Related papers: On a Theorem by Bezboruah & Shepherdson

We prove a new fixed point theorem of Schauder-type which applies to discontinuous operators in non-compact domains. In order to do so, we present a modification of a recent Schauder-type theorem due to Pouso. We apply our result to…

The standard interpretation of first-order number theory (PA), according to the generally accepted view, associates well-defined set-theoretic entities with each and every well-formed formula of this system. But this implies that the class…

This article demonstrates the invalidity of Theorem VI of Godel's monograph of 1931, showing that propositions (15) and (16), derived from definition (8.1), in its proof, are false in PA. This is achieved in two steps. First, the predicate…

We present a relatively simple description of binary, definable subsets of models of weakly quasi-o-minimal theories. In particular, we closely describe definable linear orders and prove a weak version of the monotonicity theorem. We also…

We study the first-order consequences of Ramsey's Theorem for $k$-colourings of $n$-tuples, for fixed $n, k \ge 2$, over the relatively weak second-order arithmetic theory $\mathrm{RCA}^*_0$. Using the Chong-Mourad coding lemma, we show…

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…

The minimal bad sequence argument due to Nash-Williams is a powerful tool in combinatorics with important implications for theoretical computer science. In particular, it yields a very elegant proof of Kruskal's theorem. At the same time,…

In Lorentz violating quantum electrodynamics (QED) it is known that a radiatively induced Chern-Simons term appears in the effective action for the gauge field, which is finite but undetermined. This ambiguity is shown to be absent in a…

We prove the consistency of the failure of the weak diamond $\Phi_\lambda$ at strongly inaccessible cardinals. On the other hand, we show that the very weak diamond $\Psi_\lambda$ is equivalent to the statement $2^{<\lambda}<2^\lambda$ and…

Based on the MRDP theorem, we introduce the ideas of the proof equation of a formula and universal proof equation of Peano Arithmetic (PA); and then, combining universal proof equation and G\"odel's Second Incompleteness Theorem, it is…

It is often said that measuring a system's position must disturb the complementary property, momentum, by some minimum amount due to the Heisenberg uncertainty principle. Using a "weak-measurement", this disturbance can be reduced. One…

Thomason \cite{Thomason74} showed that a certain modal logic $\mathbf{L}\subset \mathbf{S4}$ is incomplete with respect to Kripke semantics. Later Gerson \cite{Gerson75} showed that $\mathbf{L}$ is also incomplete with respect to…

The prevalent interpretation of G\"odel's Second Theorem states that a sufficiently adequate and consistent theory does not prove its consistency. It is however not entirely clear how to justify this informal reading, as the formulation of…

We show that the existence of a weakly compact cardinal over the Zermelo-Fraenkel's set theory is proof-theoretically reducible to iterations of Mostowski collapsings and Mahlo operations.

We show that the sequential closure of a family of probability measures on the canonical space of c{\`a}dl{\`a}g paths satisfying Stricker's uniform tightness condition is a weak${}^*$ compact set of semimartingale measures in the pairing…

Several variants of the Halpern-L\"auchli Theorem for trees of uncountable height are investigated. For $\kappa$ weakly compact, we prove that the various statements are all equivalent. We show that the strong tree version holds for one…

We prove collective versions of semi-duality theorems for sets of almost (limitedly, order) L-weakly compact operators.

In 2010, Vladimir Voevodsky gave a lecture on "What If Current Foundations of Mathematics Are Inconsistent?" Among other things he said that he was seriously suspicious that an inconsistency in PA (first-order Peano arithmetic) might…

I argue that quantum optical experiments that purport to refute Bohr's principle of complementarity (BPC) fail in their aim. Some of these experiments try to refute complementarity by refuting the so called particle-wave duality relations,…

A recent claim that finite precision in the design of real experiments ``nullifies'' the impact of the Kochen-Specker theorem, is shown to be unsupportable, because of the continuity of probabilities of measurement outcomes under slight…