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

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.