527 research outputs found
A computability theoretic equivalent to Vaught's conjecture
We prove that, for every theory which is given by an sentence, has less than many countable
models if and only if we have that, for every on a cone of
Turing degrees, every -hyperarithmetic model of has an -computable
copy. We also find a concrete description, relative to some oracle, of the
Turing-degree spectra of all the models of a counterexample to Vaught's
conjecture
On the Invariance of G\"odel's Second Theorem with regard to Numberings
The prevalent interpretation of G\"odel's Second Theorem states that a
sufficiently adequate and consistent theory does not prove its consistency. It
is however not entirely clear how to justify this informal reading, as the
formulation of the underlying mathematical theorem depends on several arbitrary
formalisation choices. In this paper I examine the theorem's dependency
regarding G\"odel numberings. I introduce deviant numberings, yielding
provability predicates satisfying L\"ob's conditions, which result in provable
consistency sentences. According to the main result of this paper however,
these "counterexamples" do not refute the theorem's prevalent interpretation,
since once a natural class of admissible numberings is singled out, invariance
is maintained.Comment: Forthcoming in The Review of Symbolic Logi
Independence, Relative Randomness, and PA Degrees
We study pairs of reals that are mutually Martin-L\"{o}f random with respect
to a common, not necessarily computable probability measure. We show that a
generalized version of van Lambalgen's Theorem holds for non-computable
probability measures, too. We study, for a given real , the
\emph{independence spectrum} of , the set of all so that there exists a
probability measure so that and is
-random. We prove that if is r.e., then no set
is in the independence spectrum of . We obtain applications of this fact to
PA degrees. In particular, we show that if is r.e.\ and is of PA degree
so that , then
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