical mathematician says: It's valid.
Constructive mathematician says: It's not valid.

Rinse, repeat. Instead of a halting problem, now you have yourself a proof-validity problem.

This is a very big deal that no one has ever done before!

PeteOlcott wrote: ↑Fri May 13, 2022 5:53 am This is a very big deal that no one has ever done before!

It's a bit too late in your life to be chasing self-confidence and validation. No?

Skepdick wrote: ↑Fri May 13, 2022 5:54 am

PeteOlcott wrote: ↑Fri May 13, 2022 5:53 am This is a very big deal that no one has ever done before!

It's a bit too late in your life to be chasing self-confidence and validation. No?

As an objective fact it is a very big deal!

PeteOlcott wrote: ↑Fri May 13, 2022 5:56 am As an objective fact it is a very big deal!

It's not an objective fact.

It's just logical implication, a consequence of your chosen axioms. You are a constructivist. You reject proofs by contradiction.

You reject the classical paradigm. The classical mathematicians don't. In their eyes your "objective fact" isn't.

PeteOlcott wrote: ↑Fri May 13, 2022 5:56 am As an objective fact it is a very big deal!

Here is a constructive statement of the halting problem in a non-classical logic.

https://cstheory.stackexchange.com/ques ... ng-problem

You should appreciate that the author of the answer is talking about time. You know, because non-halting means "infinite runtime".

Skepdick wrote: ↑Fri May 13, 2022 6:01 am

PeteOlcott wrote: ↑Fri May 13, 2022 5:56 am As an objective fact it is a very big deal!

Here is a constructive statement of the halting problem in a non-classical logic.

https://cstheory.stackexchange.com/ques ... ng-problem

You should appreciate that the author of the answer is using linear logic. A logic where time is a computational resource.

https://en.wikipedia.org/wiki/Linear_logic

So you find an unconventional unpublished proof, that may or may not be correct, (no peer review).
None-The-Less I have defeated ALL the conventional proofs.

PeteOlcott wrote: ↑Fri May 13, 2022 6:10 am So you find an unconventional unpublished proof, that may or may not be correct, (no peer review).
None-The-Less I have defeated ALL the conventional proofs.

No, you haven't

You have merely accepted proof-theoretic criteria which render the conventional proofs unacceptable to you.

It's 2022 - the peer which reviews your proofs is your proof assistant. It avoids petty human squabbles such as this one.

https://en.wikipedia.org/wiki/Proof_assistant