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

On 7/1/2026 2:01 AM, Mikko wrote:
> On 30/06/2026 16:43, olcott wrote:
>> On 6/30/2026 2:48 AM, Mikko wrote:
>>> 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.
>>
>> It tells us exactly one thing about the real world.
> 
> Only to those who already know that there are things called "cat"
> in the real world and know what kind of things the words "cat" and
> "animal" refer to but don't already know that every thing that the
> word "cat" refers to is a thing that the word "animal" refers to.
> 

It stipulates some semantic meaning to a pair of otherwise
totally meaningless finite strings. With 200 petabytes of
these "atomic facts" one knows quite a bit about the world.


-- 
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).