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Dijkstra probably hates me (Linus Torvalds, in kernel/sched.c)


devel / comp.theory / Re: Black box halt decider is NOT a partial decider

SubjectAuthor
* Black box halt decider is NOT a partial deciderMr Flibble
`* Black box halt decider is NOT a partial deciderChris M. Thomasson
 `* Black box halt decider is NOT a partial deciderDavid Brown
  `* Black box halt decider is NOT a partial deciderChris M. Thomasson
   +* Black box halt decider is NOT a partial deciderRichard Damon
   |`* Black box halt decider is NOT a partial deciderChris M. Thomasson
   | `* Black box halt decider is NOT a partial deciderRichard Damon
   |  `* Black box halt decider is NOT a partial deciderChris M. Thomasson
   |   +- Black box halt decider is NOT a partial deciderRichard Damon
   |   +* Black box halt decider is NOT a partial deciderBen Bacarisse
   |   |`* Black box halt decider is NOT a partial deciderChris M. Thomasson
   |   | +* Black box halt decider is NOT a partial deciderBen Bacarisse
   |   | |`* Black box halt decider is NOT a partial deciderChris M. Thomasson
   |   | | +- Black box halt decider is NOT a partial deciderRichard Damon
   |   | | `* Black box halt decider is NOT a partial deciderBen Bacarisse
   |   | |  `* Black box halt decider is NOT a partial deciderChris M. Thomasson
   |   | |   +* Black box halt decider is NOT a partial deciderAndré G. Isaak
   |   | |   |`* Black box halt decider is NOT a partial deciderChris M. Thomasson
   |   | |   | `* Black box halt decider is NOT a partial deciderMike Terry
   |   | |   |  `* Black box halt decider is NOT a partial deciderChris M. Thomasson
   |   | |   |   `* Black box halt decider is NOT a partial deciderChris M. Thomasson
   |   | |   |    +- Black box halt decider is NOT a partial deciderMike Terry
   |   | |   |    +* Black box halt decider is NOT a partial deciderBen Bacarisse
   |   | |   |    |+* Black box halt decider is NOT a partial deciderJeff Barnett
   |   | |   |    ||+- Black box halt decider is NOT a partial deciderJeff Barnett
   |   | |   |    ||`* Black box halt decider is NOT a partial deciderMike Terry
   |   | |   |    || +- Black box halt decider is NOT a partial deciderChris M. Thomasson
   |   | |   |    || `* Black box halt decider is NOT a partial deciderJeff Barnett
   |   | |   |    ||  `- Black box halt decider is NOT a partial deciderMike Terry
   |   | |   |    |`* Black box halt decider is NOT a partial deciderChris M. Thomasson
   |   | |   |    | `* Black box halt decider is NOT a partial deciderBen Bacarisse
   |   | |   |    |  `* Black box halt decider is NOT a partial deciderChris M. Thomasson
   |   | |   |    |   +- Black box halt decider is NOT a partial deciderChris M. Thomasson
   |   | |   |    |   `* Black box halt decider is NOT a partial deciderChris M. Thomasson
   |   | |   |    |    `- Black box halt decider is NOT a partial deciderChris M. Thomasson
   |   | |   |    `- Black box halt decider is NOT a partial deciderwij
   |   | |   +* Black box halt decider is NOT a partial deciderRichard Damon
   |   | |   |`* Black box halt decider is NOT a partial deciderChris M. Thomasson
   |   | |   | `* Black box halt decider is NOT a partial deciderRichard Damon
   |   | |   |  `- Black box halt decider is NOT a partial deciderChris M. Thomasson
   |   | |   `* Black box halt decider is NOT a partial deciderBen Bacarisse
   |   | |    +* Black box halt decider is NOT a partial deciderChris M. Thomasson
   |   | |    |`* Black box halt decider is NOT a partial deciderBen Bacarisse
   |   | |    | `* Black box halt decider is NOT a partial deciderChris M. Thomasson
   |   | |    |  `* Black box halt decider is NOT a partial deciderRichard Damon
   |   | |    |   `- Black box halt decider is NOT a partial deciderChris M. Thomasson
   |   | |    `* Black box halt decider is NOT a partial deciderAndré G. Isaak
   |   | |     +* Black box halt decider is NOT a partial deciderBen Bacarisse
   |   | |     |+- Black box halt decider is NOT a partial deciderAndré G. Isaak
   |   | |     |`* Black box halt decider is NOT a partial deciderMike Terry
   |   | |     | +* Black box halt decider is NOT a partial deciderBen Bacarisse
   |   | |     | |+* Black box halt decider is NOT a partial deciderAndy Walker
   |   | |     | ||`* Black box halt decider is NOT a partial deciderMike Terry
   |   | |     | || +* Black box halt decider is NOT a partial deciderMalcolm McLean
   |   | |     | || |+* Black box halt decider is NOT a partial decider [ H(P,P)==0 is always correct ]olcott
   |   | |     | || ||`- Black box halt decider is NOT a partial decider [ H(P,P)==0 isRichard Damon
   |   | |     | || |+* Black box halt decider is NOT a partial decider [ H(P,P)==0 is always correct ]olcott
   |   | |     | || ||+- Black box halt decider is NOT a partial decider [ H(P,P)==0 isAndré G. Isaak
   |   | |     | || ||+* Black box halt decider is NOT a partial decider [ H(P,P)==0 isRichard Damon
   |   | |     | || |||`* Black box halt decider is NOT a partial decider [ H(P,P)==0 isMalcolm McLean
   |   | |     | || ||| `* Black box halt decider is NOT a partial decider [ H(P,P)==0 isRichard Damon
   |   | |     | || |||  `- Black box halt decider is NOT a partial decider [ H(P,P)==0 isJeff Barnett
   |   | |     | || ||`- Black box halt decider is NOT a partial decider [ H(P,P)==0 is always correct ]Ben Bacarisse
   |   | |     | || |+* Black box halt decider is NOT a partial deciderBen Bacarisse
   |   | |     | || ||`* Black box halt decider is NOT a partial deciderMalcolm McLean
   |   | |     | || || `* Black box halt decider is NOT a partial decider [ paradox ratherolcott
   |   | |     | || ||  +- Black box halt decider is NOT a partial decider [ paradox ratherRichard Damon
   |   | |     | || ||  `* Black box halt decider is NOT a partial decider [ paradox ratherAndré G. Isaak
   |   | |     | || ||   `* Black box halt decider is NOT a partial decider [ H refutes Rice's Theorem ]olcott
   |   | |     | || ||    +- Black box halt decider is NOT a partial decider [ H refutesRichard Damon
   |   | |     | || ||    `* Black box halt decider is NOT a partial decider [ H refutesAndré G. Isaak
   |   | |     | || ||     `* Black box halt decider is NOT a partial decider [ H refutes Rice's Theorem ]olcott
   |   | |     | || ||      +* Black box halt decider is NOT a partial decider [ H refutesAndré G. Isaak
   |   | |     | || ||      |`* Black box halt decider is NOT a partial decider [ H refutesolcott
   |   | |     | || ||      | `- Black box halt decider is NOT a partial decider [ H refutesRichard Damon
   |   | |     | || ||      `* Black box halt decider is NOT a partial decider [ H refutesJeff Barnett
   |   | |     | || ||       `* Black box halt decider is NOT a partial decider [ H refutesolcott
   |   | |     | || ||        `* Black box halt decider is NOT a partial decider [ H refutesAndré G. Isaak
   |   | |     | || ||         +* Black box halt decider is NOT a partial decider [ H refutesolcott
   |   | |     | || ||         |+- Black box halt decider is NOT a partial decider [ H refutesAndré G. Isaak
   |   | |     | || ||         |`- Black box halt decider is NOT a partial decider [ H refutesRichard Damon
   |   | |     | || ||         `* Black box halt decider is NOT a partial decider [ H refutesolcott
   |   | |     | || ||          +* Black box halt decider is NOT a partial decider [ H refutesAndré G. Isaak
   |   | |     | || ||          |`* Black box halt decider is NOT a partial decider [ H refutes Rice's Theorem ]olcott
   |   | |     | || ||          | `* Black box halt decider is NOT a partial decider [ H refutesAndré G. Isaak
   |   | |     | || ||          |  `* Black box halt decider is NOT a partial decider [ H refutesolcott
   |   | |     | || ||          |   +- Black box halt decider is NOT a partial decider [ H refutesAndré G. Isaak
   |   | |     | || ||          |   +- Black box halt decider is NOT a partial decider [ H refutesRichard Damon
   |   | |     | || ||          |   `* _Black_box_halt_decider_is_NOT_a_partial_decider_[_André_doesn't_know_Rice's_Theolcott
   |   | |     | || ||          |    +* _Black_box_halt_decider_is_NOT_a_partial_decider_[André G. Isaak
   |   | |     | || ||          |    |`* _Black_box_halt_decider_is_NOT_a_partial_decider_[olcott
   |   | |     | || ||          |    | +* _Black_box_halt_decider_is_NOT_a_partial_decider_[André G. Isaak
   |   | |     | || ||          |    | |`* _Black_box_halt_decider_is_NOT_a_partial_decider_Malcolm McLean
   |   | |     | || ||          |    | | `* _André_doesn't_know_Rice's_Theorem_[_Malcolm_]olcott
   |   | |     | || ||          |    | |  +* _André_doesn't_know_Rice's_Theorem_[_MalcRichard Damon
   |   | |     | || ||          |    | |  |`* _André_doesn't_know_Rice's_Theorem_[_Malcolcott
   |   | |     | || ||          |    | |  | `* _André_doesn't_know_Rice's_Theorem_[_MalcRichard Damon
   |   | |     | || ||          |    | |  |  `* _André_doesn't_know_Rice's_Theorem_[_Malcolm_](_attention_deficit_disorder_)olcott
   |   | |     | || ||          |    | |  |   `* _André_doesn't_know_Rice's_Theorem_[_MalcRichard Damon
   |   | |     | || ||          |    | |  |    `* _André_doesn't_know_Rice's_Theorem_[_Malcolcott
   |   | |     | || ||          |    | |  |     +- _André_doesn't_know_Rice's_Theorem_[_MalcRichard Damon
   |   | |     | || ||          |    | |  |     +* _André_doesn't_know_Rice's_Theorem_[_Malcolm_](_attention_deficit_disorder_)olcott
   |   | |     | || ||          |    | |  |     `* André doesn't know Rice's Theorem [ MalcolmBen Bacarisse
   |   | |     | || ||          |    | |  +* _André_doesn't_know_Rice's_Theorem_[_MalcAndré G. Isaak
   |   | |     | || ||          |    | |  `- _André_doesn't_know_Rice's_Theorem_[_MalcJeff Barnett
   |   | |     | || ||          |    | +- _Black_box_halt_decider_is_NOT_a_partial_decider_[Richard Damon
   |   | |     | || ||          |    | `* _Black_box_halt_decider_is_NOT_a_partial_decider_[_André_doesn't_know_Rice's_Theolcott
   |   | |     | || ||          |    `- _Black_box_halt_decider_is_NOT_a_partial_decider_[Richard Damon
   |   | |     | || ||          `- Black box halt decider is NOT a partial decider [ H refutesRichard Damon
   |   | |     | || |`* Black box halt decider is NOT a partial deciderMike Terry
   |   | |     | || `- Black box halt decider is NOT a partial deciderAndy Walker
   |   | |     | |`* Black box halt decider is NOT a partial deciderMike Terry
   |   | |     | `* Black box halt decider is NOT a partial deciderwij
   |   | |     `- Black box halt decider is NOT a partial deciderChris M. Thomasson
   |   | `* Black box halt decider is NOT a partial deciderRichard Damon
   |   `* Black box halt decider is NOT a partial deciderMalcolm McLean
   `* Black box halt decider is NOT a partial deciderJeff Barnett

