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

On 6/27/2026 6:18 PM, olcott wrote:
> On 6/27/2026 5:15 PM, dbush wrote:
>> On 6/27/2026 6:11 PM, olcott wrote:
>>> 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.
>>
>> Why are you so reluctant to provide a simple term?
>>
> 
> OFF-TOPIC <is> THE TERM.

So we've established that "off-topic" means "a statement that has *any* 
sequence of inference steps, either finite or infinite, between it and 
the axioms of the system".

So going back to Wittgenstein, using his terminology and the term you 
provided:

A system is incomplete if it contains a statement that is off-topic but 
not true.

> 
>> It seems you're attempting to engage in Newspeak.
>>
>> https://en.wikipedia.org/wiki/Newspeak
>>
>>>
>>>> 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.
>>>
>>>
>>>
>>
> 
>