From Newsgroup: comp.theory

At least two types of undecidable problems.
Notably, Prop5 may exlain why Collatz Problem cannot be 'officially' proved following syllogistic production-arule. https://sourceforge.net/projects/cscall/files/MisFiles/Coll-proof-en.txt/download
https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber2-zh.txt/download
...[cut]
Prop 4: Procedural statement contains undecidable statements.
Proof:
bool H(void(*)()); // Let H be a decision procedure. H(D)==true iff D()
// terminates normally.
// If H exists, then there is a D similar to the liar paradox, making
// H(D) undecidable.
void D() {
if(H(D) == true) for(;;){};
}
If H is considered a propositional statement, then this proposition is
undecidable.
Note: The proof of the classical Halting Problem is often summarized in the
above manner, but the content is quite different and does not involve
infinite self-references.
Note: Another type of proposition involving infinite self-reference, like the
'honest person's paradox,' is also non-terminating, such as:
bool P() { // "My words is true"
return P();
}
Prop5: Peano's axiomatic system cannot prove that infinity is a natural number.
Proof: The successor generator S does not terminate. If the meaning of 'proof'
must be in the form like syllogistic production, then the Peano
axiomatic system cannot produce the statement "reRreerao". -------------------------------
Thank olcott, who voluntarily tries the opposite, even sold his soul in the Lake
of Fire for the last >20 years for the opposite, and still failed.
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 1/26/26 12:48 PM, wij wrote:

At least two types of undecidable problems.

Notably, Prop5 may exlain why Collatz Problem cannot be 'officially' proved following syllogistic production-arule. https://sourceforge.net/projects/cscall/files/MisFiles/Coll-proof-en.txt/download

https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber2-zh.txt/download
...[cut]
Prop 4: Procedural statement contains undecidable statements.
Proof:
bool H(void(*)()); // Let H be a decision procedure. H(D)==true iff D()
// terminates normally.

// If H exists, then there is a D similar to the liar paradox, making
// H(D) undecidable.
void D() {
if(H(D) == true) for(;;){};
}
If H is considered a propositional statement, then this proposition is
undecidable.

Note: The proof of the classical Halting Problem is often summarized in the
above manner, but the content is quite different and does not involve
infinite self-references.
Note: Another type of proposition involving infinite self-reference, like the
'honest person's paradox,' is also non-terminating, such as:
bool P() { // "My words is true"
return P();
}

Prop5: Peano's axiomatic system cannot prove that infinity is a natural number.
Proof: The successor generator S does not terminate. If the meaning of 'proof'
must be in the form like syllogistic production, then the Peano
axiomatic system cannot produce the statement "reRreerao". -------------------------------

Thank olcott, who voluntarily tries the opposite, even sold his soul in the Lake
of Fire for the last >20 years for the opposite, and still failed.

But "infinity" isn't considered a natural number, as it isn't the
"successor" of any given natural number, which is how we build them, so
not being able to prove it is a good thing.

Some "alternate models" allow for other roots of succession, which allow
us to define an w (omega) as the first trans-finite "number" and build
up that first order of trans-finite numbers. But that isn't the basic PA

Note, in your two types of undecidable statements, the Halting Problem
is of the form that is inherently undecidable, as any presumed solution creates a contradiction.

The second form might be deciable if some other information is available
about the statement.
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: comp.theory

On Mon, 2026-01-26 at 13:39 -0500, Richard Damon wrote:

On 1/26/26 12:48 PM, wij wrote:

At least two types of undecidable problems.

Notably, Prop5 may exlain why Collatz Problem cannot be 'officially' proved following syllogistic production-arule. https://sourceforge.net/projects/cscall/files/MisFiles/Coll-proof-en.txt/download

https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber2-zh.txt/download
...[cut]
Prop 4: Procedural statement contains undecidable statements.
-a-a Proof:
-a-a-a-a-a-a-a-a bool H(void(*)()); // Let H be a decision procedure. H(D)==true iff D()
-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a // terminates normally.

