XPost: sci.math, comp.theory
From: polcott333@gmail.com
On 11/25/2025 10:07 PM, Python wrote:
> Le 26/11/2025 à 05:01, olcott a écrit :
>> On 11/25/2025 9:47 PM, Python wrote:
>>> Le 26/11/2025 à 04:45, olcott a écrit :
>>>> On 11/25/2025 9:26 PM, Python wrote:
>>>>> Le 26/11/2025 à 04:24, olcott a écrit :
>>>>>> When ALL *objects of thought* are defined
>>>>>> in terms of other *objects of thought* then
>>>>>> their truth and their proof is simply walking
>>>>>> the knowledge tree.
>>>>>
>>>>> A definition tree is not a proof system, Peter.
>>>>
>>>> When you have a narrow-minded view maybe not.
>>>> When a proof is any process applied to any
>>>> combination of finite strings (such as a tree
>>>> of knowledge) that makes its conclusion necessarily
>>>> true then it is a proof in the most generic sense.
>>>>
>>>> When we stipulate that "cats" "animals"
>>>> then the stipulated relation between those two
>>>> finite string is the proof that it is true.
>>>>
>>>> A tree of knowledge works this exact same
>>>> way yet the relationships can also be their
>>>> position in the inheritance hierarchy of types.
>>>>
>>>>> Walking a hierarchy does not make undecidable truths disappear — it
>>>>> just hides them from your model.
>>>>>
>>>>
>>>> A tree of knowledge makes undecidability impossible
>>>> within the entire body of knowledge that can be
>>>> expressed in language.
>>>>
>>>>> If “truth” were just “following links in a tree,” then:
>>>>>
>>>>> no arithmetic fact would require a proof,
>>>>>
>>>>
>>>> We also have semantic logical entailment from
>>>> a finite set of atomic facts. This set is not
>>>> finite.
>>>>
>>>>> no theorem would be non-trivial,
>>>>>
>>>>> no undecidable sentence would exist,
>>>>>
>>>>> and mathematics would collapse into a directory structure.
>>>>>
>>>>> But mathematics is not a filesystem.
>>>>
>>>> It makes no difference that G is not provable in PA.
>>>> Any X is provable in General_Knowledge or it is not
>>>> a member of General_Knowledge.
>>>
>>> Peter, you have quietly redefined “proof” to mean “any link I choose
>>> to place in my tree of definitions.”
>>> But that is not proof — that is classification.
>>>
>>
>> When the full and complete semantic meaning of any
>> *object of thought* is 100% entirely specified
>> by its connection to other *objects of thought*
>> then every element in this system has its provability
>> exactly the same as its truth.
>>
>>> Stipulating “cats = animals” is not a derivation; it is a definition.
>>> Definitions can build a taxonomy, but they cannot prove arithmetic
>>> truths, resolve undecidable statements, or replace inference rules.
>>>
>>
>> That is where semantic logical entailment specified
>> syntactically comes in.
>>
>>> Your “General_Knowledge” is complete only because you delete
>>> everything it cannot derive and declare it “not a member.”
>>> That is not a solution to incompleteness — it is circular pruning.
>>>
>>
>> You still do not understand how the combination
>> of a complete finite set of atomic definitions
>> along with every combination of things that
>> they entail derives the entire body of knowledge
>> that can be expressed in language.
>>
>>> You didn’t eliminate undecidability.
>>> You eliminated every sentence that would make your system face it.
>>>
>>
>> The complete body of knowledge that can be expressed
>> in language can be algorithmically compressed as I
>> have specified. It is only a finite set of atomic ideas
>> and a finite set of ways to combine these ideas together.
>> This derives an infinite set of ideas.
>
> Peter, if you claim that all knowledge expressible in language comes
> from a finite set of atomic ideas combined in finite ways, then your
> system can only generate countably many expressions,
If every verbal thought that anyone every had and every verbal
thought that anyone will ever have until the Earth is engulfed
by the Sun was written down this would be a finite set. Don't
bring religion into this it is only a distraction away form
the point.
You don't seem to understand the distinction between truth
and knowledge. True(L, x) tests for membership in the body
of General_Knowledge. Elements not in this set are
(a) False(L, x) defined as True(L, ~x)
(b) unknown
(c) semantically malformed
> whereas arithmetic
> truth involves uncountably many possible functions and a non-recursively
If the Goldbach conjecture requires an infinite proof
to resolve then its truth or falsity is not an element
of the set of General_Knowledge.
https://en.wikipedia.org/wiki/Goldbach%27s_conjecture
> enumerable set of true statements; no finite definitional base can
> produce a complete theory strong enough to capture all arithmetic
> consequences,
Of course not. That is why we also have semantic logical
inference.
> so when you insist that your finite tree “derives the
> entire body of knowledge,” what you actually have is a system that
> generates only a tiny countable fragment and discards everything it
> cannot express, which is not completeness but simply shrinking the
> universe of discourse until it fits inside your chosen model.
The body of General_Knowledge is algorithmically compressed
into a complete and finite set of atomic facts. When we add
semantic logical entailment to this any element in the set
of General_Knowledge can be derived.
"cats" "animals" is general knowledge.
"Fluffy" "Brown" "Cat" is specific knowledge.
--
Copyright 2025 Olcott
My 28 year goal has been to make
"true on the basis of meaning" computable.
This required establishing a new foundation
for correct reasoning.
--- SoupGate-Win32 v1.05
* Origin: you cannot sedate... all the things you hate (1:229/2)
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