On 20/01/2026 05:29, Richard Damon wrote:
On 1/19/26 9:39 PM, olcott wrote:
On 1/17/2026 3:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have quietly
relied on a fundamental conflation: the idea that
rCLtrue in arithmeticrCY meant rCLtrue in the standard model of rao.rCY >>>> But PA itself has no truth predicate, no internal semantics,
and no mechanism for assigning truth values.
Don't we assume it to be (implicitly) a schematic system, where the
axioms define the deduction rules?
...
^^^reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
reCx ree PA (~TruthBearer(PA, x) rei (~True(PA, x) reo (~False(PA, x))
PA doesn't have a truth predicate, because it CAN'T.
a unary truth predicate
but perhaps an operation "IsElementaryTheorem_p(system, objects...)"
for each predicate 'p' can be admitted to an extension of PA.
Perhaps importantly, I note that PA doesn't relate = with rea but both
appear in the axioms, naively avoiding the problem of "what do you mean
by 'negation'?" but leaving a problem of "what do you mean by 'contradiction'?"
What resolutions do you perceive regarding that?
On 20/01/2026 21:39, olcott wrote:
a metarCalevel system is required to stand above PA and
filter expressions before PA ever evaluates them. The
metarCasystem performs the structural work PA cannot do:
it detects cycles, blocks diagonalization, rejects
nonrCatruthrCabearers, and prevents PA from entering
infinite loops.
Then the truth predicate is a restricted truth predicate.
I think Tarski's findings don't directly apply to what Olcott is doing
as they are stated for systems with negation (of statements) carrying
the semantics of contradiction. PA doesn't seem to have that in its
axioms; then there's the matter of universal generality: is that a
predicate, connective, or an operation? Some of those take the truth "predicate" away from Elementary Theorems, some of them don't but
negation must lose its naivety as it becomes an operation.
How does one characterise PA among:
- syntactical system
- schematic system
- abstract formal system
- concrete formal system
etc...
understanding that there is some overlap.
On 1/28/2026 10:21 AM, Tristan Wibberley wrote:
On 20/01/2026 05:29, Richard Damon wrote:
On 1/19/26 9:39 PM, olcott wrote:
On 1/17/2026 3:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have quietly
relied on a fundamental conflation: the idea that
rCLtrue in arithmeticrCY meant rCLtrue in the standard model of rao.rCY >>>>> But PA itself has no truth predicate, no internal semantics,
and no mechanism for assigning truth values.
Don't we assume it to be (implicitly) a schematic system, where the
axioms define the deduction rules?
That is the conflation error of G||del's incompleteness.
It seems to be saying what you said to casual observers.
...
-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a ^^^reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
reCx ree PA (~TruthBearer(PA, x) rei (~True(PA, x) reo (~False(PA, x)) >>>>
PA doesn't have a truth predicate, because it CAN'T.
-a-a-a-a-a-a-a-a a unary truth predicate
but perhaps an operation "IsElementaryTheorem_p(system, objects...)"
for each predicate 'p' can be admitted to an extension of PA.
You just understand these things more deeply than
anyone else here.
When we refer to Haskell Curry's notion of elementary
theorems that are true then anything derived from
them is a theorem that is also true. That is the
key foundation of proof theoretic semantics:
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
*Please keep comp.theory because I am showing*
Perhaps importantly, I note that PA doesn't relate = with rea but both
appear in the axioms, naively avoiding the problem of "what do you mean
by 'negation'?" but leaving a problem of "what do you mean by
'contradiction'?"
What resolutions do you perceive regarding that?
On 1/28/2026 10:34 AM, Tristan Wibberley wrote:
On 20/01/2026 21:39, olcott wrote:
a metarCalevel system is required to stand above PA and
filter expressions before PA ever evaluates them. The
metarCasystem performs the structural work PA cannot do:
it detects cycles, blocks diagonalization, rejects
nonrCatruthrCabearers, and prevents PA from entering
infinite loops.
Then the truth predicate is a restricted truth predicate.
It is only restricted to its domain of knowledge
expressed in language. This excludes semantic nonsense
like pathological self-reference, type mismatch errors
and unknowns such as the truth value of the Goldbach
conjecture.
I think Tarski's findings don't directly apply to what Olcott is doing
as they are stated for systems with negation (of statements) carrying
the semantics of contradiction. PA doesn't seem to have that in its
axioms; then there's the matter of universal generality: is that a
predicate, connective, or an operation? Some of those take the truth
"predicate" away from Elementary Theorems, some of them don't but
negation must lose its naivety as it becomes an operation.
How does one characterise PA among:
- syntactical system
- schematic system
- abstract formal system
- concrete formal system
etc...
understanding that there is some overlap.
What I am proposing is that PA is entirely syntactic
and when we add a truth predicate anchored entirely
in the axioms of PA that this predicate itself is
at a meta-level. When we explicitly add this predicate
then we can really see what is actually true in PA
itself and this has always been provable in PA.
Incompleteness only arose because what was true
outside of PA could not be proved inside PA. This
was a mere conflation error all along.
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