Related papers: On consistency and existence in mathematics
We prove the consistency result from the title. By forcing we construct a model of g=aleph_1, b=cf(Sym(omega))=aleph_2.
An algebraic formalism for quantum decoherence in systems with continuous evolution spectrum is introduced. A certain subalgebra, dense in the characteristic algebra of the system, is defined in such a way that Riemann-Lebesgue theorem can…
Although the categorical arithmetic is not effectively axiomatizable, the belief that the incompleteness Theorems can be apply to it is fairly common. Furthermore, the so-called "essential" (or "inherent") semantic incompleteness of the…
We study the faithfulness of an explanation system to the underlying prediction model. We show that this can be captured by two properties, consistency and sufficiency, and introduce quantitative measures of the extent to which these hold.…
Consistency is the theoretical property of a meta learning algorithm that ensures that, under certain assumptions, it can adapt to any task at test time. An open question is whether and how theoretical consistency translates into practice,…
Taking several statistical examples, in particular one involving a choice of experiment, as points of departure, and making symmetry assumptions, the link towards quantum theory developed in Helland (2005a,b) is surveyed and clarified. The…
We prove a generalization of Gotzmann's persistence theorem in the case of modules with constant Hilbert polynomial. As a consequence, we show that the defining equations that give the embedding of a Quot scheme of points into a…
Consistency properties of concurrent computations, e.g., sequential consistency, linearizability, or eventual consistency, are essential for devising correct concurrent algorithms. In this paper, we present a logical formalization of such…
We present a self-contained elementary and detailed exposition of Mertens' own proof of his theorem on the divergence of the series of the reciprocals of the primes and compare it with the modern proofs. His proof contains explicit…
Godelian sentences of a sufficiently strong and recursively enumerable theory, constructed in Godel's 1931 groundbreaking paper on the incompleteness theorems, are unprovable if the theory is consistent; however, they could be refutable.…
We note that Pillay's result on the stability of an algebraically closed field with a predicate for a group of Lang type implies that number uniformity follows formally from the finiteness results analogous to Faltings' Theorem.
It is well known that the resolution method (for propositional logic) is complete. However, completeness proofs found in the literature use an argument by contradiction showing that if a set of clauses is unsatisfiable, then it must have a…
We give a proof of the inconsistency of PM arithmetic, classical set theory and related systems, incidentally exposing an error in Goedel's own proof of Goedel's Theorems. The inconsistency proof, that formulae of the form R and ~R occur as…
Besides their use for efficient computation, quantum computers are a base for studying quantum systems that create valid physical theories using mathematics and physics. An essential part of the validation process for quantum mechanics is…
We use algorithmic methods from online learning to explore some important objects at the intersection of model theory and combinatorics, and find natural ways that algorithmic methods can detect and explain (and improve our understanding…
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…
Recently 't Hooft demonstrated that ``For any quantum system there exists at least one deterministic model that reproduces all its dynamics after prequantization''. An extension is presented here which covers quantum systems that are…
In this paper we prove Chaitin's ``heuristic principle'', {\it the theorems of a finitely-specified theory cannot be significantly more complex than the theory itself}, for an appropriate measure of complexity. We show that the measure is…
Is it possible to encompass the full extent of the universe within a theory based on a finite set of first principles and inference rules? The r\^{o}le of observers and observations in physics theories is considered here in the light of…
In this thesis we study the principle that extremal objects in differential geometry correspond to stable objects in algebraic geometry. In our introduction we survey the most famous instances of this principle with a view towards the…