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Re: Black box halt decider is NOT a partial decider

<sdfh1k$rvk$1@dont-email.me>

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https://www.novabbs.com/devel/article-flat.php?id=18812&group=comp.theory#18812

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Path: i2pn2.org!i2pn.org!eternal-september.org!reader02.eternal-september.org!.POSTED!not-for-mail
From: agis...@gm.invalid (André G. Isaak)
Newsgroups: comp.theory
Subject: Re: Black box halt decider is NOT a partial decider
Date: Fri, 23 Jul 2021 16:49:24 -0600
Organization: Christians and Atheists United Against Creeping Agnosticism
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 by: André G. Isaak - Fri, 23 Jul 2021 22:49 UTC

On 2021-07-23 16:28, Chris M. Thomasson wrote:

> Sum[n=1..oo] 1/n^24 converges on 1.

That's certainly news to me!

André

--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.

Re: Black box halt decider is NOT a partial decider

<sdfi86$1ca3$1@gioia.aioe.org>

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https://www.novabbs.com/devel/article-flat.php?id=18813&group=comp.theory#18813

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From: chris.m....@gmail.com (Chris M. Thomasson)
Newsgroups: comp.theory
Subject: Re: Black box halt decider is NOT a partial decider
Date: Fri, 23 Jul 2021 16:09:58 -0700
Organization: Aioe.org NNTP Server
Message-ID: <sdfi86$1ca3$1@gioia.aioe.org>
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 by: Chris M. Thomasson - Fri, 23 Jul 2021 23:09 UTC

On 7/23/2021 3:49 PM, André G. Isaak wrote:
> On 2021-07-23 16:28, Chris M. Thomasson wrote:
>
>> Sum[n=1..oo] 1/n^24 converges on 1.
>
> That's certainly news to me!

https://www.wolframalpha.com/input/?i=Sum%5Bn%3D1..oo%5D+1%2Fn%5E24

Bugger in Wolfram?

Re: Black box halt decider is NOT a partial decider

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Subject: Re: Black box halt decider is NOT a partial decider
Newsgroups: comp.theory
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From: Rich...@Damon-Family.org (Richard Damon)
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 by: Richard Damon - Fri, 23 Jul 2021 23:10 UTC

On 7/23/21 3:28 PM, Chris M. Thomasson wrote:
>
> Sum[n=1..oo] 1/n^24 converges on 1.

Can't be.

First Term is 1, and then it get larger, so it will never get back down
to 1.

