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

On 01/07/2026 18:13, olcott wrote:
> On 7/1/2026 2:13 AM, Mikko wrote:
>> On 30/06/2026 17:22, olcott wrote:
>>> On 6/30/2026 3:43 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.
>>>>
>>>> What I said applies to logical tautologies, too. That cats are
>>>> defined to be animals only tells somthing (but not much) about
>>>> the meanings of the words but nothing about the real world.
>>>
>>> When a complete set of general "atomic facts" of the
>>> actual world is encoded as axioms along with all of
>>> the semantic entailment relations between these facts
>>> then every element of general knowledge that can be
>>> expressed in language is known by this system.
>>
>> In order to achieve that the atomic facts must be non-tautologies.

> That "cats" <are> "animals" is a semantic tautology.

That depends on the semantic system. Often the meaning of the word
"cat" indeed involves that what is called a "cat" is also called
an "animal" but there are other posiibilities. Conseqently, the
sentence "cats are animals" can be a semantic tautology that tells
nothing about the real world, or it can be a consequence of the
definitions of "cat" and "animal" and terms used in these definitions
that tells nothing about the real world, or it can be a statement
about the real world that cannot be inferred from definitions alne.

> "atomic facts" that correspond to things in the world
> only have stipulation as their basis in truth within
> the formal system.
One usually says "assumption" instead of "stipulation". The latter
usually means a rfusal to proceed before the stipulation is accepted
or an alternative agreed.

-- 
Mikko