On 12/05/2026 16:32, olcott wrote:
On 5/12/2026 2:05 AM, Mikko wrote:
On 11/05/2026 14:44, olcott wrote:
On 5/11/2026 2:24 AM, Mikko wrote:
On 10/05/2026 22:06, olcott wrote:
On 5/10/2026 2:10 AM, Mikko wrote:
On 09/05/2026 15:13, olcott wrote:
On 5/9/2026 3:30 AM, Mikko wrote:
On 08/05/2026 19:58, olcott wrote:
On 5/8/2026 11:06 AM, dart200 wrote:
On 5/8/26 12:19 AM, Mikko wrote:
On 07/05/2026 12:00, dart200 wrote:
On 5/7/26 12:18 AM, Mikko wrote:A machine is not a "diagonal".
On 06/05/2026 22:40, dart200 wrote:the H machine defined on p247 from his paper /on computable >>>>>>>>>>>>> numbers/
On 5/6/26 12:55 AM, Mikko wrote:it. Any use of the word before the definition is nonsense,. >>>>>>>>>>>>
On 05/05/2026 12:28, dart200 wrote:
On 5/5/26 1:25 AM, Mikko wrote:
On 04/05/2026 10:53, dart200 wrote:
On 5/3/26 11:15 PM, Mikko wrote:
On 03/05/2026 12:09, dart200 wrote:i can't read if u can't explain
On 5/3/26 12:53 AM, Mikko wrote:
On 02/05/2026 23:39, dart200 wrote: >>>>>>>>>>>>>>>>>>>>>>> On 4/19/26 10:58 AM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 4/19/26 1:21 PM, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 4/19/2026 3:59 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 18/04/2026 15:58, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>dunno what ur saying here.
Turing proved that there are universal Turing >>>>>>>>>>>>>>>>>>>>>> machines. An universalTuring machine halts with >>>>>>>>>>>>>>>>>>>>>> some inputs and doesn't halt with any other >>>>>>>>>>>>>>>>>>>>>> input. Every Turing machine that can be given the >>>>>>>>>>>>>>>>>>>>>> same input as anrichard richard richard, that is in-correct. >>>>>>>>>>>>>>>>>>>>>>>Unknown truths are not elements of the body of >>>>>>>>>>>>>>>>>>>>>>>>>>> knowledge is a semantic tautology. Did you think >>>>>>>>>>>>>>>>>>>>>>>>>>> that things that are unknown are known? >>>>>>>>>>>>>>>>>>>>>>>>>>No, but that measn that for some sentences X >>>>>>>>>>>>>>>>>>>>>>>>>> True(X) is unknown and there >>>>>>>>>>>>>>>>>>>>>>>>>> is no method to find out.
I don't know about philosophers but >>>>>>>>>>>>>>>>>>>>>>>>>> mathematicians and logicians don't >>>>>>>>>>>>>>>>>>>>>>>>>> find it interesting if all you can say that >>>>>>>>>>>>>>>>>>>>>>>>>> all knowledge is knowable
and everything else is not. >>>>>>>>>>>>>>>>>>>>>>>>>>
Ross Finlayson, seemed to endlessly hedge on >>>>>>>>>>>>>>>>>>>>>>>>> whether
or not the truth value of the Goldbach >>>>>>>>>>>>>>>>>>>>>>>>> conjecture was
known. He seemed to think that there are >>>>>>>>>>>>>>>>>>>>>>>>> alternative
analytical frameworks that make the question of >>>>>>>>>>>>>>>>>>>>>>>>> whether
or not its truth value is known an ambiguous >>>>>>>>>>>>>>>>>>>>>>>>> question.
I needed to refer to unknown truth values >>>>>>>>>>>>>>>>>>>>>>>>> specifically
because all "undecidability" when construed >>>>>>>>>>>>>>>>>>>>>>>>> correctly
falls into one of two categories. >>>>>>>>>>>>>>>>>>>>>>>>> (a) Semantic incoherence
(b) Unknown truth values.
