On 2024-10-16 14:27:09 +0000, olcott said:

The whole notion of undecidability is anchored in ignoring the fact that
some expressions of language are simply not truth bearers.

A formal theory is undecidable if there is no Turing machine that
determines whether a formula of that theory is a theorem of that
theory or not. Whether an expression is a truth bearer is not
relevant. Either there is a valid proof of that formula or there
is not. No third possibility.

--
Mikko

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 10/16/24 1:55 PM, olcott wrote:

On 10/16/2024 12:31 PM, olcott wrote:

On 10/16/2024 11:37 AM, Mikko wrote:

On 2024-10-16 14:27:09 +0000, olcott said:

The whole notion of undecidability is anchored in ignoring the fact
that
some expressions of language are simply not truth bearers.

A formal theory is undecidable if there is no Turing machine that
determines whether a formula of that theory is a theorem of that
theory or not.

When the question: Is finite string X a theory of L?
has no correct answer from YES and NO, then the question
is rejected as not a truth bearer.

I did not say that exactly correctly.

When the question:
Is finite string X a theory of L?
has no correct answer from YES and NO,

then the statements:
(a) Finite string X is a theory of L
and
(b) Finite string X is NOT a theory of L

are rejected as not a truth bearers.

How can there not be a Yes or No answer?

Either X IS or it IS NOT a theory of L, as either a proof exists or it
doesn't.

If X is non-sense, then it isn't a theory of L, as you can't prove
non-sense to be true in a non-contradictory L.

So, how can THOSE questions not be a truth bearers?

You don't seem to understad what Truth actually is.

I guess your logic is that there is no such thing as a non-contradictory
field of study.

But that is just because you don't seem to understand how logic actually
works.

Whether an expression is a truth bearer is not
relevant. Either there is a valid proof of that formula or there
is not. No third possibility.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 10/16/24 8:51 PM, olcott wrote:

On 10/16/2024 7:47 PM, Richard Damon wrote:

On 10/16/24 6:34 PM, olcott wrote:

On 10/16/2024 11:37 AM, Mikko wrote:

On 2024-10-16 14:27:09 +0000, olcott said:

The whole notion of undecidability is anchored in ignoring the fact
that
some expressions of language are simply not truth bearers.

A formal theory is undecidable if there is no Turing machine that
determines whether a formula of that theory is a theorem of that
theory or not. Whether an expression is a truth bearer is not
relevant. Either there is a valid proof of that formula or there
is not. No third possibility.

*I still said that wrong*
(1) There is a finite set of expressions of language
that are stipulated to be true (STBT) in theory L.

(2) There is a finite set of true preserving operations
(TPO) that can be applied to this finite set in theory L.

When formula x cannot be derived by applying the TPO
of L to STBT of L then x is not a theorem of L.

A theorem is a statement that can be demonstrated to be
true by accepted mathematical operations and arguments.
https://mathworld.wolfram.com/Theorem.html

How can there not be a Yes or No answer to it being a statement that
can be proven true?

I didn't say anything like that in this post.

You said "The whole notion of undecidabioiut is anchord in ignoring the
fat that some expressions of language are simply not truth bearers"

As explain, "undeciability" of a system is based on the question of if
there are some expressions in it that can not be determined if they are
a provable theorem in the system (the only kind of theorems that exist)
or not.

The question "Is X a Theorem of L" can not be a statement without a
truth value, as X either CAN be proven or it can not (we might not KNOW
if it is provable, which is what leads to undecidability, but in fact,
it either is or it isn;t).

IF x is a statement without a truth value, the answer to the quesiton
about x will just be false, as no consistant system can prove a non-truthbearer.

Thus, you DID says something like that, but are apparently too stupid to undertstand that you did.

My only conclusion from your remarks is that you must be assuming that
all logic system are inconsistant, so the question of the provability of
some statements doesn't have a truth value because the statement might
be both provable and not provable at the same time.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 10/17/24 10:53 AM, olcott wrote:

On 10/16/2024 7:47 PM, Richard Damon wrote:

On 10/16/24 6:34 PM, olcott wrote:

On 10/16/2024 11:37 AM, Mikko wrote:

On 2024-10-16 14:27:09 +0000, olcott said:

The whole notion of undecidability is anchored in ignoring the fact
that
some expressions of language are simply not truth bearers.

A formal theory is undecidable if there is no Turing machine that
determines whether a formula of that theory is a theorem of that
theory or not. Whether an expression is a truth bearer is not
relevant. Either there is a valid proof of that formula or there
is not. No third possibility.