Re: Black box halt decider is NOT a partial decider

<sdfigv$1gil$1@gioia.aioe.org>

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https://www.novabbs.com/devel/article-flat.php?id=18815&group=comp.theory#18815

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From: chris.m....@gmail.com (Chris M. Thomasson)
Newsgroups: comp.theory
Subject: Re: Black box halt decider is NOT a partial decider
Date: Fri, 23 Jul 2021 16:14:39 -0700
Organization: Aioe.org NNTP Server
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 by: Chris M. Thomasson - Fri, 23 Jul 2021 23:14 UTC

On 7/23/2021 4:10 PM, Richard Damon wrote:
> On 7/23/21 3:28 PM, Chris M. Thomasson wrote:
>>
>> Sum[n=1..oo] 1/n^24 converges on 1.
>
> Can't be.
>
> First Term is 1, and then it get larger, so it will never get back down
> to 1.
>

I think I found a bug in Wolfram. Why do I trust it from time to time?
Yikes!

https://i.ibb.co/mvDgLyt/Untitled.png

Re: Black box halt decider is NOT a partial decider

<xrIKI.58912$h8.47633@fx47.iad>

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https://www.novabbs.com/devel/article-flat.php?id=18816&group=comp.theory#18816

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Subject: Re: Black box halt decider is NOT a partial decider
Newsgroups: comp.theory
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From: Rich...@Damon-Family.org (Richard Damon)
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 by: Richard Damon - Fri, 23 Jul 2021 23:40 UTC

On 7/23/21 4:14 PM, Chris M. Thomasson wrote:
> On 7/23/2021 4:10 PM, Richard Damon wrote:
>> On 7/23/21 3:28 PM, Chris M. Thomasson wrote:
>>>
>>> Sum[n=1..oo] 1/n^24 converges on 1.
>>
>> Can't be.
>>
>> First Term is 1, and then it get larger, so it will never get back down
>> to 1.
>>
>
> I think I found a bug in Wolfram. Why do I trust it from time to time?
> Yikes!
>
> https://i.ibb.co/mvDgLyt/Untitled.png

I think the issue is that the second term is 2^-24 which is very small,
and might be below their defined tolerance.

I also note that the first line says it is APPOXIMATELY 1, and gives it
as a rationnal multiplier of pi, whcih CAN'T be EXACTLY 1.

Re: Black box halt decider is NOT a partial decider

<87zgucfux4.fsf@bsb.me.uk>

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https://www.novabbs.com/devel/article-flat.php?id=18819&group=comp.theory#18819

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From: ben.use...@bsb.me.uk (Ben Bacarisse)
Newsgroups: comp.theory
Subject: Re: Black box halt decider is NOT a partial decider
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 by: Ben Bacarisse - Fri, 23 Jul 2021 23:54 UTC

"Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:

> On 7/23/2021 3:15 PM, Ben Bacarisse wrote:
>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>
>>> On 7/22/2021 2:10 PM, Ben Bacarisse wrote:
>>>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>>>
>>>>> On 7/21/2021 5:17 AM, Ben Bacarisse wrote:
>>>>>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>>>>>
>>>>>>> For some stupid reason I always thought that the halting problem
>>>>>>> extends to all problems, and all programs.
>>>>>> It is a problem in mathematics and, like all mathematics, it is based on
>>>>>> sound definitions. It says nothing about "all programs" unless "all
>>>>>> programs" is a well-specified set. In that case, "all programs" might
>>>>>> well have a halting problem, and might even have a halting theorem.
>>>>>> The problem is always grounded to the context of some formal model of
>>>>>> computation like Turing machines, recursive functions or the lambda
>>>>>> calculus. I say "always" but that has to exclude comp.theory which is
>>>>>> dominated by silly ideas coming from people who don't know much about
>>>>>> such format models.
>>>>>>
>>>>>>> Well, here is some pure sarcasm:
>>>>>>>
>>>>>>> Every program halts because they will not get to run
>>>>>>> forever... Eventually, Earth will die.
>>>>>> This is not just sarcasm. It's a plain fact.
>>>>>>
>>>>>
>>>>> Well, based on this fact, can we say everything on Earth halts because
>>>>> the planet will eventually die?
>>>>>
>>>>> Humm... It should be "helpful" to think about infinity, where the
>>>>> program is being executed on a system that never dies. Its output is
>>>>> being observed by beings that never die.
>>>>>
>>>>> Does the program halt in the realm of the infinite?
>>>> I don't know what this means. I can answer about the mathematics, but I
>>>> don't know about these hypothetical computations that take time and are
>>>> "observed" by mythical creatures.
>>>
>>> I was just wondering if the algorihtm might halt in a finite number of
>>> steps, but the number of steps would take a very long time. The Earth
>>> would be dead long before it gets a chance to halt.
>> Would Sum[n=1..oo] 1/n^2 still be = (pi^2)/6 even if every addition took
>> a year?
>
> Sure. It converges on that value. Gaining precision every step.
>
> Sum[n=1..oo] 1/n^24 converges on 1.

Putting aside the value, the point is that the sum simply "is". It's
just daft to talk about addition taking time.

The problem with TMs is the language suggests action, but there is
none. A TM computation is simply an iterated partial sate-transition
function. The sequence generated is either finite or it is not finite.
That is fact about the function and the initial conditions just like a
converging sum's value is a fact about limits.

--
Ben.

Re: Black box halt decider is NOT a partial decider

<sdfl4d$86d$1@gioia.aioe.org>

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https://www.novabbs.com/devel/article-flat.php?id=18820&group=comp.theory#18820

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From: chris.m....@gmail.com (Chris M. Thomasson)
Newsgroups: comp.theory
Subject: Re: Black box halt decider is NOT a partial decider
Date: Fri, 23 Jul 2021 16:59:09 -0700
Organization: Aioe.org NNTP Server
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 by: Chris M. Thomasson - Fri, 23 Jul 2021 23:59 UTC

On 7/23/2021 4:54 PM, Ben Bacarisse wrote:
> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>
>> On 7/23/2021 3:15 PM, Ben Bacarisse wrote:
>>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>>
>>>> On 7/22/2021 2:10 PM, Ben Bacarisse wrote:
>>>>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>>>>
>>>>>> On 7/21/2021 5:17 AM, Ben Bacarisse wrote:
>>>>>>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>>>>>>
>>>>>>>> For some stupid reason I always thought that the halting problem
>>>>>>>> extends to all problems, and all programs.
>>>>>>> It is a problem in mathematics and, like all mathematics, it is based on
>>>>>>> sound definitions. It says nothing about "all programs" unless "all
>>>>>>> programs" is a well-specified set. In that case, "all programs" might
>>>>>>> well have a halting problem, and might even have a halting theorem.
>>>>>>> The problem is always grounded to the context of some formal model of
>>>>>>> computation like Turing machines, recursive functions or the lambda
>>>>>>> calculus. I say "always" but that has to exclude comp.theory which is
>>>>>>> dominated by silly ideas coming from people who don't know much about
>>>>>>> such format models.
>>>>>>>
>>>>>>>> Well, here is some pure sarcasm:
>>>>>>>>
>>>>>>>> Every program halts because they will not get to run
>>>>>>>> forever... Eventually, Earth will die.
>>>>>>> This is not just sarcasm. It's a plain fact.
>>>>>>>
>>>>>>
>>>>>> Well, based on this fact, can we say everything on Earth halts because
>>>>>> the planet will eventually die?
>>>>>>
>>>>>> Humm... It should be "helpful" to think about infinity, where the
>>>>>> program is being executed on a system that never dies. Its output is
>>>>>> being observed by beings that never die.
>>>>>>
>>>>>> Does the program halt in the realm of the infinite?
>>>>> I don't know what this means. I can answer about the mathematics, but I
>>>>> don't know about these hypothetical computations that take time and are
>>>>> "observed" by mythical creatures.
>>>>
>>>> I was just wondering if the algorihtm might halt in a finite number of
>>>> steps, but the number of steps would take a very long time. The Earth
>>>> would be dead long before it gets a chance to halt.
>>> Would Sum[n=1..oo] 1/n^2 still be = (pi^2)/6 even if every addition took
>>> a year?
>>
>> Sure. It converges on that value. Gaining precision every step.
>>
>> Sum[n=1..oo] 1/n^24 converges on 1.
>
> Putting aside the value, the point is that the sum simply "is". It's
> just daft to talk about addition taking time.

No problem. There in an instant. Makes me think of hyper processes.
However, I use a lot of math that takes intervals or steps to reach a
final result. Say, to draw a fractal.

