Related papers: Hilbert's Program and Infinity
We show that including degrees of a particular kind of provability in the search target for any theorem-prover in sufficiently powerful formal systems over finite-sized statements preserves well-definition and a sufficient consistency while…
We bring forward a logical system of transition algebras that enhances many-sorted first-order logic using features from dynamic logics. The sentences we consider include compositions, unions, and transitive closures of transition…
A principled approach to the design of program verification and con- struction tools is applied to separation logic. The control flow is modelled by power series with convolution as separating conjunction. A generic construction lifts…
We propose an incremental approach for safety proofs that decomposes a proof with a complex inductive invariant into a sequence of simpler proof steps. Our proof system combines rules for (i) forward reasoning using inductive invariants,…
In reductive proof search, proofs are naturally generalized by solutions, comprising all possibly infinite structures generated by locally correct, bottom-up application of inference rules. We propose an extension of the Curry-Howard…
This paper discusses the semantics and proof theory of Nilsson's probabilistic logic, outlining both the benefits of its well-defined model theory and the drawbacks of its proof theory. Within Nilsson's semantic framework, we derive a set…
Proof search has been used to specify a wide range of computation systems. In order to build a framework for reasoning about such specifications, we make use of a sequent calculus involving induction and co-induction. These proof principles…
We lay out novel foundations for the computer-aided verification of guaranteed bounds on expected outcomes of imperative probabilistic programs featuring (i) general loops, (ii) continuous distributions, and (iii) conditioning. To handle…
The usual mathematical formalism of quantum field theory is non-rigorous because it contains divergences that can only be renormalized by non-rigorous mathematical methods. The purpose of this paper is to present a method of subtraction of…
In this short paper we present an elementary proof of the infinitude of primes. Our proof is similar in spirit to Euler's proof that the reciprocals of primes diverges and only uses tools from elementary number theory and calculus. In…
In this paper, we used the principle of sieve function transformation to improve sieve method and the prime number theorem in the arithmetic sequence.For this, we proved General Riemann Hypothesis and Riemann Hypothesis to be true. further,…
Let F be a totally real field and p a rational prime unramified in F. We prove a partial classicality theorem for overconvergent Hilbert modular forms: when the slope is small compared to certain but not all weights, an overconvergent form…
The usual nonnegative modulus function is based on addition. A natural different modulus function on the set of positive reals is introduced. Arguments for results for series through the usual modulus function are transformed to arguments…
In proof-theoretic semantics, model-theoretic validity is replaced by proof-theoretic validity. Validity of formulae is defined inductively from a base giving the validity of atoms using inductive clauses derived from proof-theoretic rules.…
We consider the explicit fragment of the basic justification stit logic introduced in earlier publications. We define a Hilbert-style axiomatic system for this logic and show that this system is strongly complete relative to the intended…
Two main algorithmic approaches are known for making Hironaka's proof of resolution of singularities in characteristic zero constructive. Their main difference is the use of different notions of transforms during the resolution process and…
Non-wellfounded proof theory results from allowing proofs of infinite height in proof theory. To guarantee that there is no vicious infinite reasoning, it is usual to add a constraint to the possible infinite paths appearing in a proof.…
The criterion for an affine primary algebra over the field to be integral, is proven. Using this criterion we give a simple proof that Hilbert scheme of 0-dimensional subschemes of length $l$ of nonsingular $d$-dimensional algebraic variety…
To obtain the highest confidence on the correction of numerical simulation programs implementing the finite element method, one has to formalize the mathematical notions and results that allow to establish the soundness of the method.…
The principle which allows to construct new physical theories on the basis of classical mechanics by reduction of the number of its axiom without engaging new postulates is formulated. The arising incompleteness of theory manifests itself…