From Newsgroup: comp.theory

The following claim from p246 of Turing’s seminal paper On Computable Numbers is a fallacy:

/the problem of enumerating computable sequences is equivalent to the
problem of finding out whether a given number is the D.N of a circle-
free machine, and we have no general process for doing this in a finite
number of steps/

For any given computable sequence, there are _infinite_ circle-free
machines which compute that particular sequence. Not only can various
machines differ significantly in the specific steps to produce the same output, machines can be changed in superficial ways that do not
meaningfully affect the steps of computation, akin to modern no-op
statements or unreachable code

The problem of enumerating computable sequences, however, only depends
on successfully identifying _one_ circle-free machine that computes any
given computable sequences. While identifying more than one can
certainly be done, it is _not_ a requirement for enumerating computable sequences, as _one_ machine computing a sequence /suffices to output any
and all digits of that sequence/

The problem of enumerating computable sequences is therefore _not_
actually equivalent to a _general process_ of enumerating circle-free machines, as there is no need to identify all circle-free machines which compute any given computable sequence

Said problem is only equivalent to a _limited process_ of enumerating circle-free machines. The machine which identifies circle-free machines
only needs the limited power of determining _at least one_ circle-free
machine for any given computable sequence, _not all_ machines for any
given computable sequence

Because of this fallacy, the proof found on the following p247, where an ill-defined machine 𝓗 (which attempts and fails to compute the direct diagonal β’) is found to be undecidable in respect to circle-free
decider 𝓓; does not then prove an impossibility for enumerating
computable sequences. As the problem of enumerating /all circle-free
machines/ is _not_ equivalent to that of enumerating /just computable sequences/
--
arising us out of the computing dark ages,
please excuse my pseudo-pyscript,
~ the little crank that could

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From Newsgroup: comp.theory

On 10/03/2026 16:51, dart200 wrote:

The following claim from p246 of Turing’s seminal paper On Computable Numbers is a fallacy:

/the problem of enumerating computable sequences is equivalent to the
problem of finding out whether a given number is the D.N of a circle-
free machine, and we have no general process for doing this in a finite number of steps/

I am delighted to see that your usage of forward-slashes is valid as an
example of usenet text formatting markup but frustrated that it's so
close to being an example of the linguist's forward-slash markup to
indicate a wrong example.

Alas, the meaning of the sentence is what is wrong, although perhaps the
syntax is technically wrong somehow too, and perhaps some intrinsic
semantic problem is present too so we can say they are the linguists
error markers too after all.

Annoyingly, the linguists markup of a leading asterisk is also taken in
usenet formatting markup for a bullet mark:

* this nonsense sentence is

/too crap this is sentence a/
--
Tristan Wibberley

The message body is Copyright (C) 2026 Tristan Wibberley except
citations and quotations noted. All Rights Reserved except that you may,
of course, cite it academically giving credit to me, distribute it
verbatim as part of a usenet system or its archives, and use it to
promote my greatness and general superiority without misrepresentation
of my opinions other than my opinion of my greatness and general
superiority which you _may_ misrepresent. You definitely MAY NOT train
any production AI system with it but you may train experimental AI that
will only be used for evaluation of the AI methods it implements.

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From Newsgroup: comp.theory

Tristan Wibberley <[email protected]> wrote or quoted:

Annoyingly, the linguists markup of a leading asterisk is also taken in >usenet formatting markup for a bullet mark:

I see. But we usually can tell from context.

However, for multi-line quotations, the most educated and
noble-minded Usenet authors prefer a "|" at the start of
/every line/, so that it does not get lost when only some
lines of the quotation are quoted later by someone else.


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From Newsgroup: comp.theory

[ Followup-To: set ]
In comp.theory dart200 <[email protected]d> wrote:

The following claim from p246 of Turing’s seminal paper On Computable Numbers is a fallacy:

You've missed something out. That something is something like "If I
understand correctly" or "As far as I can see". Without such a
qualification, your statement just looks like extreme hubris.
It is vanishingly unlikely that you have understood correctly or you
have seen far enough. If there were a flaw in Turing's 1936 paper, and
it were subtle enough to escape detection by the millions of specialists
who have verified it, it would certainly be beyond your powers as
somebody lacking education in the subject to spot it.