Nope.
Undecidability can not come from Semantic >>>>>>>>>>>>>>>>>>>>>>>> Incoherence, as the definition of Undecidability >>>>>>>>>>>>>>>>>>>>>>>> ia based on there being a coherent answer, just >>>>>>>>>>>>>>>>>>>>>>>> not one that can be determined by a computation. >>>>>>>>>>>>>>>>>>>>>>>
the undecidable problem turing described (as well >>>>>>>>>>>>>>>>>>>>>>> as the basic halting problem) involves a >>>>>>>>>>>>>>>>>>>>>>> situations that have _no_ coherent answer, not >>>>>>>>>>>>>>>>>>>>>>> just one that can be known by not computed ... >>>>>>>>>>>>>>>>>>>>>>
universal Turing machine either fails to accept >>>>>>>>>>>>>>>>>>>>>> some input with which
that universal Turing machine halts or fails to >>>>>>>>>>>>>>>>>>>>>> reject some input with
which that universal Turing macnie does not halt. >>>>>>>>>>>>>>>>>>>>>
There is a way to find out if you can read. >>>>>>>>>>>>>>>>>>>
I can't explain the art of reading Common Language. >>>>>>>>>>>>>>>>>>
It is impossible to have a Turing machine that >>>>>>>>>>>>>>>>>> computes a number thatyes, but then he argues it's impossible to compute >>>>>>>>>>>>>>>>>>> the diagonal because of the paradox that ensues when >>>>>>>>>>>>>>>>>>> naively running the classifier on the diagonal itself >>>>>>>>>>>>>>>>>> inturing hypothesized a diagonal computation that >>>>>>>>>>>>>>>>>>>>> tries to put the Nth digit from the Nth circle-free >>>>>>>>>>>>>>>>>>>>> machine as the Nth digit on this diagonal across >>>>>>>>>>>>>>>>>>>>> all circle- free machine...
That is possible because there nither the machines >>>>>>>>>>>>>>>>>>>> nor digit positions
are more numerous than natural numbers. >>>>>>>>>>>>>>>>>>>
no Turing machine can compute. But you can compute it >>>>>>>>>>>>>>>>>> if you can use
all (infinitely many) Turing machines.
no you can't.
Hard to test as I han't infinite many Turing machines. >>>>>>>>>>>>>>>> But it is
u don't need to test it, you can't define a total >>>>>>>>>>>>>>> dovetailing machine to compute turing's diagonal, >>>>>>>>>>>>>> You should not say anything about the diagonal before you >>>>>>>>>>>>>> have defined
the machine supposes to compute the "turing's diagonal"
across circle- free sequences, otherwise labeled as +#' in the >>>>>>>>>>> paper, defined at the bottom of p246
Anything that any machine can possibly compute can
be computed by applying a finite set of finite string
transformation rules to a finite set of finite strings.
Everything else is simply out-of-scope for computation
like making a silk purse from a sow's ear.
That "everything else" includes many thigns that would be
useful to
know. In particular, whether some useful function can be
computed is
in that "everything else".
Like the truth value of: "This sentence is not true"
that has no truth value.
I don't think knowing the truth value of that would be useful. At >>>>>>> least
not for any important purpose.
Knowing that all undecidability is merely semantic
incoherence enables:
"true on the basis of meaning expressed in language"
to be reliably computable for the entire body of knowledge.
We don't have the knowledge that all undecidability is merely semantic >>>>> incoherence, and can't know because we already know that there is
undecidability that is not semantic incoherence. FOr example the
axiom system
reCx (1riax = x)
1.5 != 5 re| you are wrong and I am couinting the rest as gibberish
Middle dot is a commonly used mathematical operator. In this context
where the purpose of the operation is not specified some other symbols
are often used instead, like rey or reO, or operands are just put side by >>> side with no operator between.
