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

On 29/06/2026 06:12, olcott wrote:
> On 6/28/2026 4:31 AM, Mikko wrote:
>> On 27/06/2026 22:40, olcott wrote:
>>> On 6/27/2026 2:23 PM, dbush wrote:
>>>> On 6/27/2026 3:16 PM, olcott wrote:
>>>>> On 6/27/2026 2:04 PM, dbush wrote:
>>>>>> On 6/27/2026 3:01 PM, olcott wrote:
>>>>>>> On 6/27/2026 1:39 PM, dbush wrote:
>>>>>>>> On 6/27/2026 2:38 PM, olcott wrote:
>>>>>>>>> On 6/27/2026 1:29 PM, dbush wrote:
>>>>>>>>>> On 6/27/2026 2:27 PM, olcott wrote:
>>>>>>>>>>> On 6/27/2026 1:01 PM, dbush wrote:
>>>>>>>>>>>> On 6/27/2026 11:43 AM, polcott wrote:
>>>>>>>>>>>>> On 6/27/2026 2:35 AM, Mikko wrote:
>>>>>>>>>>>>>> On 26/06/2026 16:10, olcott wrote:
>>>>>>>>>>>>>>> On 6/26/2026 1:39 AM, Mikko wrote:
>>>>>>>>>>>>>>>> On 25/06/2026 19:14, olcott wrote:
>>>>>>>>>>>>>>>>> On 6/25/2026 2:21 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>> On 24/06/2026 23:26, olcott wrote:
>>>>>>>>>>>>>>>>>>> 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.
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> If you mean not looking elsewhere that may indeed 
>>>>>>>>>>>>>>>>>> prevent loops.
>>>>>>>>>>>>>>>>>> In most cases that also prevents finding the proof.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> Truth Conditional Semantics (TCS) <is> incoherent
>>>>>>>>>>>>>>>>> compared to Proof Theoretic Semantics (PTS). Essentially
>>>>>>>>>>>>>>>>> PTS just coherently connects the semantic meanings
>>>>>>>>>>>>>>>>> expressed in language together into one coherent body
>>>>>>>>>>>>>>>>> of general knowledge. It does this without undecidability
>>>>>>>>>>>>>>>>> or mathematical incompleteness.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Looking for a proof does not need any semantics so it is 
>>>>>>>>>>>>>>>> not obvious
>>>>>>>>>>>>>>>> how switching to another semantics could improve it.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> In proof theoretic semantics an expression only gains
>>>>>>>>>>>>>>> semantic meaning by finding a proof.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> It should be obvious that finding a proof does not happen 
>>>>>>>>>>>>>> before
>>>>>>>>>>>>>> looking for a proof.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> If there is no sequence of inference steps in Q from
>>>>>>>>>>>>> ~∃x x=S(x) to the axioms of Q 
>>>>>>>>>>>>
>>>>>>>>>>>> There are, but that sequence is infinite
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> If there is no FINITE 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.
>>>>>>>>>>
>>>>>>>>>> i.e., ~∃x x=S(x) is unprovable is Q, as is commonly known.
>>>>>>>>>>
>>>>>>>>> Is it commonly known that ~∃x x=S(x) 
>>>>>>>>
>>>>>>>> Which has the semantic meaning "no number is equal to its 
>>>>>>>> successor" as per the definition of Q.
>>>>>>>>
>>>>>>>
>>>>>>> Since there are no steps in Q that affirm ~∃x x=S(x)
>>>>>>> in Q it is an open question in Q and not a confirmed
>>>>>>> statement in Q. 
>>>>>>
>>>>>> In other words, unproven as is commonly known.
>>>>>>
>>>>> Yet never gets to undecidable or in any sense of incomplete.
>>>>>
>>>>
>>>> False, as by definition, Q is incomplete because ~∃x x=S(x) is 
>>>> unprovable / out-of-scope / not semantically grounded in Q.
>>>
>>> Proof theoretic semantics DOES NOT DO IT THAT WAY !!!
>>
>> Irrelevant. The statement that both ∃x x=S(x) and ~∃x x=S(x) are
>> sentences of Q but neither is a rheorem or Q does not depend on
>> any semantics.
> 
> The entire body of knowledge expressed in language
> can be represented as a semantic tautology in an
> acyclic directed graph. That knowledge is a DAG was
> my very thought on this subject more than 30 years ago.
> This single idea gets rid of all undecidability
> within the entire body of knowledge.

No, it cannot. The usual meaning of knoledge excludes tautologies
because the determination of their truth does not need any knowledge
beyond a method to determine whether a string is a tautology in the
relevant language.

When the word "knowledge" is used it usually means knowing about the
real world something that cannot be determined without observation.

-- 
Mikko