A new reading and comparative interpretation of Gödel’s completeness (1930) and incompleteness (1931) theorems

Peano arithmetic cannot serve as the ground of mathematics for it is inconsistent to infinity, and infinity is necessary for its foundation. Though Peano arithmetic cannot be complemented by any axiom of infinity, there exists at least one (logical) axiomatics consistent to infinity. That is nothing else than a new reading at issue and comparative interpretation of Gödel’s papers (1930; 1931) meant here. Peano arithmetic admits anyway generalizations consistent to infinity and thus to some addable axiom(s) of infinity. The most utilized example of those generalizations is the complex Hilbert space. Any generalization of Peano arithmetic consistent to infinity, e.g. the complex Hilbert space, can serve as a foundation for mathematics to found itself and by itself

Infinity

This essay surveys the different types of infinity that occur in pure and applied mathematics, with emphasis on: 1. the contrast between potential infinity and actual infinity; 2. Cantor's distinction between transfinite sets and absolute infinity; 3. the constructivist view of infinite quantifiers and the meaning of constructive proof; 4. the concept of feasibility and the philosophical problems surrounding feasible arithmetic; 5. Zeno's paradoxes and modern paradoxes of physical infinity involving supertasks

A Mathematical Model of Divine Infinity

Mathematics is obviously important in the sciences. And so it is likely to be equally important in any effort that aims to understand God in a scientifically significant way or that aims to clarify the relations between science and theology. The degree to which God has any perfection is absolutely infinite. We use contemporary mathematics to precisely define that absolute infinity. For any perfection, we use transfinite recursion to define an endlessly ascending series of degrees of that perfection. That series rises to an absolutely infinite degree of that perfection. God has that absolutely infinite degree. We focus on the perfections of knowledge, power, and benevolence. Our model of divine infinity thus builds a bridge between mathematics and theology

Admissible initial growth for diffusion equations with weakly superlinear absorption

We study the admissible growth at infinity of initial data of positive solutions of \prt\_t u-\Gd u+f(u)=0 in \BBR\_+\ti\BBR^N when $f(u)$ is a continuous function, {\it mildly} superlinear at infinity, the model case being f(u)=u\ln^\ga (1+u) with 1\textless{}\ga\textless{}2. We prove in particular that if the growth of the initial data at infinity is too strong, there is no more diffusion and the corresponding solution satisfies the ODE problem \prt\_t \gf+f(\gf)=0 on \BBR\_+ with \gf(0)=\infty.Comment: Communications in Contemporary Mathematics, to appea

Limit Cases of the p-Laplace Operator via Mosco Convergence

In the classic theory, p-Laplace operator (1 \u3c p \u3c infinity) joined several main parts of the mathematics in a fruitful way, and one important principle of mathematics is that extreme cases reveal interesting structure. Looking at p-Laplace operator as subgradients of a sequence of convex functionals Ep, as p goes to 1 and to infinity, we study the connection of the dual problem between 1-Laplace operator and infinity-Laplace operator using tools from convex analysis and the notion of Mosco convergence