相关论文: On G\"odel's second incompleteness theorem

We present a simple short proof of the Fundamental Theorem of Algebra, without complex analysis and with a minimal use of topology. It can be taught in a first year calculus class.

It is argued that Goedel's incompleteness theorem should be seen as self-evident, rather than unexpected or surprising.

We give a short proof for the Hartogs's extension theorem on (n-1)-complete complex spaces.

This note gives an informal overview of the proof in our paper "Borel Conjecture and Dual Borel Conjecture", see arXiv:1105.0823.

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 overarching theme of the following pages is that mathematical logic -- centered around the incompleteness theorems -- is first and foremost an investigation of $\textit{computation}$, not arithmetic. Guided by this intuition we will…

Many proofs of the Fundamental Theorem of Algebra, including various proofs based on the theory of analytic functions of a complex variable, are known. To the best of our knowledge, this proof is different from the existing ones.

We prove some constructive results that on first and maybe even on second glance seem impossible.

G\"odel proved in the 1930s in his famous Incompleteness Theorems that not all statements in mathematics can be proven or disproven from the accepted ZFC axioms. A few years later he showed the celebrated result that Cantor's Continuum…

A proof is given of Rosenthal's \(\ell_1\) theorem.

We show that first-order logic can be translated into a very simple and weak logic, and thus set theory can be formalized in this weak logic. This weak logical system is equivalent to the equational theory of Boolean algebras with three…

The purpose of this short note is to present a simplified proof of Serre's modularity conjecture using the strong modularity lifting results currently available. This second version includes extra details on definitions and proofs than the…

After highlighting the cases in which the semantics of a language cannot be mechanically reproduced (in which case it is called inherent), the main epistemological consequences of the first incompleteness Theorem for the two fundamental…

We present a mathematical framework for mapping second-order logic relations onto a simple state vector algebra. Using this algebra, basic theorems of set theory can be proven in an algorithmic way, hence by an expert system. We illustrate…

In this note we give a short, direct proof of the well known Combinatorial Nullstellensatz.

A very short proof of Kneser's theorem via transversal is given.

Analogous to G\"odel's incompleteness theorems is a theorem in physics to the effect that the set of explanations of given evidence is uncountably infinite. An implication of this theorem is that contact between theory and experiment…

We give a new proof of a classical theorem on approximation of continuous functions on totally real sets

The fundamental aim of the paper is to correct an harmful way to interpret a Goedel's erroneous remark at the Congress of Koenigsberg in 1930. Despite the Goedel's fault is rather venial, its misreading has produced and continues to produce…

For an arbitrary partially ordered set $P$ its {\em dual} $P^*$ is built as the collection of all monotone mappings $P\to\2$ where $\2=\{0,1\}$ with $0<1$. The set of mappings $P^*$ is proved to be a complete lattice with respect to the…