On 22/04/2026 10:45, olcott wrote:>On 4/22/2026 2:03 AM, Mikko wrote:>On 21/04/2026 16:22, olcott wrote:>On 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?
The truth value of the Goldbach conjecture might
be unknowable if it is true and the only way to
prove it is true is an infinite number of steps.
Peano arithmetic is unsolvable, i.e., there is no method to find
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.
>
Before getting into Peano arithmetic, or after, it might make
sense to get into the Archimedean, and the Archimedean field,
then about non-Archimedean, yet really about whether the
Archimedean properties hold. The Archimedean property is usually
given that there are infinitely-many numbers, yet no infinitely-grand
numbers. So, besides non-Archimedean fields like Conway's surreal
numbers, or, other similar sorts of ideas, and besides non-standard
models of integers like Robinsohn's hyper-integers, are included
models of integers that are non-standard yet countable, like Boucher's F
or mostly a compactification of the integers, and for example have a
single point at infinity, or, for example, one infinite number for each
finite number.
So, about Peano successors, when those are the world, and they're
represented as sets by containing their predecessor, then quantifying
over those brings brings Russell's paradox, which states that the
gesammelt collection of those would contain itself, which is the
usual model of an inductive set or omega the constant in the language
of set theory. So, "naively", considered alone, Peano's numbers
have an infinitely-grand member.
So, the Archimedean property of numbers, is at least two properties.
And, Peano's numbers themselves bring Russell's paradox for free.
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