-a-a-a-a-a-a-a-a // If H exists, then there is a D similar to the liar paradox, making
-a-a-a-a-a-a-a-a // H(D) undecidable.
-a-a-a-a-a-a-a-a void D() {
-a-a-a-a-a-a-a-a-a-a if(H(D) == true) for(;;){};
-a-a-a-a-a-a-a-a }
-a-a-a-a-a-a-a-a If H is considered a propositional statement, then this proposition is
-a-a-a-a-a-a-a-a undecidable.

-a-a Note: The proof of the classical Halting Problem is often summarized in the
-a-a-a-a-a-a-a-a above manner, but the content is quite different and does not involve
-a-a-a-a-a-a-a-a infinite self-references.
-a-a Note: Another type of proposition involving infinite self-reference, like the
-a-a-a-a-a-a-a-a 'honest person's paradox,' is also non-terminating, such as:
-a-a-a-a-a-a-a-a-a-a-a bool P() {-a-a-a // "My words is true" -a-a-a-a-a-a-a-a-a-a-a-a-a return P();
-a-a-a-a-a-a-a-a-a-a-a }

Prop5: Peano's axiomatic system cannot prove that infinity is a natural number.
-a-a Proof: The successor generator S does not terminate. If the meaning of 'proof'
-a-a-a-a-a-a-a-a must be in the form like syllogistic production, then the Peano
-a-a-a-a-a-a-a-a axiomatic system cannot produce the statement "reRreerao". -------------------------------

Thank olcott, who voluntarily tries the opposite, even sold his soul in the Lake
of Fire for the last >20 years for the opposite, and still failed.

But "infinity" isn't considered a natural number, as it isn't the "successor" of any given natural number, which is how we build them, so
not being able to prove it is a good thing.

It not the that problem of what you 'consider'. You need to prove it in PA! Successor only produces 1,S(1),SS(1),SSS(1),... PA does not specify termination condition, otherwise I think there will be a problem as an axiom. Particularly,-a
the usual 'natural number' is not strictly N.

Some "alternate models" allow for other roots of succession, which allow
us to define an w (omega) as the first trans-finite "number" and build
up that first order of trans-finite numbers. But that isn't the basic PA

I don't know that "alternate models" is. But, if you say "that isn't the basic PA",-a
then I may safely ignore this section.

Note, in your two types of undecidable statements, the Halting Problem
is of the form that is inherently undecidable, as any presumed solution creates a contradiction.

I have reason to see Undecidable is based on TM model, not traditional logic.

The second form might be deciable if some other information is available about the statement.

How does proposition "My words is true" is decidable (infinite self-reference)? --- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 1/26/26 2:54 PM, wij wrote:

On Mon, 2026-01-26 at 13:39 -0500, Richard Damon wrote:

On 1/26/26 12:48 PM, wij wrote:

At least two types of undecidable problems.

Notably, Prop5 may exlain why Collatz Problem cannot be 'officially' proved >>> following syllogistic production-arule.
https://sourceforge.net/projects/cscall/files/MisFiles/Coll-proof-en.txt/download


https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber2-zh.txt/download
...[cut]
Prop 4: Procedural statement contains undecidable statements.
-a-a Proof:
-a-a-a-a-a-a-a-a bool H(void(*)()); // Let H be a decision procedure. H(D)==true iff D()
-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a // terminates normally.

-a-a-a-a-a-a-a-a // If H exists, then there is a D similar to the liar paradox, making
-a-a-a-a-a-a-a-a // H(D) undecidable.
-a-a-a-a-a-a-a-a void D() {
-a-a-a-a-a-a-a-a-a-a if(H(D) == true) for(;;){};
-a-a-a-a-a-a-a-a }
-a-a-a-a-a-a-a-a If H is considered a propositional statement, then this proposition is
-a-a-a-a-a-a-a-a undecidable.