>
> The problem with TMs is the language suggests action, but there is
> none. A TM computation is simply an iterated partial sate-transition
> function. The sequence generated is either finite or it is not finite.
> That is fact about the function and the initial conditions just like a
> converging sum's value is a fact about limits.
>

Re: Black box halt decider is NOT a partial decider

<IYidnYTUwOis82b9nZ2dnUU78LfNnZ2d@brightview.co.uk>

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Subject: Re: Black box halt decider is NOT a partial decider
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From: news.dea...@darjeeling.plus.com (Mike Terry)
Date: Sat, 24 Jul 2021 02:34:40 +0100
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 by: Mike Terry - Sat, 24 Jul 2021 01:34 UTC

On 24/07/2021 00:09, Chris M. Thomasson wrote:
> On 7/23/2021 3:49 PM, André G. Isaak wrote:
>> On 2021-07-23 16:28, Chris M. Thomasson wrote:
>>
>>> Sum[n=1..oo] 1/n^24 converges on 1.
>>
>> That's certainly news to me!
>
> https://www.wolframalpha.com/input/?i=Sum%5Bn%3D1..oo%5D+1%2Fn%5E24
>
> Bugger in Wolfram?
>
>

The n=1 term is 1/1^24 = 1, and then there's all those other positive
terms, so obviously the limit can't be 1. :)

The Wolfram page is saying the limit is close to 1, which makes sense as
all terms in the series apart from the first are quite small!

Mike.

Re: Black box halt decider is NOT a partial decider

<sdhquv$1n0f$1@gioia.aioe.org>

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https://www.novabbs.com/devel/article-flat.php?id=18869&group=comp.theory#18869

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From: chris.m....@gmail.com (Chris M. Thomasson)
Newsgroups: comp.theory
Subject: Re: Black box halt decider is NOT a partial decider
Date: Sat, 24 Jul 2021 12:50:54 -0700
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 by: Chris M. Thomasson - Sat, 24 Jul 2021 19:50 UTC

On 7/23/2021 4:40 PM, Richard Damon wrote:
> On 7/23/21 4:14 PM, Chris M. Thomasson wrote:
>> On 7/23/2021 4:10 PM, Richard Damon wrote:
>>> On 7/23/21 3:28 PM, Chris M. Thomasson wrote:
>>>>
>>>> Sum[n=1..oo] 1/n^24 converges on 1.
>>>
>>> Can't be.
>>>
>>> First Term is 1, and then it get larger, so it will never get back down
>>> to 1.
>>>
>>
>> I think I found a bug in Wolfram. Why do I trust it from time to time?
>> Yikes!
>>
>> https://i.ibb.co/mvDgLyt/Untitled.png
>
> I think the issue is that the second term is 2^-24 which is very small,
> and might be below their defined tolerance.

That simply has to be the issue.

>
> I also note that the first line says it is APPOXIMATELY 1, and gives it
> as a rationnal multiplier of pi, whcih CAN'T be EXACTLY 1.
>

:^D

Re: Black box halt decider is NOT a partial decider

<87pmv7ffed.fsf@bsb.me.uk>

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https://www.novabbs.com/devel/article-flat.php?id=18880&group=comp.theory#18880

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From: ben.use...@bsb.me.uk (Ben Bacarisse)
Newsgroups: comp.theory
Subject: Re: Black box halt decider is NOT a partial decider
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 by: Ben Bacarisse - Sat, 24 Jul 2021 23:42 UTC

"Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:

> On 7/23/2021 4:54 PM, Ben Bacarisse wrote:
>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>
>>> On 7/23/2021 3:15 PM, Ben Bacarisse wrote:
>>>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>>>
>>>>> On 7/22/2021 2:10 PM, Ben Bacarisse wrote:
>>>>>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>>>>>
>>>>>>> On 7/21/2021 5:17 AM, Ben Bacarisse wrote:
>>>>>>>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>>>>>>>
>>>>>>>>> For some stupid reason I always thought that the halting problem
>>>>>>>>> extends to all problems, and all programs.
>>>>>>>> It is a problem in mathematics and, like all mathematics, it is based on
>>>>>>>> sound definitions. It says nothing about "all programs" unless "all
>>>>>>>> programs" is a well-specified set. In that case, "all programs" might
>>>>>>>> well have a halting problem, and might even have a halting theorem.
>>>>>>>> The problem is always grounded to the context of some formal model of
>>>>>>>> computation like Turing machines, recursive functions or the lambda
>>>>>>>> calculus. I say "always" but that has to exclude comp.theory which is
>>>>>>>> dominated by silly ideas coming from people who don't know much about
>>>>>>>> such format models.
>>>>>>>>
>>>>>>>>> Well, here is some pure sarcasm:
>>>>>>>>>
>>>>>>>>> Every program halts because they will not get to run
>>>>>>>>> forever... Eventually, Earth will die.
>>>>>>>> This is not just sarcasm. It's a plain fact.
>>>>>>>>
>>>>>>>
>>>>>>> Well, based on this fact, can we say everything on Earth halts because
>>>>>>> the planet will eventually die?
>>>>>>>
>>>>>>> Humm... It should be "helpful" to think about infinity, where the
>>>>>>> program is being executed on a system that never dies. Its output is
>>>>>>> being observed by beings that never die.
>>>>>>>
>>>>>>> Does the program halt in the realm of the infinite?
>>>>>> I don't know what this means. I can answer about the mathematics, but I
>>>>>> don't know about these hypothetical computations that take time and are
>>>>>> "observed" by mythical creatures.
>>>>>
>>>>> I was just wondering if the algorihtm might halt in a finite number of
>>>>> steps, but the number of steps would take a very long time. The Earth
>>>>> would be dead long before it gets a chance to halt.
>>>> Would Sum[n=1..oo] 1/n^2 still be = (pi^2)/6 even if every addition took
>>>> a year?
>>>
>>> Sure. It converges on that value. Gaining precision every step.
>>>
>>> Sum[n=1..oo] 1/n^24 converges on 1.
>> Putting aside the value, the point is that the sum simply "is". It's
>> just daft to talk about addition taking time.
>
> No problem. There in an instant. Makes me think of hyper
> processes.

It's not a view I like. 2 + 2 = 4 is simply a fact. It takes no time.

> However, I use a lot of math that takes intervals or steps
> to reach a final result. Say, to draw a fractal.

You certainly can take lots of time to find out which facts are the case
in any given situation. And sometimes you can't be sure to find out in
any finite time.

>> The problem with TMs is the language suggests action, but there is
>> none. A TM computation is simply an iterated partial state-transition
>> function. The sequence generated is either finite or it is not finite.
>> That is fact about the function and the initial conditions just like a
>> converging sum's value is a fact about limits.

--
Ben.

Re: Black box halt decider is NOT a partial decider

<sdi9vb$r9b$1@dont-email.me>

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From: agis...@gm.invalid (André G. Isaak)
Newsgroups: comp.theory
Subject: Re: Black box halt decider is NOT a partial decider
Date: Sat, 24 Jul 2021 18:07:05 -0600
Organization: Christians and Atheists United Against Creeping Agnosticism
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 by: André G. Isaak - Sun, 25 Jul 2021 00:07 UTC

On 2021-07-23 17:54, Ben Bacarisse wrote:
> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:

>> Sum[n=1..oo] 1/n^24 converges on 1.
>
> Putting aside the value, the point is that the sum simply "is". It's
> just daft to talk about addition taking time.

A problem with convergent series is that there is a common misconception
that such series represent a procedure for calculating a value rather
than the value itself.

While infinite series certainly suggest an algorithm for calculating a
decimal approximation of some value to arbitrary precision, that's not
really what they are. They are simple expressions of a mathematical
equality.

