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

On 27/06/2026 23:04, olcott wrote:
> On 6/27/2026 2:54 PM, dbush wrote:
>> On 6/27/2026 3:40 PM, 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 !!!
>>
>> And because PTS claims the semantically valid sentence in Q "no number 
>> is equal to its successor" is not semantically valid, it must be 
>> discarded as useless.
> 
> No you are just not bothering to pay 100% totally
> complete attention to every single word.
> 
> Wittgenstein
> 'True in Russell's system' means, as was said:
> proved in Russell's system; and 'false in Russell's
> system' means: the opposite has been proved
> in Russell's system

We reject Wittgenstain's statement as a violation of the current
rules of the language game.

> Proof Theoretic Semantics has almost gotten there.
> For the most part they stop at semantically grounded
> and never quite get all the way to True.

We have almost gotten to the rejection of Proof Theoretic Semantics.

Russell's solution to the problems in his system was to give
up and focus to politics.

-- 
Mikko