XPost: sci.math, comp.theory
From: polcott333@gmail.com
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.
--
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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