Related papers: Observations concerning G\"odel's 1931

Explicit counterexamples to Theorem 1 of R. Turyn's and J. Storer's often-cited paper "On Binary Sequences" [Proc. Amer. Math. Soc., vol. 12, no. 3, pp. 394-399, 1961] are given. Theorem 2 of their paper is well known; it states that Barker…

A semantic analysis of formal systems is undertaken, wherein the duality of their symbolic definition based on the "State of Doing" and "State of Being" is brought out. We demonstrate that when these states are defined in a way that opposes…

This paper was withdrawn by the authors. Lemma 5.1 is wrong.

It has been commonly argued, on the basis of Goedel's theorem and related mathematical results, that true artificial intelligence cannot exist. Penrose has further deduced from the existence of human intelligence that fundamental changes in…

We establish an irreducibility property for the characters of finite dimensional, irreducible representations of simple Lie algebras (or simple algebraic groups) over the complex numbers, i.e., that the characters of irreducible…

If no optimal propositional proof system exists, we (and independently Pudl\'ak) prove that ruling out length $t$ proofs of any unprovable sentence is hard. This mapping from unprovable to hard-to-prove sentences powerfully translates facts…

In this paper, we give a detailed account of Goldfeld's proof of Siegel's theorem. Particularly, we present complete proofs of the nontrivial assumptions made in his paper.

Hilbert and Ackermann asked for a method to consistently extend incomplete theories to complete theories. G\"odel essentially proved that any theory capable of encoding its own statements and their proofs contains statements that are true…

WITHDRAWN: The proof contains an uncorrectable gap in the proof of theorem 7 on page 11. A proof of the Krzyz conjecture is presented, based on the application of the variational method, as well as on the use of two classical results and…

We answer in the negative Siegel's problem for $G$-functions, as formulated by Fischler and Rivoal. Roughly, we prove that there are $G$-functions that cannot be written as polynomial expressions in algebraic pullbacks of hypergeometric…

We study the Borel-reducibility of isomorphism relations of complete first order theories and show the consistency of the following: For all such theories T and T', if T is classifiable and T' is not, then the isomorphism of models of T' is…

The famous G\"odel incompleteness theorem states that for every consistent sufficiently rich formal theory T there exist true statements that are unprovable in T. Such statements would be natural candidates for being added as axioms, but…

Remarks on the Cantor's nondenumerability proof of 1891 that the real numbers are noncountable will be given. By the Cantor's diagonal procedure, it is not possible to build numbers that are different from all numbers in a general assumed…

Any stretching of Ringel's non-Pappus pseudoline arrangement when projected into the Euclidean plane, implicitly contains a particular arrangement of nine triangles. This arrangement has a complex constraint involving the sines of its…

A famous theorem of Zudilin states that at least one of the Riemann zeta values $\zeta(5), \zeta(7), \zeta(9), \zeta(11)$ is irrational. In this paper, we establish the $p$-adic analogue of Zudilin's theorem. As a weaker form of our result,…

Glivenko's theorem says that, in propositional logic, classical provability of a formula entails intuitionistic provability of double negation of that formula. We generalise Glivenko's theorem from double negation to an arbitrary nucleus,…

Asaf Karagila pointed out that Remark 3.4 [1], directly contradicts Theorem 3.3 (c) [2] which was incorrectly stated. This note contains a proof of this remark. [1] Miller, Arnold W.; A Dedekind Finite Borel Set, Arch. Math. Logic 50…

A recent paper by Ag\'elas [Generalized Riemann Hypothesis, 2019, hal-00747680v3] claims to prove the Generalized Riemann Hypothesis (GRH) and, as a special case, the Riemann Hypothesis (RH). We show that the proof given by Ag\'elas…

In this paper, we firstly point out, by a counter example, that Proposition 6.4 of Section 6 in Bump's book (Algebraic Geometry) is error, and then give a correct statement with proof. We finally point out a gap in the proof of Theorem 3,…

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…