Re: Readings in (some of the) foundations of mathematics --- tree of knowledge

On 6/24/2026 5:00 AM, Mikko wrote:
> On 23/06/2026 17:48, olcott wrote:
>> On 6/23/2026 1:06 AM, Mikko wrote:
>>> On 22/06/2026 15:10, olcott wrote:
>>>> On 6/22/2026 1:49 AM, Mikko wrote:
>>>>> On 22/06/2026 02:02, olcott wrote:
>>>>>> On 6/21/2026 4:08 PM, André G. Isaak wrote:
>>>>>>> On 2026-06-21 14:42, olcott wrote:
>>>>>>>> On 6/21/2026 3:04 PM, Alan Mackenzie wrote:
>>>>>>>>> [ Followup-To: set ]
>>>>>>>>>
>>>>>>>>> In comp.theory olcott <[email protected]> wrote:
>>>>>>>>>> On 6/21/2026 6:26 AM, Alan Mackenzie wrote:
>>>>>>>>>>> In comp.theory olcott <[email protected]> wrote:
>>>>>>>>>>>> I just found the term:
>>>>>>>>>>>> "grounding in a proof theoretic atomic base" yesterday.
>>>>>>>>>
>>>>>>>>>>> You can find any number of terms.  That doesn't mean you're 
>>>>>>>>>>> capable of
>>>>>>>>>>> understanding them.
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>> The above is the key reason why under PTS Gödel 1931 
>>>>>>>>>> incompleteness
>>>>>>>>>> fails.
>>>>>>>>>
>>>>>>>>> I don't believe you.  You have no respect for or understanding 
>>>>>>>>> of the
>>>>>>>>> truth.  If you really want to persuade anybody that PTS somehow 
>>>>>>>>> causes
>>>>>>>>> Gödel's theorem not to hold, then cite an academic expert 
>>>>>>>>> who'll have
>>>>>>>>> some credibility.
>>>>>>>>>
>>>>>>>>>> If they are mere gibberish words to you then you will not 
>>>>>>>>>> understand.
>>>>>>>>>
>>>>>>>>> You don't understand Proof-theoritic Semantics, and you 
>>>>>>>>> certainly don't
>>>>>>>>> understand Gödel's Theorem, neither the theorem itself nor any 
>>>>>>>>> proof of
>>>>>>>>> it.
>>>>>>>>>
>>>>>>>> It is a verified fact that Gödel's G is ungrounded
>>>>>>>> in the atomic base of PA. That you do not understand
>>>>>>>> what: "grounded in the atomic base" means is less
>>>>>>>> than no rebuttal at all.
>>>>>>>
>>>>>>> "grounded in the atomic base of PA" is an expression used only by 
>>>>>>> you, and it is one which you have never explicitly defined, so 
>>>>>>> the fault here certainly doesn't lie with Alan. It's certainly 
>>>>>>> not a 'verified fact' when you haven't even adequately explained 
>>>>>>> what it is that you mean.
>>>>>
>>>>>> All of knowledge expressed in language is structured as a tree of 
>>>>>> semantic relations specified syntactically between finite strings.
>>>>>
>>>>> What makes you believe semantic relations that can be structured as
>>>>> a tree are sufficient to contain all knowledge that is exressed in
>>>>> some language?
>>>>
>>>> The CycL language and the Cyc Project.
>>>
>>> They use a tree structure for concepts. But why would one try to
>>> put knowledge in a tree structure?
>>
>> It must at least be a directed acyclic graph or
>> the proof gets stuck in an infinite loop and never
>> completes.
> 
> How can any ordering of knowledge prevent getting stuck in a loop
> when looking for a proof?
> 

By looking upward in a type hierarchy.

-- 
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).