/the problem of enumerating computable sequences is equivalent to the problem of finding out whether a given number is the D.N of a circle-
free machine, and we have no general process for doing this in a finite number of steps/
For any given computable sequence, there are _infinite_ circle-free
machines ....

What's an "infinite circle-free machine"? All the theory we are talking
about deals only with finite machines.

.... which compute that particular sequence. Not only can various
machines differ significantly in the specific steps to produce the
same output, machines can be changed in superficial ways that do not meaningfully affect the steps of computation, akin to modern no-op
statements or unreachable code
The problem of enumerating computable sequences, however, only depends
on successfully identifying _one_ circle-free machine that computes any given computable sequences. While identifying more than one can
certainly be done, it is _not_ a requirement for enumerating computable sequences, as _one_ machine computing a sequence /suffices to output any
and all digits of that sequence/

I think you'll need to prove that you can identify a machine that
computes a given computable sequence. By definition of "computable"
there is such a machine, but how do you "identify" it? Suppose you had
a computable number and a machine, how would you test whether or not
that machine generates the number?

The problem of enumerating computable sequences is therefore _not_
actually equivalent to a _general process_ of enumerating circle-free machines, as there is no need to identify all circle-free machines which compute any given computable sequence

Your given reason fails to rule out the equivalence. For a start, under
what relationship does/doesn't this equivalence hold?

Said problem is only equivalent to a _limited process_ of enumerating circle-free machines. The machine which identifies circle-free machines
only needs the limited power of determining _at least one_ circle-free machine for any given computable sequence, _not all_ machines for any
given computable sequence

Again, can this machine exist? It seems to me that the "limited power"
you describe is not at all limited. It is known that there is no
machine which can ascertain whether or not two turing machines produce
the same output. So the ability to discard machines redundant in this
sense will not exist either.

Because of this fallacy, the proof found on the following p247, where an ill-defined machine 𝓗 (which attempts and fails to compute the direct diagonal β’) is found to be undecidable in respect to circle-free
decider 𝓓; does not then prove an impossibility for enumerating computable sequences. As the problem of enumerating /all circle-free machines/ is _not_ equivalent to that of enumerating /just computable sequences/

If this were a fallacy, you should write it up properly, get it
published and become famous. Far more likely, however, is that you have
just failed to understand what Turing's paper means.

--
arising us out of the computing dark ages,
please excuse my pseudo-pyscript,
~ the little crank that could

--
Alan Mackenzie (Nuremberg, Germany).
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From Newsgroup: comp.theory

On 3/10/26 11:24 AM, Alan Mackenzie wrote:

[ Followup-To: set ]

In comp.theory dart200 <[email protected]d> wrote:

The following claim from p246 of Turing’s seminal paper On Computable
Numbers is a fallacy:

You've missed something out. That something is something like "If I understand correctly" or "As far as I can see". Without such a qualification, your statement just looks like extreme hubris.

It is vanishingly unlikely that you have understood correctly or you
have seen far enough. If there were a flaw in Turing's 1936 paper, and
it were subtle enough to escape detection by the millions of specialists
who have verified it, it would certainly be beyond your powers as
somebody lacking education in the subject to spot it.

hopefully u eventually recognize what an _origin fallacy_ is

/the problem of enumerating computable sequences is equivalent to the
problem of finding out whether a given number is the D.N of a circle-
free machine, and we have no general process for doing this in a finite
number of steps/

For any given computable sequence, there are _infinite_ circle-free
machines ....

What's an "infinite circle-free machine"?

infinite circle-free machine_s_, ei there are countably infinite
circle-free machines that compute any given "computable sequence"

All the theory we are talking about deals only with finite machines.

errr, turing's proof specifically is in regards to computing infinite sequences aka "a computable number",

the decision paradox presented on p247 of his paper is deciding on non-terminating machines that compute infinite sequences, and the
decision paradox is between machines that are

- circular (eventually resolving to a repeating looping sequence - which
can be represented by a finite-length rational number)

OR

- circle-free (which doesn't loop back on itself, and can only be
represented by an infinite-length irrational number)

turing was concerned with computing a diagonal across "computable
numbers" that can only be computed by circle-free machines

.... which compute that particular sequence. Not only can various
machines differ significantly in the specific steps to produce the
same output, machines can be changed in superficial ways that do not
meaningfully affect the steps of computation, akin to modern no-op
statements or unreachable code