*I still said that wrong*
(1) There is a finite set of expressions of language
that are stipulated to be true (STBT) in theory L.

(2) There is a finite set of true preserving operations
(TPO) that can be applied to this finite set in theory L.

When formula x cannot be derived by applying the TPO
of L to STBT of L then x is not a theorem of L.

A theorem is a statement that can be demonstrated to be
true by accepted mathematical operations and arguments.
https://mathworld.wolfram.com/Theorem.html

How can there not be a Yes or No answer to it being a statement that
can be proven true?

I didn't say anything like that in the words shown
immediately above. Maybe the reason that you get
so confused is that you never respond to the exact
words that I just said right now.

Then what are you referring to if other than your initial claim?

What statement are you saying simply not being a truth bearer makes the definition of undecidability incorrect?

I reply to your WHOLE message, as context matters.

Your statements (1) and (2) are just clearification that you understand
the problem, but then how can the fact that we can show that there can
be some statements we can not know if they are provable or not, not be a
valid proof of the system being undecidable?

Note, that the fact that we haven't been able to demonstrate that a
proof exists, is not in itself a proof that no such proof exists. If the
Turing Machine existed, then all True Statements would be provable, all
False statements refutable, and all non-truthbears detectable for being
that.

The fact that it can be shown that there can exist statements in a
language L, that are TRUE, but not provable in that language, show that
there exist language Ls that are undecidable.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 10/18/24 7:43 PM, olcott wrote:

On 10/18/2024 6:17 PM, Richard Damon wrote:

On 10/17/24 10:53 AM, olcott wrote:

On 10/16/2024 7:47 PM, Richard Damon wrote:

On 10/16/24 6:34 PM, olcott wrote:

On 10/16/2024 11:37 AM, Mikko wrote:

On 2024-10-16 14:27:09 +0000, olcott said:

The whole notion of undecidability is anchored in ignoring the
fact that
some expressions of language are simply not truth bearers.

A formal theory is undecidable if there is no Turing machine that
determines whether a formula of that theory is a theorem of that
theory or not. Whether an expression is a truth bearer is not
relevant. Either there is a valid proof of that formula or there
is not. No third possibility.

*I still said that wrong*
(1) There is a finite set of expressions of language
that are stipulated to be true (STBT) in theory L.

(2) There is a finite set of true preserving operations
(TPO) that can be applied to this finite set in theory L.

When formula x cannot be derived by applying the TPO
of L to STBT of L then x is not a theorem of L.

A theorem is a statement that can be demonstrated to be
true by accepted mathematical operations and arguments.
https://mathworld.wolfram.com/Theorem.html

How can there not be a Yes or No answer to it being a statement that
can be proven true?

I didn't say anything like that in the words shown
immediately above. Maybe the reason that you get
so confused is that you never respond to the exact
words that I just said right now.

Then what are you referring to if other than your initial claim?

What statement are you saying simply not being a truth bearer makes
the definition of undecidability incorrect?

I reply to your WHOLE message, as context matters.

Your statements (1) and (2) are just clearification that you
understand the problem, but then how can the fact that we can show
that there can be some statements we can not know if they are provable
or not, not be a valid proof of the system being undecidable?

Note, that the fact that we haven't been able to demonstrate that a
proof exists, is not in itself a proof that no such proof exists.

When one thinks of proofs as finite string transformation
rules then one finite string can be transformed into another
according to the transformation rules or not.

Right, and it has been proven that for a sufficiently powerful system,
it is possible to create a statement, that is true in the system, but no
finite sequence of transformations makes a proof of the statement, but
it is only established by an infinite string of transformations, which
is enough to create a truth, but not a proof.

This has been explained to you many times, and the fact you still don't
get it just shows your stupidity and ignorance of the subject.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 2024-10-16 17:55:55 +0000, olcott said:

On 10/16/2024 12:31 PM, olcott wrote:

On 10/16/2024 11:37 AM, Mikko wrote:

On 2024-10-16 14:27:09 +0000, olcott said:

The whole notion of undecidability is anchored in ignoring the fact that >>>> some expressions of language are simply not truth bearers.

A formal theory is undecidable if there is no Turing machine that
determines whether a formula of that theory is a theorem of that
theory or not.

When the question: Is finite string X a theory of L?
has no correct answer from YES and NO, then the question
is rejected as not a truth bearer.

I did not say that exactly correctly.

