Related papers: Tarski's Undefinability Theorem and Diagonal Lemma
We present a version of G\"odel's Second Incompleteness Theorem for recursively enumerable consistent extensions of a fixed axiomatizable theory, by incorporating some bi-theoretic version of the derivability conditions. We also argue that…
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.
This paper examines the application of Tarski's Undefinability Theorem to first-order arithmetic. The generally accepted view is that for this case the Theorem establishes that arithmetic truth is not arithmetic. A careful examination of…
Our main result (Theorem A) shows the incompleteness of any consistent sequential theory T formulated in a finite language such that T is axiomatized by a collection of sentences of bounded quantifier-alternation-depth. Our proof employs an…
The aim of this text is to present, in a technically accessible way, Tarski's definition of truth, the indefinability theorem, and to discuss two aspects of Tarski's work on truth, namely, whether or not the definition captures the notion…
We unify Godel's First Incompleteness Theorem (1931), Tarski's Undefinability Theorem (1933), Godel-Carnap's Diagonal Lemma (1934), and Rosser's (strengthening of Godel's first) Incompleteness Theorem (1936), whose proofs resemble much and…
We prove the following version of the first incompleteness theorem that simultaneously strengthens Mostowski's theorem and Vaught's theorem: For any c.e. family $\{ T_i \}_{i \in \omega}$ of consistent extensions of Tarski, Mostowski and…
Tarski's undefinability theorem states that a formal system based on conventional predicate logic (PL) cannot talk about its own truth predicate. PL is, however, not the only formal language imaginable. In this paper, it will be shown that…
One of the greatest problems in philosophy is that of meaning. The turning point in thinking on meaning was Tarski's definition of truth, and the rapid development of logical semantics and model theory was a consequence of this achievement.…
Godel's First Incompleteness Theorem is generalized to definable theories, which are not necessarily recursively enumerable, by using a couple of syntactic-semantic notions, one is the consistency of a theory with the set of all true…
The fact that the famous Godel incompleteness theorem and the archetype of all logical paradoxes, that of the Liar, are related closely is, of course, not only well known, but is a part of the common knowledge of logician community.…
We give a reframing of Godel's first and second incompleteness theorems that applies even to some undefinable theories of arithmetic. The usual Hilbert-Bernays provability conditions and the diagonal lemma are replaced by a more direct…
We present a general relational semantics framework which, by varying the axiomatization and components of the relational structures, provides a uniform semantics for sentential logics, classical and non-classical alike. The approach we…
G\"odel's second incompleteness theorem is standardly understood as showing that no sufficiently strong, consistent theory of arithmetic can prove its own consistency, a result typically interpreted against a model-theoretic background in…
We argue that Godel's completeness theorem is equivalent to completability of consistent theories, and Godel's incompleteness theorem is equivalent to the fact that this completion is not constructive, in the sense that there are some…
Considering classical first-order logic with equality, we give a "fully syntactic" construction of the (weak) syntactic category $\text{Syn}(T)$ associated to a consistent theory $T$; we show it is a consistent coherent category; and we…
The problem of defining and locating free will (FW) in physics is studied. On basis of logical paradoxes, we argue that FW has a meta-theoretic character, like the concept of truth in Tarski's undefinability theorem. Free will exists…
The Theory of Everything ($S_{\text{ToE}}$) seeks to unify all fundamental forces of nature, including quantum gravity, into a single theoretical framework. This theory would be defined internally using a set of axioms, and this paper…
I outline a new theory of truth that resolves the classical and constructive versions of the liar paradox. The theory features a provably consistent axiomatization of a global self-applicative truth predicate. Truth is defined using…
Goedel's completeness theorem is concerned with provability, while Girard's theorem in ludics (as well as full completeness theorems in game semantics) are concerned with proofs. Our purpose is to look for a connection between these two…