I think the view of 'series as procedures' is one of the reasons why so
many crackp^H^H^H^H^H^Hpeople have problems accepting, e.g., that
0.999... = 1.

André

--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.

Re: Black box halt decider is NOT a partial decider

<87eebnfc8c.fsf@bsb.me.uk>

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From: ben.use...@bsb.me.uk (Ben Bacarisse)
Newsgroups: comp.theory
Subject: Re: Black box halt decider is NOT a partial decider
Date: Sun, 25 Jul 2021 01:50:27 +0100
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 by: Ben Bacarisse - Sun, 25 Jul 2021 00:50 UTC

André G. Isaak <agisaak@gm.invalid> writes:

> On 2021-07-23 17:54, Ben Bacarisse wrote:
>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>
>>> Sum[n=1..oo] 1/n^24 converges on 1.
>> Putting aside the value, the point is that the sum simply "is". It's
>> just daft to talk about addition taking time.
>
> A problem with convergent series is that there is a common
> misconception that such series represent a procedure for calculating a
> value rather than the value itself.
>
> While infinite series certainly suggest an algorithm for calculating a
> decimal approximation of some value to arbitrary precision, that's not
> really what they are. They are simple expressions of a mathematical
> equality.
>
> I think the view of 'series as procedures' is one of the reasons why
> so many crackp^H^H^H^H^H^Hpeople have problems accepting, e.g., that
> 0.999... = 1.

Good point. Does this confusion occur in people with a maths
background? In maths, one is taught that

lim_{x->oo} f(x) = y

is simply a statement about a property of y and f, albeit a complicated
one:

∀ e > 0 [∃ c [∀ z > c [ |f(z) - y| < e ]]].

There's no notion of process, or steps, or calculation. Just something
that either is the case (lim 1/x = 0) or is not the case (lim 1/x = 1).

--
Ben.

Re: Black box halt decider is NOT a partial decider

<sdik04$d6a$1@dont-email.me>

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From: agis...@gm.invalid (André G. Isaak)
Newsgroups: comp.theory
Subject: Re: Black box halt decider is NOT a partial decider
Date: Sat, 24 Jul 2021 20:58:12 -0600
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 by: André G. Isaak - Sun, 25 Jul 2021 02:58 UTC

On 2021-07-24 18:50, Ben Bacarisse wrote:
> André G. Isaak <agisaak@gm.invalid> writes:
>
>> On 2021-07-23 17:54, Ben Bacarisse wrote:
>>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>
>>>> Sum[n=1..oo] 1/n^24 converges on 1.
>>> Putting aside the value, the point is that the sum simply "is".
>>> It's just daft to talk about addition taking time.
>>
>> A problem with convergent series is that there is a common
>> misconception that such series represent a procedure for
>> calculating a value rather than the value itself.
>>
>> While infinite series certainly suggest an algorithm for
>> calculating a decimal approximation of some value to arbitrary
>> precision, that's not really what they are. They are simple
>> expressions of a mathematical equality.
>>
>> I think the view of 'series as procedures' is one of the reasons
>> why so many crackp^H^H^H^H^H^Hpeople have problems accepting, e.g.,
>> that 0.999... = 1.
>
> Good point. Does this confusion occur in people with a maths
> background? In maths, one is taught that

I'd hope not, though that's really a question that someone with a maths
background should answer.

André

--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.

Re: Black box halt decider is NOT a partial decider

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Subject: Re: Black box halt decider is NOT a partial decider
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From: news.dea...@darjeeling.plus.com (Mike Terry)
Date: Sun, 25 Jul 2021 04:03:47 +0100
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 by: Mike Terry - Sun, 25 Jul 2021 03:03 UTC

On 25/07/2021 01:50, Ben Bacarisse wrote:
> André G. Isaak <agisaak@gm.invalid> writes:
>
>> On 2021-07-23 17:54, Ben Bacarisse wrote:
>>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>
>>>> Sum[n=1..oo] 1/n^24 converges on 1.
>>> Putting aside the value, the point is that the sum simply "is". It's
>>> just daft to talk about addition taking time.
>>
>> A problem with convergent series is that there is a common
>> misconception that such series represent a procedure for calculating a
>> value rather than the value itself.
>>
>> While infinite series certainly suggest an algorithm for calculating a
>> decimal approximation of some value to arbitrary precision, that's not
>> really what they are. They are simple expressions of a mathematical
>> equality.
>>
>> I think the view of 'series as procedures' is one of the reasons why
>> so many crackp^H^H^H^H^H^Hpeople have problems accepting, e.g., that
>> 0.999... = 1.
>
> Good point. Does this confusion occur in people with a maths
> background? In maths, one is taught that
>
> lim_{x->oo} f(x) = y
>
> is simply a statement about a property of y and f, albeit a complicated
> one:
>
> ∀ e > 0 [∃ c [∀ z > c [ |f(z) - y| < e ]]].
>
> There's no notion of process, or steps, or calculation. Just something
> that either is the case (lim 1/x = 0) or is not the case (lim 1/x = 1).
>

Surely someone studying for degree level mathematics should not be
confused by claims 0.9999... != 1 [I'll add the obvious proviso "with
normal mathematical meanings for the terms"]. Such students would have
studied properties of the reals (in particular their Archimedean
property), and would properly understand what the notation used was
claiming. But while at high school I did have a disagreement with one
of my (A-level) maths teachers, who was firmly of the opinion that
0.9999... < 1 by an infinitesimal amount (without being too clear on
exactly what that meant!).

In any case, discussion of limits involves /abstract/ concepts and
definitions, so there's NO CHANCE PO could properly understand any of
that. I really do believe he is (pretty much) totally incapable of
dealing with abstract reasoning and concepts. He would simply have one
of his familiar "intuitions" as to how he thinks it all should work, and
he'd repeat those over and over without any backing of coherent
reasoning, even when he has led himself into some blatant contradiction!
:) For PO I would expect "limit" in his mind to become some kind of
computer program calculation loop or something.

[More generally I'd hope maths students would look at results,
definitions etc. as static or timeless in their nature as you're
suggesting, e.g. should see TMs / computations as abstract "structures"
related in particular ways without anthropomorphising them or confusing
them with concrete (time-based) physical processes.]

Mike.

Re: Black box halt decider is NOT a partial decider

<87tukjdqmi.fsf@bsb.me.uk>

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From: ben.use...@bsb.me.uk (Ben Bacarisse)
Newsgroups: comp.theory
Subject: Re: Black box halt decider is NOT a partial decider
Date: Sun, 25 Jul 2021 04:22:29 +0100
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 by: Ben Bacarisse - Sun, 25 Jul 2021 03:22 UTC

Mike Terry <news.dead.person.stones@darjeeling.plus.com> writes:

> On 25/07/2021 01:50, Ben Bacarisse wrote:
>> André G. Isaak <agisaak@gm.invalid> writes:
>>
>>> On 2021-07-23 17:54, Ben Bacarisse wrote:
>>>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>>
>>>>> Sum[n=1..oo] 1/n^24 converges on 1.
>>>> Putting aside the value, the point is that the sum simply "is". It's
>>>> just daft to talk about addition taking time.
>>>
>>> A problem with convergent series is that there is a common
>>> misconception that such series represent a procedure for calculating a
>>> value rather than the value itself.
>>>
>>> While infinite series certainly suggest an algorithm for calculating a
>>> decimal approximation of some value to arbitrary precision, that's not
>>> really what they are. They are simple expressions of a mathematical
>>> equality.
>>>
>>> I think the view of 'series as procedures' is one of the reasons why
>>> so many crackp^H^H^H^H^H^Hpeople have problems accepting, e.g., that
>>> 0.999... = 1.
>> Good point. Does this confusion occur in people with a maths
>> background? In maths, one is taught that
>> lim_{x->oo} f(x) = y
>> is simply a statement about a property of y and f, albeit a complicated
>> one:
>> ∀ e > 0 [∃ c [∀ z > c [ |f(z) - y| < e ]]].
>> There's no notion of process, or steps, or calculation. Just something
>> that either is the case (lim 1/x = 0) or is not the case (lim 1/x = 1).
>
> Surely someone studying for degree level mathematics should not be
> confused by claims 0.9999... != 1 [I'll add the obvious proviso "with
> normal mathematical meanings for the terms"]. Such students would
> have studied properties of the reals (in particular their Archimedean
> property), and would properly understand what the notation used was
> claiming.

