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

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?

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.
> 
> 
>