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

On 29/06/2026 16:38, olcott wrote:
> On 6/29/2026 1:23 AM, Mikko wrote:
>> 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
> 
> You are not paying close enough attention. I did not say logical
> tautology. I said semantic tautology. That cats are defined to
> be animals is a semantic tautology. That cats are defined to be
> cats is a logical tautology. Here is a definition of that fits
> my definition of semantic tautology.
> 
> tautology, in logic, a statement so framed that
> it cannot be denied without inconsistency. Thus,
> “All humans are mammals” is held to assert with
> regard to anything whatsoever that either it is
> not a human or it is a mammal.
> https://www.britannica.com/topic/tautology
> 
> https://en.wikipedia.org/wiki/Tautology_(logic)
>> 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.
> 
> One can know that "cats are animals" when this is
> stipulated as an axiom.

That axiom only realtes the words "cat" and "animal". It does not tell
anything about the real world.

-- 
Mikko