I was not clear. The confusion I was referring to was whether people
with a maths background think of limits as processes. After giving the
definition, I think many teachers would then give a sort of "operational
explanation" that might stick in some student's minds as what's "really
going on". That's not how I learned it, but then I might be an outlier.

> In any case, discussion of limits involves /abstract/ concepts and
> definitions, so there's NO CHANCE PO could properly understand any of
> that. I really do believe he is (pretty much) totally incapable of
> dealing with abstract reasoning and concepts.

I hope so. The only other explanation is that he knows he's wrong and
he's now just lying.

But I'm not 100% sure he's honestly baffled. Ages ago, I presented a
challenge: "Show a function F such that F(P, I) is true if, and only if,
P(I) is a finite computation and false otherwise". He did not reject
this definition (as he now does) but in fact presented a candidate
function as a code sketch.

--
Ben.

Re: Black box halt decider is NOT a partial decider

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Subject: Re: Black box halt decider is NOT a partial decider
From: wyni...@gmail.com (wij)
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 by: wij - Sun, 25 Jul 2021 03:35 UTC

On Sunday, 25 July 2021 at 11:03:50 UTC+8, Mike Terry wrote:
> On 25/07/2021 01:50, Ben Bacarisse wrote:
> > André G. Isaak <agi...@gm.invalid> writes:
> >
> >> On 2021-07-23 17:54, Ben Bacarisse wrote:
> >>> "Chris M. Thomasson" <chris.m.t...@gmail.com> writes:
> >>
> >>>> Sum[n=1..oo] 1/n^24 converges on 1.
> >>> Putting aside the value, the point is that the sum simply "is". It's
> >>> just daft to talk about addition taking time.
> >>
> >> A problem with convergent series is that there is a common
> >> misconception that such series represent a procedure for calculating a
> >> value rather than the value itself.
> >>
> >> While infinite series certainly suggest an algorithm for calculating a
> >> decimal approximation of some value to arbitrary precision, that's not
> >> really what they are. They are simple expressions of a mathematical
> >> equality.
> >>
> >> I think the view of 'series as procedures' is one of the reasons why
> >> so many crackp^H^H^H^H^H^Hpeople have problems accepting, e.g., that
> >> 0.999... = 1.
> >
> > Good point. Does this confusion occur in people with a maths
> > background? In maths, one is taught that
> >
> > lim_{x->oo} f(x) = y
> >
> > is simply a statement about a property of y and f, albeit a complicated
> > one:
> >
> > ∀ e > 0 [∃ c [∀ z > c [ |f(z) - y| < e ]]].
> >
> > There's no notion of process, or steps, or calculation. Just something
> > that either is the case (lim 1/x = 0) or is not the case (lim 1/x = 1).
> >
> Surely someone studying for degree level mathematics should not be
> confused by claims 0.9999... != 1 [I'll add the obvious proviso "with
> normal mathematical meanings for the terms"]. Such students would have
> studied properties of the reals (in particular their Archimedean
> property), and would properly understand what the notation used was
> claiming. But while at high school I did have a disagreement with one
> of my (A-level) maths teachers, who was firmly of the opinion that
> 0.9999... < 1 by an infinitesimal amount (without being too clear on
> exactly what that meant!).

It is very clear to me that 0.999...≠1, and lim(x->∞) 1/x=0 is wrong.
1. It is known limit arithmetic is very 'telepathic', you can find a lot of
mysterious/inconsistent examples by yourself or in Quora.
2. repeating decimals are not rational number
Simple explanation: repeating decimal occurs only while non-zero remainder exists.
https://sourceforge.net/projects/cscall/files/MisFiles/Infinity-zh.txt/download

> In any case, discussion of limits involves /abstract/ concepts and
> definitions, so there's NO CHANCE PO could properly understand any of
> that. I really do believe he is (pretty much) totally incapable of
> dealing with abstract reasoning and concepts. He would simply have one
> of his familiar "intuitions" as to how he thinks it all should work, and
> he'd repeat those over and over without any backing of coherent
> reasoning, even when he has led himself into some blatant contradiction!
> :) For PO I would expect "limit" in his mind to become some kind of
> computer program calculation loop or something.
>
> [More generally I'd hope maths students would look at results,
> definitions etc. as static or timeless in their nature as you're
> suggesting, e.g. should see TMs / computations as abstract "structures"
> related in particular ways without anthropomorphising them or confusing
> them with concrete (time-based) physical processes.]
>
> Mike.

Re: Black box halt decider is NOT a partial decider

<87o8aqen1a.fsf@bsb.me.uk>

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From: ben.use...@bsb.me.uk (Ben Bacarisse)
Newsgroups: comp.theory
Subject: Re: Black box halt decider is NOT a partial decider
Date: Sun, 25 Jul 2021 10:54:41 +0100
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 by: Ben Bacarisse - Sun, 25 Jul 2021 09:54 UTC

wij <wyniijj@gmail.com> writes:

> It is very clear to me that 0.999...≠1, and lim(x->∞) 1/x=0 is wrong.

Do you know what lim(x->∞) 1/x means?

--
Ben.

Re: Black box halt decider is NOT a partial decider

<sdjlo4$1oct$1@gioia.aioe.org>

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From: anw...@cuboid.co.uk (Andy Walker)
Newsgroups: comp.theory
Subject: Re: Black box halt decider is NOT a partial decider
Date: Sun, 25 Jul 2021 13:34:12 +0100
Organization: Not very much
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 by: Andy Walker - Sun, 25 Jul 2021 12:34 UTC

On 25/07/2021 04:22, Ben Bacarisse wrote:
> Mike Terry <news.dead.person.stones@darjeeling.plus.com> writes:
>> Surely someone studying for degree level mathematics should not be
>> confused by claims 0.9999... != 1 [I'll add the obvious proviso "with
>> normal mathematical meanings for the terms"]. Such students would
>> have studied properties of the reals (in particular their Archimedean
>> property), and would properly understand what the notation used was
>> claiming.

It's not usually mathematicians who are confused about 0.999...,
but what we might call para-mathematicians; eg, engineers/physicists
who have learned quite a lot of maths as part of their degree, but have
not studied it as an abstraction. A lot of other people have at least
a passing interest in maths, perhaps puzzle-solving as a recreation,
come across things like 0.999... and wonder what is going on. They
[both groups, paras and puzzlers] can usually be convinced by one of
the "optical" arguments, such as "You agree that 0.333... == 1/3;
multiply both sides by 3" [this is perhaps not the place to discuss
the (in)validity of that argument!]. OTOH, they are also the ones most
likely to propose that 0.999...5 lies strictly between 0.999... and 1.

I think Mike is a bit optimistic about the extent to which
maths students study properties of the reals. I checked my old books
and notes. Most don't even mention the Archimedean property. In
Hardy's "Pure Mathematics", it's a footnote. The tendency is to
take standard properties of arithmetic as "given", and to press on
ASAP with the more interesting stuff to do with limits, calculus,
bounds, convergence, blah. Most maths students in the UK [it may be
different elsewhere] then go on to quite different parts of maths
and never re-visit foundational stuff.

