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

On 27/06/2026 21:38, 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) is
> semantic nonsense in Q? All of logic took
> a psychotic break from reality when they
> took semantics out of logic and put it in
> a separate model.

All of mathematics and logic is disconnected from reality. Proof 
theoretic semantics is just a way to emphasize the disconnection.
The connection is made when one wants to apply logic or mathematics
to description of the real world or to solving real world problems.

-- 
Mikko