If commonly used mathematics is gibberish to you then we many safely
conclude that you have nothing useful to offer to the groups you
posted to.
reCx (xria1 = x)
reCxreCyreCz (xria(yriaz) = (xriay)riaz)
reCxreay (xriay = 1)
reCxreay (yriax = 1)
is useful for many purposes. But there are sentences like
reCxreay (xriay = yriax)
that are undecidable in that system. But there is notiong semantically >>>>> incoherent in that example or similar ones.
The truth about climate change and election fraud could
be computed.
Not without real world information.
Yes, so what?
So the biggest problem is not how to compute but how to get and verify
the relevant real world infomration.
A set of "atomic facts" does exist. We only nee to write> them all down.
THat is a lot of work that can never be completed.
The place to start would be published textbooks.
How long does it take to read all texbooks published in one year?
Then peer reviewed papers.
Now long does it take to read all reviewed papers published in one year?
Then published newspaper articles.
How long does it take to read all newspaper articles publised in one
year?
And what to do with the "atomic facts" that conflict each other?
All knowledge that can be expressed in language would
include the exact (x,y,z) coordinates of every atom of your
body relative to the exact center of the Earth every millisecond.
Because this degree of detail is not physically implementable
to implement my system we exclude most specific details. It
is a system of general knowledge of about 200 petabytes.
Of course you may postulate that
the climate is immutable or that there was massimbe undetected fraud >>>>> in the last or some earlier election but that is not what the word
"knowledge" means.
The key elements of election fraud are two things:
(a) There was no actual evidence of election fraud
that could have possibly changed the results.
The key element of elction fraud is to find a vulnerability in the
election procedure.
There has only been 1620 total documented cases in the USA
in the last 30 years. Every investigation into fraud by
audits found a few more votes against Trump than prior to
the audit. Each of the 60 election fraud cases Trump lost.
He lost because his lawyers could not tell the wild stories
that he was telling the public without getting themselves
convicted of perjury.
These these fact were unconvincing seems to prove that
his supporters are in a cult.
But not that there were no fraud performed skillfully enough to avoid detection. What was found was only small unintentional errors that
nobody tried to hide.
All self-reference "paradox" is equivalent to the
Liar Paradox and can be resolved by disallowing it
like ZFC disallowed Russell's "Paradox".
Whether something is a self-reference depends on interpretation. >>>>>>> In an
uninterpreted formal language there are no references and
therefore no
self-references, which is the simplest way to avoid paradoxes by >>>>>>> self-
reference.
Even without any self-reference a theory can be inconsistent.
Russell's paradox is simply an inconsistency.
Likewise with all undecidability within the body
of knowledge that can be expressed as language.
No, per definition an inconcistency is decidable so it is not an
undecidability.
RP is merely the only instance of pathological
self-reference (PSR) that was correctly rejected.
Russell's paradox is not an undecidability.
It was until ZFC refuted it.
No, it never was. The possibility to decide it was always there. That
nobody discovered it before Russell is irrelevant.
The fact is that we did at one time have Russell's
self reference "paradox" and we no longer has it
because its incoherence was rejected by ZFC.
We still have it and its relatives. The naive set theory is still inconsistent. We have ZF and other new set tehories that don't
have it. Sometimes it would be nice to have an unversal set but
there is none in ZF.
HP counter-example input is another instance of PSR.
The halting problem counter-example is neither an undecidability
nor an
incconsistency.
In computability theory and computational complexity
theory, an undecidable problem is a decision problem
for which it is proved to be impossible to construct
an algorithm that always leads to a correct yes-or-no answer.
https://en.wikipedia.org/wiki/Undecidable_problem
There is a note on the top of the page noting that the article needs
additional citations for verification. This means that the page is
not as relable as typical Wikipedia pages. In particular, the quoted
text is not quite correct.
But the quoted text is irrelevant anyway. In logic "undecidability"
means that a sentence and its negation are unprovable in some thoery.
