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

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.

-- 
Mikko