-a-a Note: The proof of the classical Halting Problem is often summarized in the
-a-a-a-a-a-a-a-a above manner, but the content is quite different and does not involve
-a-a-a-a-a-a-a-a infinite self-references.
-a-a Note: Another type of proposition involving infinite self-reference, like the
-a-a-a-a-a-a-a-a 'honest person's paradox,' is also non-terminating, such as:
-a-a-a-a-a-a-a-a-a-a-a bool P() {-a-a-a // "My words is true"
-a-a-a-a-a-a-a-a-a-a-a-a-a return P();
-a-a-a-a-a-a-a-a-a-a-a }

Prop5: Peano's axiomatic system cannot prove that infinity is a natural number.
-a-a Proof: The successor generator S does not terminate. If the meaning of 'proof'
-a-a-a-a-a-a-a-a must be in the form like syllogistic production, then the Peano
-a-a-a-a-a-a-a-a axiomatic system cannot produce the statement "reRreerao".
-------------------------------

Thank olcott, who voluntarily tries the opposite, even sold his soul in the Lake
of Fire for the last >20 years for the opposite, and still failed.

But "infinity" isn't considered a natural number, as it isn't the
"successor" of any given natural number, which is how we build them, so
not being able to prove it is a good thing.

It not the that problem of what you 'consider'. You need to prove it in PA! Successor only produces 1,S(1),SS(1),SSS(1),... PA does not specify termination
condition, otherwise I think there will be a problem as an axiom. Particularly,
the usual 'natural number' is not strictly N.

It doesn't specify a "termination" condition, as it doesn't terminate.

The set of Natural Numbers is "infinite" in size, thus the generator
doesn't need to stop.

Some "alternate models" allow for other roots of succession, which allow
us to define an w (omega) as the first trans-finite "number" and build
up that first order of trans-finite numbers. But that isn't the basic PA

I don't know that "alternate models" is. But, if you say "that isn't the basic PA",
then I may safely ignore this section.

I guess you don't want to learn the truth then,

I say "basic PA", as there are many variations on PA where a few of the
axioms are tweeked, like to remove induction as an axiom, but add first
order sttatement to try to "fake" it.

Note, in your two types of undecidable statements, the Halting Problem
is of the form that is inherently undecidable, as any presumed solution
creates a contradiction.

I have reason to see Undecidable is based on TM model, not traditional logic.

Which are?

Part of the problem you will run into is that the concept of a "Proof"
and that of a Computation can be shown to be relatable.

The second form might be deciable if some other information is available
about the statement.

How does proposition "My words is true" is decidable (infinite self-reference)?

Because by itself, it isn't a truth bearing statement, and thus not a candidate to be undeciablity.

Since we can show that status with a finite proof, it is a decidable statement, the decision is that it is not a truth bearer.
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 26/01/2026 19:48, wij wrote:

Prop5: Peano's axiomatic system cannot prove that infinity is a
natural number.

In the language of PA you can't even say that infinity is a number.
But every natural number except 0 is the successor of some other
natural number so if infinity is not 0 it is the successor of some
other natural number.
--
Mikko
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: comp.theory

On Tue, 2026-01-27 at 14:50 -0600, olcott wrote:

On 1/26/2026 11:48 AM, wij wrote:

At least two types of undecidable problems.

reR is not a number it is the process of more.

Agree. So we can count N, from 1 to .... more, endless more.

Notably, Prop5 may exlain why Collatz Problem cannot be 'officially' proved following syllogistic production-arule. https://sourceforge.net/projects/cscall/files/MisFiles/Coll-proof-en.txt/download

https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber2-zh.txt/download
...[cut]
Prop 4: Procedural statement contains undecidable statements.
-a-a Proof:
-a-a-a-a-a-a-a-a bool H(void(*)()); // Let H be a decision procedure. H(D)==true iff D()
-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a // terminates normally.

-a-a-a-a-a-a-a-a // If H exists, then there is a D similar to the liar paradox, making
-a-a-a-a-a-a-a-a // H(D) undecidable.
-a-a-a-a-a-a-a-a void D() {
-a-a-a-a-a-a-a-a-a-a if(H(D) == true) for(;;){};
-a-a-a-a-a-a-a-a }
-a-a-a-a-a-a-a-a If H is considered a propositional statement, then this proposition is
-a-a-a-a-a-a-a-a undecidable.

