Re: Within Proof Theoretic Semantics Gödel's G h as no meaning in PA

On 6/29/2026 1:14 AM, Mikko wrote:
> On 29/06/2026 05:52, olcott wrote:
>> On 6/28/2026 3:39 AM, Mikko wrote:
>>> On 27/06/2026 17:50, polcott wrote:
>>>> On 6/27/2026 1:53 AM, Tristan Wibberley wrote:
>>>>> On 20/06/2026 18:32, olcott wrote:
>>>>>
>>>>>> A proof theoretic expression is known to be true when
>>>>>> it is fully grounded in its atomic base. Only two
>>>>>> PTS semantics researchers deal with true Dag Prawitz
>>>>>> is the one that began this. PTS previously only dealt
>>>>>> with semantic meaning and never got around to true(L,x).
>>>>>
>>>>> That's surprising, disregard for axioms?
>>>>
>>>> If there is no sequence of inference steps in Q from
>>>> ~∃x x=S(x) to the axioms of Q then ~∃x x=S(x) is
>>>> ungrounded in the PTS atomic base of Q.
>>>>
>>>> This does not mean undecidable or incomplete
>>>> it means that ~∃x x=S(x) is out-of-scope for Q.
>>>
>>> It comes close. If ∃x x=S(x) is likewise "ungrounded" but in the
>>> language of Q then ~∃x x=S(x) and ∃x x=S(x) are both undecidable
>>> and Q is incomplete, bcause that is what the words mean.
>>
>> Q also can't bake a birthday cake, this does not make
>> Q in any way "incomplete" relative to what it was
>> defined to do. Incomplete only counts relative to
>> its intended purpose. A car without an engine is
>> incomplete relative to a mode of transportation.
> 
> Irrelevant. The definition of completeness 

It a misnomer and does not literally mean (as it implies)
that something is missing that could be added to make
it complete. The way that terms-of-the-art are formed
is very misleading and as much as intentionally deceptive.

"undecidable" often means input is semantically
incoherent. Whenever the input is semantically
incoherent it is more accurately called incoherent
rather than undecidable.

Food is called inedible for many reasons such as
spoilage. We could also call a railroad tie inedible
ignoring the type mismatch error. A railroad tie
is inedible for the same kind of reason that some
expressions are undecidable.

> inly refers to the
> sentences in the language of the theory. To bake a cake is an
> action, not a sentence, so irrelevant.
> 

Q intentionally defined to not be able to prove ~∃x x=S(x).
A car that had its engine removed to make it undrivable
is not incomplete relative to its intended purpose.

-- 
Copyright 2026 Olcott

My 28 year goal has been to make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
The complete structure of this system is now defined.

The entire body of knowledge expressed in language is
comprised of two types of relations between finite strings:
(a) *Axioms* Expressions of language that are stipulated to be true.

My system bridges the analytic/synthetic distinction by
expressly encoding all empirical "atomic facts" in a formal
language such as CycL of the Cyc project.

(b) *Inference Rules* Expressions of language that are semantically
entailed syntactically from (a) and/or (b).