When we switch the proof theoretic semantics then
unprovable means ungrounded thus meaningless.
When we use a theory for some serious purpose then inability to solve
a problem means we must think harder and wider. For example, it may
turn out that a partial solution is sufficient, or we can determine
that our soultion method, although not proven correct, works in every
case we have tested, so we can trust it does not fail too often.
Going outside of PA in a separate model of PA
has always only been a mere ruse.
What we really need to know is not PA but the natural numbers. If
PA does not answer some question then it is better to look for an
asnwer elsewhere. A set theory might be a good place because an
important use of natural numbers is cardinalities of finite sets.
On 05/13/2026 02:14 AM, Mikko wrote:
On 12/05/2026 16:32, olcott wrote:
On 5/12/2026 2:05 AM, Mikko wrote:
On 11/05/2026 14:44, olcott wrote:
On 5/11/2026 2:24 AM, Mikko wrote:
On 10/05/2026 22:06, olcott wrote:
On 5/10/2026 2:10 AM, Mikko wrote:
On 09/05/2026 15:13, olcott wrote:
On 5/9/2026 3:30 AM, Mikko wrote:
On 08/05/2026 19:58, olcott wrote:
On 5/8/2026 11:06 AM, dart200 wrote:
On 5/8/26 12:19 AM, Mikko wrote:
On 07/05/2026 12:00, dart200 wrote:
On 5/7/26 12:18 AM, Mikko wrote:A machine is not a "diagonal".
On 06/05/2026 22:40, dart200 wrote:the H machine defined on p247 from his paper /on computable >>>>>>>>>>>>>> numbers/
On 5/6/26 12:55 AM, Mikko wrote:it. Any use of the word before the definition is nonsense,. >>>>>>>>>>>>>
On 05/05/2026 12:28, dart200 wrote:
On 5/5/26 1:25 AM, Mikko wrote:
On 04/05/2026 10:53, dart200 wrote:
On 5/3/26 11:15 PM, Mikko wrote:
On 03/05/2026 12:09, dart200 wrote: >>>>>>>>>>>>>>>>>>>>>> On 5/3/26 12:53 AM, Mikko wrote:i can't read if u can't explain
On 02/05/2026 23:39, dart200 wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 4/19/26 10:58 AM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 4/19/26 1:21 PM, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 4/19/2026 3:59 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>> On 18/04/2026 15:58, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>dunno what ur saying here.
Turing proved that there are universal Turing >>>>>>>>>>>>>>>>>>>>>>> machines. An universalTuring machine halts with >>>>>>>>>>>>>>>>>>>>>>> some inputs and doesn't halt with any other >>>>>>>>>>>>>>>>>>>>>>> input. Every Turing machine that can be given the >>>>>>>>>>>>>>>>>>>>>>> same input as anrichard richard richard, that is in-correct. >>>>>>>>>>>>>>>>>>>>>>>>Unknown truths are not elements of the body of >>>>>>>>>>>>>>>>>>>>>>>>>>>> knowledge is a semantic tautology. Did you >>>>>>>>>>>>>>>>>>>>>>>>>>>> thinkNo, but that measn that for some sentences X >>>>>>>>>>>>>>>>>>>>>>>>>>> True(X) is unknown and there >>>>>>>>>>>>>>>>>>>>>>>>>>> is no method to find out. >>>>>>>>>>>>>>>>>>>>>>>>>>>
that things that are unknown are known? >>>>>>>>>>>>>>>>>>>>>>>>>>>
I don't know about philosophers but >>>>>>>>>>>>>>>>>>>>>>>>>>> mathematicians and logicians don't >>>>>>>>>>>>>>>>>>>>>>>>>>> find it interesting if all you can say that >>>>>>>>>>>>>>>>>>>>>>>>>>> all knowledge is knowable >>>>>>>>>>>>>>>>>>>>>>>>>>> and everything else is not. >>>>>>>>>>>>>>>>>>>>>>>>>>>
Ross Finlayson, seemed to endlessly hedge on >>>>>>>>>>>>>>>>>>>>>>>>>> whether
or not the truth value of the Goldbach >>>>>>>>>>>>>>>>>>>>>>>>>> conjecture was
known. He seemed to think that there are >>>>>>>>>>>>>>>>>>>>>>>>>> alternative
analytical frameworks that make the question of >>>>>>>>>>>>>>>>>>>>>>>>>> whether
or not its truth value is known an ambiguous >>>>>>>>>>>>>>>>>>>>>>>>>> question.
