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

On 6/27/2026 4:50 PM, dbush wrote:
> 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?
> 

"true on the basis of meaning expressed in language"
reliably computable for the entire body of GENERAL knowledge.
Is the limit of the topic of all my posts.

Infinite inference steps are off topic.

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

Does not have any proof finite or infinite?
That would be untrue and possibly nonsense.



-- 
Copyright 2026 Olcott

My 28 year goal has been to make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
The complete structure of this system is now defined.

The entire body of knowledge expressed in language is
comprised of two types of relations between finite strings:
(a) *Axioms* Expressions of language that are stipulated to be true.

My system bridges the analytic/synthetic distinction by
expressly encoding all empirical "atomic facts" in a formal
language such as CycL of the Cyc project.

(b) *Inference Rules* Expressions of language that are semantically
entailed syntactically from (a) and/or (b).