-a-a Note: The proof of the classical Halting Problem is often summarized in the
-a-a-a-a-a-a-a-a above manner, but the content is quite different and does not involve
-a-a-a-a-a-a-a-a infinite self-references.
-a-a Note: Another type of proposition involving infinite self-reference, like the
-a-a-a-a-a-a-a-a 'honest person's paradox,' is also non-terminating, such as:
-a-a-a-a-a-a-a-a-a-a-a bool P() {-a-a-a // "My words is true" -a-a-a-a-a-a-a-a-a-a-a-a-a return P();
-a-a-a-a-a-a-a-a-a-a-a }

Prop5: Peano's axiomatic system cannot prove that infinity is a natural number.
-a-a Proof: The successor generator S does not terminate. If the meaning of 'proof'
-a-a-a-a-a-a-a-a must be in the form like syllogistic production, then the Peano
-a-a-a-a-a-a-a-a axiomatic system cannot produce the statement "reRreerao". -------------------------------

Thank olcott, who voluntarily tries the opposite, even sold his soul in the Lake
of Fire for the last >20 years for the opposite, and still failed.

--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 1/27/2026 3:04 PM, wij wrote:

On Tue, 2026-01-27 at 14:50 -0600, olcott wrote:

On 1/26/2026 11:48 AM, wij wrote:

At least two types of undecidable problems.

reR is not a number it is the process of more.

Agree. So we can count N, from 1 to .... more, endless more.

Yes.

Notably, Prop5 may exlain why Collatz Problem cannot be 'officially' proved >>> following syllogistic production-arule.
https://sourceforge.net/projects/cscall/files/MisFiles/Coll-proof-en.txt/download


https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber2-zh.txt/download
...[cut]
Prop 4: Procedural statement contains undecidable statements.
-a-a Proof:
-a-a-a-a-a-a-a-a bool H(void(*)()); // Let H be a decision procedure. H(D)==true iff D()
-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a // terminates normally.

-a-a-a-a-a-a-a-a // If H exists, then there is a D similar to the liar paradox, making
-a-a-a-a-a-a-a-a // H(D) undecidable.
-a-a-a-a-a-a-a-a void D() {
-a-a-a-a-a-a-a-a-a-a if(H(D) == true) for(;;){};
-a-a-a-a-a-a-a-a }
-a-a-a-a-a-a-a-a If H is considered a propositional statement, then this proposition is
-a-a-a-a-a-a-a-a undecidable.

-a-a Note: The proof of the classical Halting Problem is often summarized in the
-a-a-a-a-a-a-a-a above manner, but the content is quite different and does not involve
-a-a-a-a-a-a-a-a infinite self-references.
-a-a Note: Another type of proposition involving infinite self-reference, like the
-a-a-a-a-a-a-a-a 'honest person's paradox,' is also non-terminating, such as:
-a-a-a-a-a-a-a-a-a-a-a bool P() {-a-a-a // "My words is true"
-a-a-a-a-a-a-a-a-a-a-a-a-a return P();
-a-a-a-a-a-a-a-a-a-a-a }

Prop5: Peano's axiomatic system cannot prove that infinity is a natural number.
-a-a Proof: The successor generator S does not terminate. If the meaning of 'proof'
-a-a-a-a-a-a-a-a must be in the form like syllogistic production, then the Peano
-a-a-a-a-a-a-a-a axiomatic system cannot produce the statement "reRreerao".
-------------------------------

Thank olcott, who voluntarily tries the opposite, even sold his soul in the Lake
of Fire for the last >20 years for the opposite, and still failed.

--
Copyright 2026 Olcott<br><br>

My 28 year goal has been to make <br>
"true on the basis of meaning expressed in language"<br>
reliably computable for the entire body of knowledge.<br><br>

This required establishing a new foundation<br>
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 27/01/2026 22:23, wij wrote:

On Tue, 2026-01-27 at 10:25 +0200, Mikko wrote:

On 26/01/2026 19:48, wij wrote:

Prop5: Peano's axiomatic system cannot prove that infinity is a

natural number.

In the language of PA you can't even say that infinity is a number.
But every natural number except 0 is the successor of some other
natural number so if infinity is not 0 it is the successor of some
other natural number.

Agree.