I needed to refer to unknown truth values >>>>>>>>>>>>>>>>>>>>>>>>>> specifically
because all "undecidability" when construed >>>>>>>>>>>>>>>>>>>>>>>>>> correctly
falls into one of two categories. >>>>>>>>>>>>>>>>>>>>>>>>>> (a) Semantic incoherence
(b) Unknown truth values.
Nope.
Undecidability can not come from Semantic >>>>>>>>>>>>>>>>>>>>>>>>> Incoherence, as the definition of Undecidability >>>>>>>>>>>>>>>>>>>>>>>>> ia based on there being a coherent answer, just >>>>>>>>>>>>>>>>>>>>>>>>> not one that can be determined by a computation. >>>>>>>>>>>>>>>>>>>>>>>>
the undecidable problem turing described (as well >>>>>>>>>>>>>>>>>>>>>>>> as the basic halting problem) involves a >>>>>>>>>>>>>>>>>>>>>>>> situations that have _no_ coherent answer, not >>>>>>>>>>>>>>>>>>>>>>>> just one that can be known by not computed ... >>>>>>>>>>>>>>>>>>>>>>>
universal Turing machine either fails to accept >>>>>>>>>>>>>>>>>>>>>>> some input with which
that universal Turing machine halts or fails to >>>>>>>>>>>>>>>>>>>>>>> reject some input with
which that universal Turing macnie does not halt. >>>>>>>>>>>>>>>>>>>>>>
There is a way to find out if you can read. >>>>>>>>>>>>>>>>>>>>
I can't explain the art of reading Common Language. >>>>>>>>>>>>>>>>>>>
It is impossible to have a Turing machine that >>>>>>>>>>>>>>>>>>> computes a number thatyes, but then he argues it's impossible to compute >>>>>>>>>>>>>>>>>>>> the diagonal because of the paradox that ensues when >>>>>>>>>>>>>>>>>>>> naively running the classifier on the diagonal itself >>>>>>>>>>>>>>>>>>> inturing hypothesized a diagonal computation that >>>>>>>>>>>>>>>>>>>>>> tries to put the Nth digit from the Nth circle-free >>>>>>>>>>>>>>>>>>>>>> machine as the Nth digit on this diagonal across >>>>>>>>>>>>>>>>>>>>>> all circle- free machine...
That is possible because there nither the machines >>>>>>>>>>>>>>>>>>>>> nor digit positions
are more numerous than natural numbers. >>>>>>>>>>>>>>>>>>>>
no Turing machine can compute. But you can compute it >>>>>>>>>>>>>>>>>>> if you can use
all (infinitely many) Turing machines.
no you can't.
Hard to test as I han't infinite many Turing machines. >>>>>>>>>>>>>>>>> But it is
u don't need to test it, you can't define a total >>>>>>>>>>>>>>>> dovetailing machine to compute turing's diagonal, >>>>>>>>>>>>>>> You should not say anything about the diagonal before you >>>>>>>>>>>>>>> have defined
the machine supposes to compute the "turing's diagonal" >>>>>>>>>>>> across circle- free sequences, otherwise labeled as +#' in the >>>>>>>>>>>> paper, defined at the bottom of p246
Anything that any machine can possibly compute can
be computed by applying a finite set of finite string
transformation rules to a finite set of finite strings.
Everything else is simply out-of-scope for computation
like making a silk purse from a sow's ear.
