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

On 6/27/2026 7:27 PM, olcott wrote:
> On 6/27/2026 5:53 PM, dbush wrote:
>> On 6/27/2026 6:44 PM, olcott wrote:
>>> 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.
>>
>> And since x = "no number is equal to its successor" is semantically 
>> valid as per the definition of Q, PTS must be discarded as useless.
>>
> 
> So you also have no idea what Stipulative definition is.
> 
> A stipulative definition is a type of definition in which
> a new or currently existing term is given a new specific meaning
> 
> https://en.wikipedia.org/wiki/Stipulative_definition

Not allowed, as "semantically valid" is already defined, and "no number 
is equal to its successor" meets that definition.