11 research outputs found
From Tarski to G\"odel. Or, how to derive the Second Incompleteness Theorem from the Undefinability of Truth without Self-reference
In this paper, we provide a fairly general self-reference-free proof of the
Second Incompleteness Theorem from Tarski's Theorem of the Undefinability of
Truth.Comment: 7 page
The Surprise Examination Paradox and the Second Incompleteness Theorem
We give a new proof for Godel's second incompleteness theorem, based on
Kolmogorov complexity, Chaitin's incompleteness theorem, and an argument that
resembles the surprise examination paradox. We then go the other way around and
suggest that the second incompleteness theorem gives a possible resolution of
the surprise examination paradox. Roughly speaking, we argue that the flaw in
the derivation of the paradox is that it contains a hidden assumption that one
can prove the consistency of the mathematical theory in which the derivation is
done; which is impossible by the second incompleteness theorem.Comment: 8 page
Tarski's Undefinability Theorem and Diagonal Lemma
We prove the equivalence of the semantic version of Tarski's theorem on the
undefinability of truth with a semantic version of the Diagonal Lemma, and also
show the equivalence of syntactic Tarski's Undefinability Theorem with a weak
syntactic diagonal lemma. We outline two seemingly diagonal-free proofs for
these theorems from the literature, and show that syntactic Tarski's theorem
can deliver G\"odel-Rosser's Incompleteness Theorem.Comment: 8 page
An incompleteness theorem via ordinal analysis
We present an analogue of G\"{o}del's second incompleteness theorem for
systems of second-order arithmetic. Whereas G\"{o}del showed that sufficiently
strong theories that are -sound and -definable do not
prove their own -soundness, we prove that sufficiently strong theories
that are -sound and -definable do not prove their own
-soundness. Our proof does not involve the construction of a
self-referential sentence but rather relies on ordinal analysis
Provability Logic: models within models in Peano Arithmetic
In 1994 Jech gave a model theoretic proof of G\"odel's second incompleteness
theorem for Zermelo-Fraenkel set theory in the following form: ZF does not
prove that ZF has a model. Kotlarski showed that Jech's proof can be adapted to
Peano Arithmetic with the role of models being taken by complete consistent
extensions. In this note we take another step in the direction of replacing
proof-theoretic by model-theoretic arguments. We show, without passing through
the arithmetized completeness theorem, that the existence of a model of PA of
complexity is independent of PA, where a model is identified with
the set of formulas with parameters which hold in the model. Our approach is
based on a new interpretation of the provability logic of Peano Arithmetic with
the modal operator interpreted as truth in every -model
Finding the limit of incompleteness I
In this paper, we examine the limit of applicability of G\"{o}del's first
incompleteness theorem ( for short). We first define the notion " holds for the theory ". This paper is motivated by the following
question: can we find a theory with a minimal degree of interpretation for
which holds. To approach this question, we first examine the following
question: is there a theory such that Robinson's interprets
but does not interpret (i.e. is weaker than
w.r.t. interpretation) and holds for ? In this paper,
we show that there are many such theories based on Je\v{r}\'{a}bek's work using
some model theory. We prove that for each recursively inseparable pair , we can construct a r.e. theory such that
is weaker than w.r.t. interpretation and
holds for . As a corollary, we answer a
question from Albert Visser. Moreover, we prove that for any Turing degree
, there is a theory with Turing
degree such that holds for and is weaker than
w.r.t. Turing reducibility. As a corollary, based on Shoenfield's
work using some recursion theory, we show that there is no theory with a
minimal degree of Turing reducibility for which holds.Comment: 18 pages. Accepted and to appear in Bulletin of Symbolic Logi
On the depth of G\"{o}del's incompleteness theorem
In this paper, we use G\"{o}del's incompleteness theorem as a case study for
investigating mathematical depth. We take for granted the widespread judgment
by mathematical logicians that G\"{o}del's incompleteness theorem is deep, and
focus on the philosophical question of what its depth consists in. We focus on
the methodological study of the depth of G\"{o}del's incompleteness theorem,
and propose three criteria to account for its depth: influence, fruitfulness,
and unity. Finally, we give some explanations for our account of the depth of
G\"{o}del's incompleteness theorem.Comment: 23 pages, revised version. arXiv admin note: text overlap with
arXiv:2009.0488
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