Related papers: The Provability of Consistency
We conclude from Goedel's Theorem VII of his seminal 1931 paper that every recursive function f(x_{1}, x_{2}) is representable in the first-order Peano Arithmetic PA by a formula [F(x_{1}, x_{2}, x_{3})] which is algorithmically verifiable,…
G\"odel's second incompleteness theorem is proved for Herbrand consistency of some arithmetical theories with bounded induction, by using a technique of logarithmic shrinking the witnesses of bounded formulas, due to Z. Adamowicz [Herbrand…
In the paper we introduce a weak set theory $\mathsf{H}_{<\omega}$ . A formalization of arithmetic on finite von Neumann ordinals gives an embedding of arithmetical language into this theory. We show that $\mathsf{H}_{<\omega}$ proves a…
In this paper we prove Chaitin's ``heuristic principle'', {\it the theorems of a finitely-specified theory cannot be significantly more complex than the theory itself}, for an appropriate measure of complexity. We show that the measure is…
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…
In response to a recent rebuttal of [1] presented in [2], we defend the claim that the Consistent Histories formulation of quantum mechanics does not solve the measurement problem. In order to do so, we argue that satisfactory solutions to…
It is proved that if $T$ is a $\Sigma_{n+1}$ Definable theory which is $\Sigma_n$-sound and extends $PA$, then $T$ can not prove the sentence $\Sigma_n-sound(T)$ that expresses the $\Sigma_n$-soundness of $T$. Optimality of this result is…
Recently, Artemov [4] offered the notion of constructive consistency for Peano Arithmetic and generalized it to constructive truth and falsity in the spirit of Brouwer-Heyting-Kolmogorov semantics and its formalization, the Logic of Proofs.…
The compactness theorem for a logic states, roughly, that the satisfiability of a set of well-formed formulas can be determined from the satisfiability of its finite subsets, and vice versa. Usually, proofs of this theorem depend on the…
We investigate the theory PAI (Peano Arithmetic with Indiscernibles). Models of PAI are of the form (M, I), where M is a model of PA, I is an unbounded set of order indiscernibles over M, and (M, I) satisfies the extended induction scheme…
General mathematical reasoning is computationally undecidable, but humans routinely solve new problems. Moreover, discoveries developed over centuries are taught to subsequent generations quickly. What structure enables this, and how might…
In this paper we introduce the notion of the stability of a sequence of modules over Hecke algebras. We prove that a finitely generated consistent sequence associated with Hecke algebras is representation stable.
We consider pure equational theories that allow substitution but disallow induction, which we denote as PETS, based on recursive definition of their function symbols. We show that the Bounded Arithmetic theory $S^1_2$ proves the consistency…
We consider extensions of Peano arithmetic which include an assertibility predicate. Any such system which is arithmetically sound effectively verifies its own soundness. This leads to the resolution of a range of paradoxes involving…
Justification theory is an abstract unifying formalism that captures semantics of various non-monotonic logics. One intriguing problem that has received significant attention is the consistency problem: under which conditions are…
G\"odel's argument for the First Incompleteness Theorem is, structurally, a proof by contradiction. This article intends to reframe the argument by, first, isolating an additional assumption the argument relies on, and then, second, arguing…
We present an alternative cyclic proof system for Peano arithmetic that could be simpler than the existing ones and well-adapted both for proof analysis and for automatizing inductive proof search. In addition, we will show how various…
Feferman proved in 1962 that any arithmetical theorem is a consequence of a suitable transfinite iteration of full uniform reflection of $\mathsf{PA}$. This result is commonly known as Feferman's completeness theorem. The purpose of this…
We present a new manifestation of G\"odel's second incompleteness theorem and discuss its foundational significance, in particular with respect to Hilbert's program. Specifically, we consider a proper extension of Peano arithmetic…
We develop a comprehensive theory of the stable representation categories of several sequences of groups, including the classical and symmetric groups, and their relation to the unstable categories. An important component of this theory is…