The danger is to some extent the opposite of what is being
posited here, that querying "0.999... == 1" is a "confusion". If
you think "Well, viewed as a limit, it's obvious. Full stop. End
of debate, nothing to see here, move on.", then you miss out on a
whole slew of mathematics in which the Archimedean/Eudoxian axiom
does not hold, some of which is very interesting and practical, not
least, surreal numbers and games. Students often find these topics
difficult because they aren't sure which bits of elementary maths
they can still assume. Meanwhile, it's worth noting that there are
[lots of] games "G" such that

0.999...9 [any number of 9's] < G < 1.

This does not contradict "0.99... == 1" as "G" is not a real number;
but it does mean that it's a deeper result than we tend to assume.

> I was not clear. The confusion I was referring to was whether people
> with a maths background think of limits as processes. After giving the
> definition, I think many teachers would then give a sort of "operational
> explanation" that might stick in some student's minds as what's "really
> going on". That's not how I learned it, but then I might be an outlier.

We talk about [eg] "the limit /as x tends/ to 1". That
phraseology suggests a process, though not necessarily one rooted
in time ["do this, then do that, then ..."]. It doesn't seem to
me to be important; what matters, and what is often more difficult,
is to find out what the limit /is/. But worth noting that there is
a large branch of mathematics, numerical analysis, which concerns
itself with looking at the processes, and making them as efficient
and as accurate as possible. NA is one of the principal points of
contact between maths [and so the real world] and computing.

--
Andy Walker, Nottingham.
Andy's music pages: www.cuboid.me.uk/andy/Music
Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Chwatal

Re: Black box halt decider is NOT a partial decider

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Subject: Re: Black box halt decider is NOT a partial decider
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From: news.dea...@darjeeling.plus.com (Mike Terry)
Date: Sun, 25 Jul 2021 15:10:14 +0100
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 by: Mike Terry - Sun, 25 Jul 2021 14:10 UTC

On 25/07/2021 04:22, Ben Bacarisse wrote:
> Mike Terry <news.dead.person.stones@darjeeling.plus.com> writes:
>
>> On 25/07/2021 01:50, Ben Bacarisse wrote:
>>> André G. Isaak <agisaak@gm.invalid> writes:
>>>
>>>> On 2021-07-23 17:54, Ben Bacarisse wrote:
>>>>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>>>
>>>>>> Sum[n=1..oo] 1/n^24 converges on 1.
>>>>> Putting aside the value, the point is that the sum simply "is". It's
>>>>> just daft to talk about addition taking time.
>>>>
>>>> A problem with convergent series is that there is a common
>>>> misconception that such series represent a procedure for calculating a
>>>> value rather than the value itself.
>>>>
>>>> While infinite series certainly suggest an algorithm for calculating a
>>>> decimal approximation of some value to arbitrary precision, that's not
>>>> really what they are. They are simple expressions of a mathematical
>>>> equality.
>>>>
>>>> I think the view of 'series as procedures' is one of the reasons why
>>>> so many crackp^H^H^H^H^H^Hpeople have problems accepting, e.g., that
>>>> 0.999... = 1.
>>> Good point. Does this confusion occur in people with a maths
>>> background? In maths, one is taught that
>>> lim_{x->oo} f(x) = y
>>> is simply a statement about a property of y and f, albeit a complicated
>>> one:
>>> ∀ e > 0 [∃ c [∀ z > c [ |f(z) - y| < e ]]].
>>> There's no notion of process, or steps, or calculation. Just something
>>> that either is the case (lim 1/x = 0) or is not the case (lim 1/x = 1).
>>
>> Surely someone studying for degree level mathematics should not be
>> confused by claims 0.9999... != 1 [I'll add the obvious proviso "with
>> normal mathematical meanings for the terms"]. Such students would
>> have studied properties of the reals (in particular their Archimedean
>> property), and would properly understand what the notation used was
>> claiming.
>
> I was not clear. The confusion I was referring to was whether people
> with a maths background think of limits as processes. After giving the
> definition, I think many teachers would then give a sort of "operational
> explanation" that might stick in some student's minds as what's "really
> going on". That's not how I learned it, but then I might be an outlier.
>

I first came across limits in layman maths books (Martin Gardner's
writings and the like), and I don't recall how I thought about them.
I'm pretty sure I never considered them to be "really" some kind of
operational process, but I was always quite good at thinking of things
"rigorously" and when I came across the metric space definition of limit
I just thought, yeah that captures my intuition for what a limit (for
real numbers at least) should be. An equation such as
1/2 + 1/4 + 1/8 + ... = 1
I would have explained as saying that the lhs and rhs both represented
the number 1, but of course to make sense of that you have to understand
and apply the definition of limit, and the notation only makes sense
because the definition of limit precludes a sequence converging to two
different real numbers, so it is not 100% simple.

I think my university was noted as being quite abstract in its
mathematics teaching, with students expected to think up their own
deeper meanings for the most part. Of course we all make up internal
ways of thinking about a concept to guide our intuitions for what we
expect to be true, but I would recognise that as different from actually
/proving/ stuff.

Anyway I would be with you on the "static" vs. operational (temporal)
process question, but we might both be outliers I guess. [Something you
said a few years back raised the possibility in my mind that perhaps we
had both studied at the same university, possibly even around the same
time. (Cambridge, 1979-1981.)]

Mike.

Re: Black box halt decider is NOT a partial decider

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Subject: Re: Black box halt decider is NOT a partial decider
From: wyni...@gmail.com (wij)
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 by: wij - Sun, 25 Jul 2021 14:10 UTC

On Sunday, 25 July 2021 at 17:54:44 UTC+8, Ben Bacarisse wrote:
> wij <wyn...@gmail.com> writes:
>
> > It is very clear to me that 0.999...≠1, and lim(x->∞) 1/x=0 is wrong.
> Do you know what lim(x->∞) 1/x means?
>
> --
> Ben.

limit: an academically disguised word for 'approach'.

Re: Black box halt decider is NOT a partial decider

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From: ben.use...@bsb.me.uk (Ben Bacarisse)
Newsgroups: comp.theory
Subject: Re: Black box halt decider is NOT a partial decider
Date: Sun, 25 Jul 2021 15:26:05 +0100
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 by: Ben Bacarisse - Sun, 25 Jul 2021 14:26 UTC

Mike Terry <news.dead.person.stones@darjeeling.plus.com> writes:

> Anyway I would be with you on the "static" vs. operational (temporal)
> process question, but we might both be outliers I guess. [Something
> you said a few years back raised the possibility in my mind that
> perhaps we had both studied at the same university, possibly even
> around the same time. (Cambridge, 1979-1981.)]

Blimey, yes. Peterhouse.

--
Ben.

Re: Black box halt decider is NOT a partial decider

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From: anw...@cuboid.co.uk (Andy Walker)
Newsgroups: comp.theory
Subject: Re: Black box halt decider is NOT a partial decider
Date: Sun, 25 Jul 2021 16:56:01 +0100
Organization: Not very much
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 by: Andy Walker - Sun, 25 Jul 2021 15:56 UTC

On 25/07/2021 15:26, Ben Bacarisse wrote:
> Mike Terry <news.dead.person.stones@darjeeling.plus.com> writes:
>> [...] (Cambridge, 1979-1981.)]
> Blimey, yes. Peterhouse.

Bloomin' youngsters! [Sidney, 1961-65.]

