On 4/22/2026 2:03 AM, Mikko wrote:Peano arithmetic is unsolvable, i.e., there is no method to findOn 21/04/2026 16:22, olcott wrote:The truth value of the Goldbach conjecture mightOn 4/21/2026 1:30 AM, Mikko wrote:>On 20/04/2026 16:31, olcott wrote:>On 4/20/2026 3:49 AM, Mikko wrote:>On 19/04/2026 20:21, olcott wrote:>On 4/19/2026 3:59 AM, Mikko wrote:>On 18/04/2026 15:58, olcott wrote:>>>
Unknown truths are not elements of the body of
knowledge is a semantic tautology. Did you think
that things that are unknown are known?
No, but that measn that for some sentences X True(X) is unknown and there
is no method to find out.
>
I don't know about philosophers but mathematicians and logicians don't
find it interesting if all you can say that all knowledge is knowable
and everything else is not.
Ross Finlayson, seemed to endlessly hedge on whether
or not the truth value of the Goldbach conjecture was
known. He seemed to think that there are alternative
analytical frameworks that make the question of whether
or not its truth value is known an ambiguous question.
>
I needed to refer to unknown truth values specifically
because all "undecidability" when construed correctly
falls into one of two categories.
(a) Semantic incoherence
(b) Unknown truth values.
A centence can be said to be undecidable when it is known that neither
the sentence nor its negation is a theorem.
When we skip model theory and and define True and False
as the existence of a back chained sequence of inference
steps of expressions x or ~x reaching axioms
It is not useful to define new terms for comcepts that already have
good terms.
The result of undecidability proves that the current
foundations are incoherent in the same way that
Russell's paradox proved that naive set theory had
a glitch.
Hardly the same way as Russell's paradox proves that there is no
undecidability in the naive set theory.
>>If the sequence of inference steps is restricted to>
valid inferences the term "True" as defined above then "sentence is
true" is just another way to say "sentence is a theorem".
>then it is a yes or no question that has no correct yes>
or no answer within the formal system.
Even if a question has no answer within a formal theory of natural
numbers it may have an answer in the natural numbers themselves.
>My system is based on simple type theory and formalized>
natural language.
>
This makes it a yes or no question that has no
correct yes or no answer at all anywhere, thus
an incorrect polar question.
How does your system handle questions that are not known to have a
yes or no answer but k´nor known to lack such answer, either, e.g. Goldbach's conjecture ?
out-of-scope of the body of knowledge.
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
So the question whether something is in the scope of your system
is not in the scope of your system? OK, but shoudn't such questions
be answerable anyway?
be unknowable if it is true and the only way to
prove it is true is an infinite number of steps.
out whether a particular sentence (for exmaple Goldbach conjecture)
is provable or not. If you find a proof then you know it but it is
possible that you never find, no matter how much you search.
--
Mikko
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