That "everything else" includes many thigns that would be
useful to
know. In particular, whether some useful function can be
computed is
in that "everything else".
Like the truth value of: "This sentence is not true"
that has no truth value.
I don't think knowing the truth value of that would be useful. At >>>>>>>> least
not for any important purpose.
Knowing that all undecidability is merely semantic
incoherence enables:
"true on the basis of meaning expressed in language"
to be reliably computable for the entire body of knowledge.
We don't have the knowledge that all undecidability is merely
semantic
incoherence, and can't know because we already know that there is
undecidability that is not semantic incoherence. FOr example the
axiom system
reCx (1riax = x)
1.5 != 5 re| you are wrong and I am couinting the rest as gibberish
Middle dot is a commonly used mathematical operator. In this context
where the purpose of the operation is not specified some other symbols >>>> are often used instead, like rey or reO, or operands are just put side by >>>> side with no operator between.
If commonly used mathematics is gibberish to you then we many safely
conclude that you have nothing useful to offer to the groups you
posted to.
reCx (xria1 = x)
reCxreCyreCz (xria(yriaz) = (xriay)riaz)
reCxreay (xriay = 1)
reCxreay (yriax = 1)
is useful for many purposes. But there are sentences like
reCxreay (xriay = yriax)
that are undecidable in that system. But there is notiong
semantically
incoherent in that example or similar ones.
The truth about climate change and election fraud could
be computed.
Not without real world information.
Yes, so what?
So the biggest problem is not how to compute but how to get and verify >>>> the relevant real world infomration.
A set of "atomic facts" does exist. We only nee to write> them all down.
THat is a lot of work that can never be completed.
The place to start would be published textbooks.
How long does it take to read all texbooks published in one year?
Then peer reviewed papers.
Now long does it take to read all reviewed papers published in one year?
Then published newspaper articles.
How long does it take to read all newspaper articles publised in one
year?
And what to do with the "atomic facts" that conflict each other?
All knowledge that can be expressed in language would
include the exact (x,y,z) coordinates of every atom of your
body relative to the exact center of the Earth every millisecond.
Because this degree of detail is not physically implementable
to implement my system we exclude most specific details. It
is a system of general knowledge of about 200 petabytes.
Of course you may postulate that
the climate is immutable or that there was massimbe undetected fraud >>>>>> in the last or some earlier election but that is not what the word >>>>>> "knowledge" means.
The key elements of election fraud are two things:
(a) There was no actual evidence of election fraud
that could have possibly changed the results.
The key element of elction fraud is to find a vulnerability in the
election procedure.
There has only been 1620 total documented cases in the USA
in the last 30 years. Every investigation into fraud by
audits found a few more votes against Trump than prior to
the audit. Each of the 60 election fraud cases Trump lost.
He lost because his lawyers could not tell the wild stories
that he was telling the public without getting themselves
convicted of perjury.
These these fact were unconvincing seems to prove that
his supporters are in a cult.
But not that there were no fraud performed skillfully enough to avoid
detection. What was found was only small unintentional errors that
nobody tried to hide.
All self-reference "paradox" is equivalent to the
Liar Paradox and can be resolved by disallowing it
like ZFC disallowed Russell's "Paradox".
Whether something is a self-reference depends on interpretation. >>>>>>>> In an
uninterpreted formal language there are no references and
therefore no
self-references, which is the simplest way to avoid paradoxes by >>>>>>>> self-
reference.
Even without any self-reference a theory can be inconsistent.
Russell's paradox is simply an inconsistency.
Likewise with all undecidability within the body
of knowledge that can be expressed as language.
No, per definition an inconcistency is decidable so it is not an
undecidability.
RP is merely the only instance of pathological
self-reference (PSR) that was correctly rejected.
Russell's paradox is not an undecidability.
It was until ZFC refuted it.
No, it never was. The possibility to decide it was always there. That
nobody discovered it before Russell is irrelevant.