The problem of enumerating computable sequences, however, only depends
on successfully identifying _one_ circle-free machine that computes any
given computable sequences. While identifying more than one can
certainly be done, it is _not_ a requirement for enumerating computable
sequences, as _one_ machine computing a sequence /suffices to output any
and all digits of that sequence/

I think you'll need to prove that you can identify a machine that

that's is a further research question i intend to answer by proving a
further thesis: given computable number can be mapped to a _at least
one_ machine that is paradox free as therefore classifiable by all
relevant classifiers (like a classic decider on some property),

but to demonstrate a fault in turing's argument that is not necessary,
as the fault needs to be _corrected_ for turing's proof to stand

computes a given computable sequence. By definition of "computable"
there is such a machine, but how do you "identify" it? Suppose you had
a computable number and a machine, how would you test whether or not

the computable number that turing is dealing with are infinitely long,
one cannot "test" them without already have a machine that computes it,
and even then one cannot "test" the whole number in finite time

that machine generates the number?

The problem of enumerating computable sequences is therefore _not_
actually equivalent to a _general process_ of enumerating circle-free
machines, as there is no need to identify all circle-free machines which
compute any given computable sequence

Your given reason fails to rule out the equivalence. For a start, under

honestly that sentence is self-evident to me, i have no idea what u
don't understand about it tbh

what relationship does/doesn't this equivalence hold?

i don't know what u mean by under what relationship does/not the
equivalence hold ... because to me, the equivalence is categorically
false...

enumerating the possible computable sequences only requires identifying
_one_, so _not all_, circle-free machines which compute it which is
therefore merely a subset of circle-free machines,

but enumerating all circle-free machines requires identifiable _all_,
and _only all_, machines that are circle free,

identifying _a subset_ of circle-free machines is just _not_ the same
problem as identifiable _all_ circle-free machines

and idk how to make that anymore clear tbh.

Said problem is only equivalent to a _limited process_ of enumerating
circle-free machines. The machine which identifies circle-free machines
only needs the limited power of determining _at least one_ circle-free
machine for any given computable sequence, _not all_ machines for any
given computable sequence

Again, can this machine exist? It seems to me that the "limited power"
you describe is not at all limited. It is known that there is no
machine which can ascertain whether or not two turing machines produce
the same output. So the ability to discard machines redundant in this
sense will not exist either.

this comment is also not relevant to proving the fault, it is a matter
for further research,

but to respond: if my further thesis proves true, that _for each_
computable number there exists _at least one_ paradox-free machine that computes it,

then we can discard _both_ machines that are provably equivalent to some machine already found _and_ machines that fail to be provable either way
to some machine already found

demonstrating that, however, is not required for demonstrating _the_ particular fault in turing's arguments i've posted about

Because of this fallacy, the proof found on the following p247, where an
ill-defined machine 𝓗 (which attempts and fails to compute the direct
diagonal β’) is found to be undecidable in respect to circle-free
decider 𝓓; does not then prove an impossibility for enumerating
computable sequences. As the problem of enumerating /all circle-free
machines/ is _not_ equivalent to that of enumerating /just computable
sequences/

If this were a fallacy, you should write it up properly, get it

this is a first draft of a proper write up, and i intend to get it
published ofc. i finally have something simple enough to be undeniable

published and become famous. Far more likely, however, is that you have
just failed to understand what Turing's paper means.

--
arising us out of the computing dark ages,
please excuse my pseudo-pyscript,
~ the little crank that could

--
arising us out of the computing dark ages,
please excuse my pseudo-pyscript,
~ the little crank that could
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From Newsgroup: comp.theory

On 10/03/2026 18:24, Alan Mackenzie wrote:

If there were a flaw in Turing's 1936 paper, and

^^^^

it were subtle enough to escape detection by the millions of specialists
who have verified it, it would certainly be beyond your powers as

^^^^^

somebody lacking education in the subject to spot it.

You left a conditional case in there, while it could be mere syntactic agreement between the two and semantic agreement of that with "If" you
could use "is" and thereby eliminate the doubt and demonstrate the same
hubris in your assessment of dart200 as he has shown in his assessment
of Turing (1936).

