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

On 6/27/2026 5:24 PM, olcott wrote:
> On 6/27/2026 3:59 PM, dbush wrote:
>> On 6/27/2026 4:52 PM, olcott wrote:
>>> On 6/27/2026 3:30 PM, dbush wrote:
>>>> On 6/27/2026 4:27 PM, olcott wrote:
>>>>> On 6/27/2026 3:22 PM, dbush wrote:
>>>>>> On 6/27/2026 4:17 PM, olcott wrote:
>>>>>>> On 6/27/2026 3:11 PM, dbush wrote:
>>>>>>>> On 6/27/2026 4:04 PM, 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
>>>>>>>>>
>>>>>>>>> Proof Theoretic Semantics has almost gotten there.
>>>>>>>>> For the most part they stop at semantically grounded
>>>>>>>>
>>>>>>>> i.e. proven
>>>>>>>>
>>>>>>>>>
>>>>>>>>> and never quite get all the way to True.
>>>>>>>>>
>>>>>>>>> For Wittgenstein's slight extension of the PTS
>>>>>>>>> notion ~∃x x=S(x) is untrue 
>>>>>>>>
>>>>>>>> i.e. unproven
>>>>>>>>
>>>>>>>>> in Q and true in PA.
>>>>>>>>> I have been saying it that way long before I ever
>>>>>>>>> heard of Wittgenstein.
>>>>>>>>
>>>>>>>> So again, you're saying the same thing as everyone else but with 
>>>>>>>> different words.
>>>>>>>
>>>>>>> So everyone says that ~∃x x=S(x) 
>>>>>>
>>>>>> which has the semantic meaning "no number is equal to its 
>>>>>> successor" as per the definition of Q
>>>>>>
>>>>>>> is simply untrue
>>>>>>
>>>>>> i.e. unprovable
>>>>>>
>>>>>>> in Q and does nor derive either undecidability or
>>>>>>> incompleteness?
>>>>>>
>>>>>> It does derive incompleteness, as by definition Q is incomplete 
>>>>>> because "no number is equal to its successor" is unprovable in Q.
>>>>>>
>>>>>>
>>>>>
>>>>> Wittgenstein (1937)
>>>>> '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
>>>>
>>>> So what term would you use to describe a sentence that has an 
>>>> infinite sequence of inference steps between it and the axioms of 
>>>> the system?
>>>>
>>>
>>> untrue and unfalse.
>>
>> What about the more general case, i.e. a term for a statement that has 
>> *any* sequence of inference steps, either finite or infinite, between 
>> it and the axioms of the system?  And what would the negation of such 
>> a statement be called?
>>
>>
> 
> The truth value of the Goldbach conjecture
> may have an infinite number of steps thus
> would be unknowable and not a member of the
> body of knowledge that can be expressed in
> language. Negation has no effect on expressions
> that are neither true no false.
> 
> Every finite string including gibberish has the
> truth value of: {True, False, Neither}.
> 
> Finite strings are a superset of expressions
> of language.
> 

That's not the definition I asked for.

What term would you use for a statement that has *any* sequence of 
inference steps, either finite or infinite, between it and the axioms of 
the system?

What term would you use for the negation of the above statement?