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

On 7/2/2026 10:16 AM, Ross Finlayson wrote:
> On 07/02/2026 07:45 AM, olcott wrote:
>> On 7/2/2026 1:44 AM, Mikko wrote:
>>> On 01/07/2026 18:13, olcott wrote:
>>>> On 7/1/2026 2:13 AM, Mikko wrote:
>>>>> On 30/06/2026 17:22, olcott wrote:
>>>>>> On 6/30/2026 3:43 AM, Mikko wrote:
>>>>>>> On 29/06/2026 16:38, olcott wrote:
>>>>>>>> On 6/29/2026 1:23 AM, Mikko wrote:
>>>>>>>>> On 29/06/2026 06:12, olcott wrote:
>>>>>>>>>> On 6/28/2026 4:31 AM, Mikko wrote:
>>>>>>>>>>> On 27/06/2026 22:40, 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 !!!
>>>>>>>>>>>
>>>>>>>>>>> Irrelevant. The statement that both ∃x x=S(x) and ~∃x x=S(x) are
>>>>>>>>>>> sentences of Q but neither is a rheorem or Q does not depend on
>>>>>>>>>>> any semantics.
>>>>>>>>>>
>>>>>>>>>> The entire body of knowledge expressed in language
>>>>>>>>>> can be represented as a semantic tautology in an
>>>>>>>>>> acyclic directed graph. That knowledge is a DAG was
>>>>>>>>>> my very thought on this subject more than 30 years ago.
>>>>>>>>>> This single idea gets rid of all undecidability
>>>>>>>>>> within the entire body of knowledge.
>>>>>>>>>
>>>>>>>>> No, it cannot. The usual meaning of knoledge excludes tautologies
>>>>>>>>
>>>>>>>> You are not paying close enough attention. I did not say logical
>>>>>>>> tautology. I said semantic tautology. That cats are defined to
>>>>>>>> be animals is a semantic tautology. That cats are defined to be
>>>>>>>> cats is a logical tautology. Here is a definition of that fits
>>>>>>>> my definition of semantic tautology.
>>>>>>>
>>>>>>> What I said applies to logical tautologies, too. That cats are
>>>>>>> defined to be animals only tells somthing (but not much) about
>>>>>>> the meanings of the words but nothing about the real world.
>>>>>>
>>>>>> When a complete set of general "atomic facts" of the
>>>>>> actual world is encoded as axioms along with all of
>>>>>> the semantic entailment relations between these facts
>>>>>> then every element of general knowledge that can be
>>>>>> expressed in language is known by this system.
>>>>>
>>>>> In order to achieve that the atomic facts must be non-tautologies.
>>>
>>>> That "cats" <are> "animals" is a semantic tautology.
>>>
>>> That depends on the semantic system. Often the meaning of the word
>>> "cat" indeed involves that what is called a "cat" is also called
>>> an "animal" but there are other posiibilities. Conseqently, the
>>> sentence "cats are animals" can be a semantic tautology that tells
>>> nothing about the real world,
>>
>> It tells us exactly one thing about the real world 1 != 0
>>
>>> or it can be a consequence of the
>>> definitions of "cat" and "animal" and terms used in these definitions
>>
>> it tells us exactly one thong about the real world
>> through the definition of terms.
>>
>>> that tells nothing about the real world, or it can be a statement
>>> about the real world that cannot be inferred from definitions alne.
>>>
>>>> "atomic facts" that correspond to things in the world
>>>> only have stipulation as their basis in truth within
>>>> the formal system.
>>
>>> One usually says "assumption" instead of "stipulation". The latter
>>> usually means a rfusal to proceed before the stipulation is accepted
>>> or an alternative agreed.
>>>
>>
>> https://en.wikipedia.org/wiki/Stipulative_definition
>> Cats are animals, if you disagree then you are necessarily incorrect.
>>
> 
> Animals are {cats, dogs, wallabies, llamas, aardvarks, crustaceans,
> ...}, a potentially infinitary expression.
> 

Not even if you do each individual at a time
that ever lived since the beginning of life
on Earth is this an infinite set.

How could a smart guy like you make such a
huge mistake?


-- 
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).