From Newsgroup: comp.lang.prolog

On 1/17/2026 3:46 AM, Mikko wrote:

On 15/01/2026 22:37, olcott wrote:

On 1/15/2026 4:02 AM, Mikko wrote:

On 15/01/2026 07:30, olcott wrote:

On 1/14/2026 9:44 PM, Richard Damon wrote:

On 1/14/26 4:36 PM, olcott wrote:

Interpreting incompleteness as a gap between mathematical truth
and proof depends on truth-conditional semantics; once this is
replaced by proof-theoretic semantics a framework not yet
sufficiently developed at the time of Gödel’s proof the notion of >>>>>> such a gap becomes unfounded.

But that isn't what Incompleteness is about, so you are just
showing your ignorance of the meaning of words.

You can't just "change" the meaning of truth in a system.

Yet that is what happens when you replace the foundational basis
from truth-conditional semantics to proof-theoretic semantics.

Gödel constructed a sentence that is correct by the rules of first
order Peano arithmetic

within truth conditional semantics and non-well-founded
in proof theoretic semantics. All of PA can be fully
expressed in proof theoretic semantics. Even G can be
expressed, yet rejected as semantically non-well-founded.

Gödel's sentence is a sentence of Peano arithmetic so its primary
meaning is its arithmetic meaning. Peano's postulates fail to
capture all of its arithmetic meaning but it is possible to add
other postulates without introducing inconsistencies to make
Gödel's sentence provable in a stronger theory of natural numbers.

Plain PA has no internal notion of truth; any truth
talk is meta‑theoretic. To work proof‑theoretically,
we must add a rule‑anchored truth predicate in the
sense of Curry, governed by elementary theorems of T.
If we then impose an object‑level well‑foundedness
constraint on truth—rejecting any cyclic truth
dependencies—Gödel’s fixed‑point sentence G becomes
syntactically non‑well‑founded and is blocked before
any truth value is assigned. In such a system,
Gödel’s G is not a deep undecidable truth, but
an ill‑formed attempt at self‑reference.
--
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.<br><br>

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

From Newsgroup: comp.lang.prolog

On 1/17/26 10:54 AM, olcott wrote:

On 1/17/2026 3:46 AM, Mikko wrote:

On 15/01/2026 22:37, olcott wrote:

On 1/15/2026 4:02 AM, Mikko wrote:

On 15/01/2026 07:30, olcott wrote:

On 1/14/2026 9:44 PM, Richard Damon wrote:

On 1/14/26 4:36 PM, olcott wrote:

Interpreting incompleteness as a gap between mathematical truth >>>>>>> and proof depends on truth-conditional semantics; once this is
replaced by proof-theoretic semantics a framework not yet
sufficiently developed at the time of Gödel’s proof the notion of >>>>>>> such a gap becomes unfounded.

But that isn't what Incompleteness is about, so you are just
showing your ignorance of the meaning of words.

You can't just "change" the meaning of truth in a system.

Yet that is what happens when you replace the foundational basis
from truth-conditional semantics to proof-theoretic semantics.

Gödel constructed a sentence that is correct by the rules of first
order Peano arithmetic

within truth conditional semantics and non-well-founded
in proof theoretic semantics. All of PA can be fully
expressed in proof theoretic semantics. Even G can be
expressed, yet rejected as semantically non-well-founded.

Gödel's sentence is a sentence of Peano arithmetic so its primary
meaning is its arithmetic meaning. Peano's postulates fail to
capture all of its arithmetic meaning but it is possible to add
other postulates without introducing inconsistencies to make
Gödel's sentence provable in a stronger theory of natural numbers.

Plain PA has no internal notion of truth; any truth
talk is meta‑theoretic. To work proof‑theoretically,
we must add a rule‑anchored truth predicate in the
sense of Curry, governed by elementary theorems of T.
If we then impose an object‑level well‑foundedness
constraint on truth—rejecting any cyclic truth
dependencies—Gödel’s fixed‑point sentence G becomes
syntactically non‑well‑founded and is blocked before
any truth value is assigned. In such a system,
Gödel’s G is not a deep undecidable truth, but
an ill‑formed attempt at self‑reference.

Sure PA has an internal notion of truth.

It knows that 2 + 2 is 4.

Just like it knows that for Relation that Godel created in it any given
number will either satisfy it or not (but none will), and thus that
either there is or there is not a number that satisfies it.

Therefore, it is a FACT in PA, that Godel's statement G is a truth bearer.

It is also a FACT, determinable in the meta system that when we test
every natural number with that relation, none will satisfy it, and thus
it is s fact in PA that none will satisfy it, even if we can't prove
that statement in PA


Note, you seem to be confused about the logic system, as there is NO
"cyclc" in the derivation of the relationship. So, I guess in your
system, statements like 1 + 1 = 2 are just not-well-founded too, as they
use nothing different than that relationship.

Note also, if you TRY to add your rule-anchored truth predicate, your
system just becomes inconsistant.

Also, logic systems don't "Block" statements, such a concept just means
you admit you system is inconsistant.

It seems that all you are able to do it prove you are just stupidly
ignorant of what you talk about, and so don't care about facts that you
ignore the errors pointed out in your statements.

--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: comp.lang.prolog

On 17/01/2026 17:54, olcott wrote:

On 1/17/2026 3:46 AM, Mikko wrote:

On 15/01/2026 22:37, olcott wrote:

On 1/15/2026 4:02 AM, Mikko wrote:

On 15/01/2026 07:30, olcott wrote:

On 1/14/2026 9:44 PM, Richard Damon wrote:

On 1/14/26 4:36 PM, olcott wrote:

Interpreting incompleteness as a gap between mathematical truth >>>>>>> and proof depends on truth-conditional semantics; once this is
replaced by proof-theoretic semantics a framework not yet
sufficiently developed at the time of Gödel’s proof the notion of >>>>>>> such a gap becomes unfounded.

But that isn't what Incompleteness is about, so you are just
showing your ignorance of the meaning of words.

You can't just "change" the meaning of truth in a system.

Yet that is what happens when you replace the foundational basis
from truth-conditional semantics to proof-theoretic semantics.

Gödel constructed a sentence that is correct by the rules of first
order Peano arithmetic

within truth conditional semantics and non-well-founded
in proof theoretic semantics. All of PA can be fully
expressed in proof theoretic semantics. Even G can be
expressed, yet rejected as semantically non-well-founded.

Gödel's sentence is a sentence of Peano arithmetic so its primary
meaning is its arithmetic meaning. Peano's postulates fail to
capture all of its arithmetic meaning but it is possible to add
other postulates without introducing inconsistencies to make
Gödel's sentence provable in a stronger theory of natural numbers.

Plain PA has no internal notion of truth; any truth
talk is meta‑theoretic.

Of course. Truth is a meta-theoretic concept. The corresponding concept
about an uninterpreted theory is theorem.

The statement that there is a sentence that is neither provable nor the negation of a provable sentence does not refer to truth.
--
Mikko
--- Synchronet 3.21b-Linux NewsLink 1.2