Not to mention it is both ad-hominem and appeal-to-authority which are
valid for your personal gambles on which information to study and check
to replace your current beliefs but which are not valid for a reasonable
usenet discussion - for that, they're trolling.
--
Tristan Wibberley

The message body is Copyright (C) 2026 Tristan Wibberley except
citations and quotations noted. All Rights Reserved except that you may,
of course, cite it academically giving credit to me, distribute it
verbatim as part of a usenet system or its archives, and use it to
promote my greatness and general superiority without misrepresentation
of my opinions other than my opinion of my greatness and general
superiority which you _may_ misrepresent. You definitely MAY NOT train
any production AI system with it but you may train experimental AI that
will only be used for evaluation of the AI methods it implements.

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From Newsgroup: comp.theory

On 10/03/2026 18:24, Alan Mackenzie wrote:

I think you'll need to prove that you can identify a machine that
computes a given computable sequence. By definition of "computable"
there is such a machine, but how do you "identify" it? Suppose you had
a computable number and a machine, how would you test whether or not
that machine generates the number?

I think "identify" is a term of art, do you mean "How do you determine
it?" or "How do you name it?" or, I think you mean, "How do you reognise
it?".

Note, the popular word "identify" such as in "Halt! Identify yourself!" literally means "Halt! Demonstrate which person there was at one of my
valid registration events that you are the same person as!": the term of
art, in fact.
--
Tristan Wibberley

The message body is Copyright (C) 2026 Tristan Wibberley except
citations and quotations noted. All Rights Reserved except that you may,
of course, cite it academically giving credit to me, distribute it
verbatim as part of a usenet system or its archives, and use it to
promote my greatness and general superiority without misrepresentation
of my opinions other than my opinion of my greatness and general
superiority which you _may_ misrepresent. You definitely MAY NOT train
any production AI system with it but you may train experimental AI that
will only be used for evaluation of the AI methods it implements.

--- Synchronet 3.21d-Linux NewsLink 1.2

From Newsgroup: comp.theory


Ar an deichiú lá de mí Márta, scríobh Tristan Wibberley:

On 10/03/2026 16:51, dart200 wrote:

The following claim from p246 of Turing’s seminal paper On Computable Numbers is a fallacy:

/the problem of enumerating computable sequences is equivalent to the problem of finding out whether a given number is the D.N of a circle-
free machine, and we have no general process for doing this in a finite number of steps/

I am delighted to see that your usage of forward-slashes is valid as an example of usenet text formatting markup but frustrated that it's so
close to being an example of the linguist's forward-slash markup to
indicate a wrong example.

Hmm? The usual linguists’ use of the forward slash is to indicate a phonemic transcription (as opposed to a phonetic transcription). E.g. the mode of transport is /tɹeɪn/ phonemically vs [tʃɹeɪn] phonetically.

Alas, the meaning of the sentence is what is wrong, although perhaps the syntax is technically wrong somehow too, and perhaps some intrinsic
semantic problem is present too so we can say they are the linguists
error markers too after all.

Annoyingly, the linguists markup of a leading asterisk is also taken in usenet formatting markup for a bullet mark:

* this nonsense sentence is

/too crap this is sentence a/

Usenet formatting is plain text, often ASCII. I admit Gnus underlined your second sentence there, I will need to look into turning that off. I agree that a leading asterisk indicates a wrong example.
--
‘As I sat looking up at the Guinness ad, I could never figure out /
How your man stayed up on the surfboard after fourteen pints of stout’
(C. Moore)
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From Newsgroup: comp.theory

On 10/03/2026 18:24, Alan Mackenzie wrote:

It is known that there is no
machine which can ascertain whether or not two turing machines produce
the same output.

"ANY two turing machines"

My superficial reading of dart200s text is that ANY is not the restriction.
--
Tristan Wibberley

The message body is Copyright (C) 2026 Tristan Wibberley except
citations and quotations noted. All Rights Reserved except that you may,
of course, cite it academically giving credit to me, distribute it
verbatim as part of a usenet system or its archives, and use it to
promote my greatness and general superiority without misrepresentation
of my opinions other than my opinion of my greatness and general
superiority which you _may_ misrepresent. You definitely MAY NOT train
any production AI system with it but you may train experimental AI that
will only be used for evaluation of the AI methods it implements.

--- Synchronet 3.21d-Linux NewsLink 1.2