....[quote]
Prop5: Peano's axiomatic system cannot prove that infinity is or is not a
natural number.
Proof: The successor generator S does not terminate. If the meaning of 'proof'
must be in the form like syllogistic production, then the Peano
axiomatic system cannot produce the statement like "reRreerao","reRreerao".
-------

I added this proposition to explain, like many other hard problems, why proving
Collatz Problem is difficult (except using algorithm (TM model)).

It is deep. IMO, many mathematical 'difficult' problems are based on illusion that "too small is zero".

In natural numbers everything smaller than 1 is zero.
--
Mikko
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 27/01/2026 22:50, olcott wrote:

On 1/26/2026 11:48 AM, wij wrote:

At least two types of undecidable problems.

reR is not a number it is the process of more.

In an extended number system reR can be a number. Sometimes it is useful
to extend a number system with one or two extra numbers.
--
Mikko
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 29/01/2026 09:21, Mikko wrote:

On 27/01/2026 22:50, olcott wrote:

On 1/26/2026 11:48 AM, wij wrote:

At least two types of undecidable problems.

reR is not a number it is the process of more.

In an extended number system reR can be a number. Sometimes it is useful
to extend a number system with one or two extra numbers.

How do you characterise an object as a number vs some other denomination?
--
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 3.21b-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 29/01/2026 12:03, Tristan Wibberley wrote:

On 29/01/2026 09:21, Mikko wrote:

In an extended number system reR can be a number. Sometimes it is useful
to extend a number system with one or two extra numbers.

How do you characterise an object as a number vs some other denomination?

By presenting a set of axioms/definitions and saying "These are the axioms/definitions of Wibberley numbers" [or whatever]. Then you can start talking about how such Wibberley numbers relate to other interesting sets of numbers, such as the "standard" reals. Beyond that, it's largely a matter
of utility. If Wibberley numbers can be used to solve interesting problems, they will come into wider use, otherwise not. That's it, really. There's nothing about Wibberley numbers that makes them specifically numbers "vs some other denomination" beyond that. Oh, plus the recognition that the standard number systems that we all learn in infancy and at school are not the whole story.

One of the reasons why this group gets snarled up is that some of
the contributors have idiosyncratic views of what numbers are, but cannot or will not explain how their axioms differ from the usual set. This is, in particular, the case with those who want to use infinity or limits in ways
that would be regarded as eccentric/wrong by conventional mathematicians.
--
Andy Walker, Nottingham.
Andy's music pages: www.cuboid.me.uk/andy/Music
Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Favarger
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 29/01/2026 14:03, Tristan Wibberley wrote:

On 29/01/2026 09:21, Mikko wrote:

On 27/01/2026 22:50, olcott wrote:

On 1/26/2026 11:48 AM, wij wrote:

At least two types of undecidable problems.

reR is not a number it is the process of more.

In an extended number system reR can be a number. Sometimes it is useful
to extend a number system with one or two extra numbers.

How do you characterise an object as a number vs some other denomination?

Being a number is not an intrinsic feature of a number. Cantor
constructed the natural numbers as sets, so 0 is a natural
number if regarded one way and a set if regarded another way.
Being a number means being a member of a number space. Whether
a space can be called a number space depends on how similar it
is to those spaces that are usuall called number spaces. This
usually mans that a number space must contain zero and one and
must have two functions that can be called + and * and have at
least most of the properties that

x + 0 = x
x + y = y + x
1 * x = x
x * 1 = x
x * (y + z) = x * y + x * z
(x + y) * z = x * z + y * z

for at least most of the members of the space.
--
Mikko
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 1/29/26 4:21 AM, Mikko wrote:

On 27/01/2026 22:50, olcott wrote:

On 1/26/2026 11:48 AM, wij wrote:

At least two types of undecidable problems.

reR is not a number it is the process of more.

In an extended number system reR can be a number. Sometimes it is useful
to extend a number system with one or two extra numbers.

I don't know of many formal number systems that make reR itself a
"number". It is at best a "limit" you go to.

When we get into the various trans-finite number systems, we tend to use something else, (and get more than one or two extra numbers) with a more precise value. The more common value is omega as the first transfinte
number beyond the natural numbers, which then creates a whole field of trans-finite numbers of the form a*w + b.
--- Synchronet 3.21b-Linux NewsLink 1.2