I noticed. You should have said "Is finite string X a theorem of T?".
The letter L usually denotes the language of the theory of T, i.e. the
set of all syntactically correct formulas. The letter T is used for a
theory including its language, logic, and postulates.

--
Mikko

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 2024-10-16 17:31:47 +0000, olcott said:

On 10/16/2024 11:37 AM, Mikko wrote:

On 2024-10-16 14:27:09 +0000, olcott said:

The whole notion of undecidability is anchored in ignoring the fact that >>> some expressions of language are simply not truth bearers.

A formal theory is undecidable if there is no Turing machine that
determines whether a formula of that theory is a theorem of that
theory or not.

When the question: Is finite string X a theory of L?
has no correct answer from YES and NO, then the question
is rejected as not a truth bearer.

As I already said:

Whether an expression is a truth bearer is not
relevant. Either there is a valid proof of that formula or there
is not. No third possibility.

--
Mikko

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 2024-10-16 22:34:51 +0000, olcott said:

On 10/16/2024 11:37 AM, Mikko wrote:

On 2024-10-16 14:27:09 +0000, olcott said:

The whole notion of undecidability is anchored in ignoring the fact that >>> some expressions of language are simply not truth bearers.

A formal theory is undecidable if there is no Turing machine that
determines whether a formula of that theory is a theorem of that
theory or not. Whether an expression is a truth bearer is not
relevant. Either there is a valid proof of that formula or there
is not. No third possibility.

*I still said that wrong*
(1) There is a finite set of expressions of language
that are stipulated to be true (STBT) in theory L.

(2) There is a finite set of true preserving operations
(TPO) that can be applied to this finite set in theory L.

When formula x cannot be derived by applying the TPO
of L to STBT of L then x is not a theorem of L.

A theorem is a statement that can be demonstrated to be
true by accepted mathematical operations and arguments. https://mathworld.wolfram.com/Theorem.html

Better. The word "theory" starts with T so instead of L the
letter T should be used as the name of a theory.

In a formal theory no set of expressions are stipuated to be
true. Instead they are defined to be the postulates of the
theory.

When discussing a formal theory the theorems are not assumed to
be true. They can be true in one interpretation and false in
another one.

Whether the inference rules of a theory are truth preserving is
a matter of separate investigation.

--
Mikko

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 2024-10-18 23:43:15 +0000, olcott said:

On 10/18/2024 6:17 PM, Richard Damon wrote:

On 10/17/24 10:53 AM, olcott wrote:

On 10/16/2024 7:47 PM, Richard Damon wrote:

On 10/16/24 6:34 PM, olcott wrote:

On 10/16/2024 11:37 AM, Mikko wrote:

On 2024-10-16 14:27:09 +0000, olcott said:

The whole notion of undecidability is anchored in ignoring the fact that
some expressions of language are simply not truth bearers.

A formal theory is undecidable if there is no Turing machine that
determines whether a formula of that theory is a theorem of that
theory or not. Whether an expression is a truth bearer is not
relevant. Either there is a valid proof of that formula or there
is not. No third possibility.

*I still said that wrong*
(1) There is a finite set of expressions of language
that are stipulated to be true (STBT) in theory L.

(2) There is a finite set of true preserving operations
(TPO) that can be applied to this finite set in theory L.

When formula x cannot be derived by applying the TPO
of L to STBT of L then x is not a theorem of L.

A theorem is a statement that can be demonstrated to be
true by accepted mathematical operations and arguments.
https://mathworld.wolfram.com/Theorem.html

How can there not be a Yes or No answer to it being a statement that
can be proven true?

I didn't say anything like that in the words shown
immediately above. Maybe the reason that you get
so confused is that you never respond to the exact
words that I just said right now.

Then what are you referring to if other than your initial claim?

What statement are you saying simply not being a truth bearer makes the
definition of undecidability incorrect?

I reply to your WHOLE message, as context matters.

Your statements (1) and (2) are just clearification that you understand
the problem, but then how can the fact that we can show that there can
be some statements we can not know if they are provable or not, not be
a valid proof of the system being undecidable?

Note, that the fact that we haven't been able to demonstrate that a
proof exists, is not in itself a proof that no such proof exists.

When one thinks of proofs as finite string transformation
rules then one finite string can be transformed into another
according to the transformation rules or not.

Typical logic systems have transformation rules that transform two
strings to one. For example, you cannot infer A ∧ B from A nor from
B but if you have both A and B then you can infer A ∧ B.

--
Mikko

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)