XPost: sci.logic
That Gödel emphasized "formal" in his paper
has to do that his "decidable" comes from an
ontology related to syntactic derivability.
"deciable" is defined on the basis of the
notion of general validity embodied as
provability. But Gödel wasn't that one dimensional,
you find also a semantic leaning terminology
in some of his papers, for example his completeness
theorem. There he uses another terminology,
which doesn't have a one-to-one mapping to
"decidable". He uses notions such as "erfüllbar"
or "statisfiable", and "unerfüllbar" or
"unsatisfiable". The main theorem here is:
an unprovable sentences A is satisfiable by a counter
model, meaning its complement has a model.
Mild Shock schrieb:
For example Gödel belongs to the generation of
logicians that use the term "undecidable".
In German the term is translated to "unentscheidbar":
Über formal unentscheidbare Sätze der Principia Mathematica und
verwandter Systeme I" ("On Formally Undecidable Propositions of
Principia Mathematica and Related Systems I") https://en.wikipedia.org/wiki/On_Formally_Undecidable_Propositions_of_Principia_Mathematica_and_Related_Systems
Mild Shock schrieb:
Since generations logicians have called sentences
which you clumsily call "not a truth-bearer",
simple called "undecidable" sentences.
A theory is incomplete, if it has undecidable
sentences. There is a small difference between
unprovable and undecidable.
An unprovable senetence A is only a sentence with:
~True(L, A).
An undecidable sentence A is a sentence with:
~True(L, A) & ~True(L, ~A)
Meaning the sentence itself and its complement
are both unprovable.
olcott schrieb:
~True(L,x) ∧ ~True(L,~x)
means that x is not a truth-bearer in L.
It does not mean that L is incomplete
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