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

On 6/27/2026 5:33 PM, dbush wrote:
> 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.

If there is no finite sequence of inference steps
between x and the axioms of Q then PTS stipulates
that x is not semantically valid in Q.

-- 
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).