--
Andy Walker, Nottingham.
Andy's music pages: www.cuboid.me.uk/andy/Music
Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Chwatal

Re: Black box halt decider is NOT a partial decider

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Subject: Re: Black box halt decider is NOT a partial decider
Newsgroups: comp.theory
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From: news.dea...@darjeeling.plus.com (Mike Terry)
Date: Sun, 25 Jul 2021 17:14:18 +0100
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 by: Mike Terry - Sun, 25 Jul 2021 16:14 UTC

On 25/07/2021 13:34, Andy Walker wrote:
> On 25/07/2021 04:22, Ben Bacarisse wrote:
>> Mike Terry <news.dead.person.stones@darjeeling.plus.com> writes:
>>> Surely someone studying for degree level mathematics should not be
>>> confused by claims 0.9999... != 1 [I'll add the obvious proviso "with
>>> normal mathematical meanings for the terms"].  Such students would
>>> have studied properties of the reals (in particular their Archimedean
>>> property), and would properly understand what the notation used was
>>> claiming.
>
>     It's not usually mathematicians who are confused about 0.999...,
> but what we might call para-mathematicians;  eg, engineers/physicists
> who have learned quite a lot of maths as part of their degree, but have
> not studied it as an abstraction.  A lot of other people have at least
> a passing interest in maths, perhaps puzzle-solving as a recreation,
> come across things like 0.999... and wonder what is going on.  They
> [both groups, paras and puzzlers] can usually be convinced by one of
> the "optical" arguments, such as "You agree that 0.333... == 1/3;
> multiply both sides by 3" [this is perhaps not the place to discuss
> the (in)validity of that argument!].  OTOH, they are also the ones most
> likely to propose that 0.999...5 lies strictly between 0.999... and 1.
>
>     I think Mike is a bit optimistic about the extent to which
> maths students study properties of the reals.  I checked my old books
> and notes.  Most don't even mention the Archimedean property.  In
> Hardy's "Pure Mathematics", it's a footnote.  The tendency is to
> take standard properties of arithmetic as "given", and to press on
> ASAP with the more interesting stuff to do with limits, calculus,
> bounds, convergence, blah.  Most maths students in the UK [it may be
> different elsewhere] then go on to quite different parts of maths
> and never re-visit foundational stuff.

You're undoubtably right about my over-optimism, but the Archimedian
property is implicit in other results a maths student studying real
analysis would have proved (probably still being too optimistic here,
butI only have my own background to go by...!), e.g. between any two
real numbers there is a rational number. There might well be no mention
of the name "Archimedian property" - I don't think there was in my case,
and I only came across that term many years later.

I'm talking about "real analysis" here, not calculus courses, and I
think someone studying maths at degree level could not avoid real
analysis(?) But the level of detail/thoroughness will obviously vary
considerably between institutions, probably having different final goals
in mind.

>
>     The danger is to some extent the opposite of what is being
> posited here, that querying "0.999... == 1" is a "confusion".  If
> you think "Well, viewed as a limit, it's obvious.  Full stop.  End
> of debate, nothing to see here, move on.", then you miss out on a
> whole slew of mathematics in which the Archimedean/Eudoxian axiom
> does not hold, some of which is very interesting and practical, not
> least, surreal numbers and games.  Students often find these topics
> difficult because they aren't sure which bits of elementary maths
> they can still assume.  Meanwhile, it's worth noting that there are
> [lots of] games "G" such that
>
>   0.999...9 [any number of 9's] < G < 1.
>
> This does not contradict "0.99... == 1" as "G" is not a real number;
> but it does mean that it's a deeper result than we tend to assume.
>

Sure. I enjoyed Conway's "On Numbers and Games" as a student, and the
easier "Winning Ways" books later on, but I would say that with all that
understood, people who think 0.999... < 1 aren't thinking at all about
surreal numbers! They are, almost universally, just confused, having
never had to rigorously think through all the consequences of their
position. (And that's perfectly reasonable - why /should/ someone spend
time thinking through such issues? No reason at all.)

If we grant that they were "really" talking about surreal numbers (or
similar) then what surreal number is 0.9999... actually supposed to
represent? Calling this s, with d = 1-s > 0 then what is 1-2d? If any
reply is forthcoming it will be something like "that's 0.9999... with an
8 on the end, which is just incoherent - at least it can't make any
sense with decimal notation being used. (We've not missed out on a
whole slew of mathematics by saying that - it's still there, but just
not so relevent to all the 0.999... < 1 arguments that go on. :) )

I think this is all a bit like Malcolm suggesting that PO is raising
"interesting ideas" that might be useful with more study, or that some
basic idea of PO's is "really quite clever...". It is NOT. PO has no
incling of those possibilities and really no interest in them. He is
simply saying things he naively thinks are true, without any logical
reasoning going on. No cleverness at all. He is not "performing a
magic trick", where he will pull a rabit out of a hat - he genuinely
believes he is refuting the Linz proof, no tricks. Pretending otherwise
may be being nice to PO, making him feel better, but is ultimately
unhelpful IMO. I'd say it seems like "dishonest niceness" to me. (But
maybe Malcolm really thinks PO is producing worthwhile results, or is on
the path to that, in which case it's just being "actually nice"!)

Mike.

Re: Black box halt decider is NOT a partial decider

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Subject: Re: Black box halt decider is NOT a partial decider
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From: news.dea...@darjeeling.plus.com (Mike Terry)
Date: Sun, 25 Jul 2021 17:18:18 +0100
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 by: Mike Terry - Sun, 25 Jul 2021 16:18 UTC

On 25/07/2021 15:26, Ben Bacarisse wrote:
> Mike Terry <news.dead.person.stones@darjeeling.plus.com> writes:
>
>> Anyway I would be with you on the "static" vs. operational (temporal)
>> process question, but we might both be outliers I guess. [Something
>> you said a few years back raised the possibility in my mind that
>> perhaps we had both studied at the same university, possibly even
>> around the same time. (Cambridge, 1979-1981.)]
>
> Blimey, yes. Peterhouse.
>
(Trinity Hall... (Probably I don't need to add my usual "not Trinity"))

Re: Black box halt decider is NOT a partial decider

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Subject: Re: Black box halt decider is NOT a partial decider
From: malcolm....@gmail.com (Malcolm McLean)
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 by: Malcolm McLean - Sun, 25 Jul 2021 17:40 UTC

On Sunday, 25 July 2021 at 17:14:20 UTC+1, Mike Terry wrote:
>
> I think this is all a bit like Malcolm suggesting that PO is raising
> "interesting ideas" that might be useful with more study, or that some
> basic idea of PO's is "really quite clever...". It is NOT. PO has no
> incling of those possibilities and really no interest in them. He is
> simply saying things he naively thinks are true, without any logical
> reasoning going on. No cleverness at all. He is not "performing a
> magic trick", where he will pull a rabit out of a hat - he genuinely
> believes he is refuting the Linz proof, no tricks. Pretending otherwise
> may be being nice to PO, making him feel better, but is ultimately
> unhelpful IMO. I'd say it seems like "dishonest niceness" to me. (But
> maybe Malcolm really thinks PO is producing worthwhile results, or is on
> the path to that, in which case it's just being "actually nice"!)
>
I've always been very clear that I haven't yet seen from PO anything that
constitutes a refutation of Linz. However when we have, for example, the
"H is the operating system" ruse, I do tend to say "that's a clever cheat"
rather than "how could you make such a simple and obvious error?".
Largely because it's a nicer way of conveying essentially the same
information.

I did say recently that PO had constructed his own paradox. It's this. If
H is simulating halt decider, and is called on H, it creates a series of
nested recursions. If it doesn't detect the situation, it never halts. If
it does detect the situation and terminates the simulations, it halts.
However if it halts, the nested recursions were not infinite.

I do wonder if this could form the nucleus of an another proof that
halting is undecidable.

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