The fact is that we did at one time have Russell's
self reference "paradox" and we no longer has it
because its incoherence was rejected by ZFC.
We still have it and its relatives. The naive set theory is still
inconsistent. We have ZF and other new set tehories that don't
have it. Sometimes it would be nice to have an unversal set but
there is none in ZF.
HP counter-example input is another instance of PSR.
The halting problem counter-example is neither an undecidability
nor an
incconsistency.
In computability theory and computational complexity
theory, an undecidable problem is a decision problem
for which it is proved to be impossible to construct
an algorithm that always leads to a correct yes-or-no answer.
https://en.wikipedia.org/wiki/Undecidable_problem
There is a note on the top of the page noting that the article needs
additional citations for verification. This means that the page is
not as relable as typical Wikipedia pages. In particular, the quoted
text is not quite correct.
But the quoted text is irrelevant anyway. In logic "undecidability"
means that a sentence and its negation are unprovable in some thoery.
When we switch the proof theoretic semantics then
unprovable means ungrounded thus meaningless.
When we use a theory for some serious purpose then inability to solve
a problem means we must think harder and wider. For example, it may
turn out that a partial solution is sufficient, or we can determine
that our soultion method, although not proven correct, works in every
case we have tested, so we can trust it does not fail too often.
Going outside of PA in a separate model of PA
has always only been a mere ruse.
What we really need to know is not PA but the natural numbers. If
PA does not answer some question then it is better to look for an
asnwer elsewhere. A set theory might be a good place because an
important use of natural numbers is cardinalities of finite sets.
Thanks for writing. One thing I though to suggest is to consider
the "deconstructive" approach. For example PA, Peano Arithmetic,
is usually given as describing a least natural number, then the
successor function making a case for induction that each following
natural number exists and is unique under the operation. Since
that basically is the same thing as cases for induction, it doesn't
say much about all the body of relation of arithmetic fact.
So, one might consider the successor operation as alike "increment",
then the idea that that's the Sumerian concept of an operation on
numbers. Then, the other basic concept of operations on numbers
is "partition", or the Egyptians. So, Sumerians start with "addition"
after "increment", while, Egyptians start with "division" after
"partition". These are familiar as "positional notation" and
"continued fractions".
Then, the idea is that the rationals or the ordered field (since
the Pythagorean and the Eudoxan, about Pythagoras and Eudoxus,
then as with regards to their Egyptian and Sumerian influences)
have addition (and subtraction) and later multiplication (and
division) as being a field after a group, a different deconstructive
account can instead make for that it's two groups, one after increment
(like Peano's, it's just increment the axiom of the operation, not
quite "addition"), the other after partition.
So, set theory is plainly a "theory-of-one-relation: elt", or "member".
That relation "elt" is a binary predicate, vis-a-vis, "contains", or
"has", which is usually considered the class relation in class theory a "theory-of-one-relation: contains".
So, familiar with accounts of theories-of-one-relation, "elt" and "has",
or "member-of" and "contains", then the descriptive account
relating to numbers (a model-theoretic account) for "increment" and "partition", then help go to show that Peano Arithmetic then later
for Peano and Presburger about the schema and the orders of the logic
about addition and multiplication, then make for why it's natural
to have for "Egyptian-Peano" and "Egyptian-Presburger", why then
the natural numbers are also as after an account of "Egyptian fractions"
then on down to "long-subtraction" (or, "long-addition"),
are for a whole set of whole numbers.
So, "relations", and, "operations", then about the _modular_, is
about how many various accounts of numbers, like the Sumerian and
the Egyptian, come to agree on what looks like counting and the
rational, and the ordered field, as from the complete ordered field,
or about: an infinity, and, a continuum.
Then various accounts of Church/Turing and Church/Rosser and
Church/Rice are much about the independence of the infinitary,
for example where the natural integers are compact, or not.
"Prime at infinity" is a usual enough way to describe that,
for what's usually given as "the fundamental th'm of arithmetic".
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