137 research outputs found
Current research on G\"odel's incompleteness theorems
We give a survey of current research on G\"{o}del's incompleteness theorems
from the following three aspects: classifications of different proofs of
G\"{o}del's incompleteness theorems, the limit of the applicability of
G\"{o}del's first incompleteness theorem, and the limit of the applicability of
G\"{o}del's second incompleteness theorem.Comment: 54 pages, final accepted version, to appear in The Bulletin of
Symbolic Logi
Turing jumps through provability
Fixing some computably enumerable theory , the
Friedman-Goldfarb-Harrington (FGH) theorem says that over elementary
arithmetic, each formula is equivalent to some formula of the form
provided that is consistent. In this paper we give various
generalizations of the FGH theorem. In particular, for we relate
formulas to provability statements which
are a formalization of "provable in together with all true
sentences". As a corollary we conclude that each is
-complete. This observation yields us to consider a recursively
defined hierarchy of provability predicates which look a lot
like except that where calls upon the
oracle of all true sentences, the recursively
calls upon the oracle of all true sentences of the form . As such we obtain a `syntax-light' characterization of
definability whence of Turing jumps which is readily extended
beyond the finite. Moreover, we observe that the corresponding provability
predicates are well behaved in that together they provide a
sound interpretation of the polymodal provability logic
The Modal Logics of Kripke-Feferman Truth
We determine the modal logic of fixed-point models of truth and their axiomatizations by Solomon Feferman via Solovay-style completeness results
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