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

On 6/27/2026 6:29 PM, olcott wrote:
> On 6/27/2026 5:21 PM, dbush wrote:
>> 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".
>>
> 
> The ones that have infinite steps out outside the
> body of knowledge and off topic for that reason.
> 

So you're claiming the sentence "no number is equal to its successor" in 
Q is "off topic".

Rejected, as it is a semantically valid sentence in the language of Q.