The reals, of course, already have an axiom system. They are a complete, linearly ordered field. This axiom system is even categorical, meaning that it completely characterizes the reals. Up to isomorphism, the reals are the only complete, linearly ordered field. Another property of axiom systems, considered to be particularly elegant ever since the birth of formal logic, is independence. In an...

The chaotic hypothesis discussed in [GC1] is tested experimentally in a simple conduction model. Besides a confirmation of the hypothesis predictions the results suggest the validity of the hypothesis in the much wider context in which, as the forcing strength grows, the attractor ceases to be an Anosov system and becomes an Axiom A attractor. A first test of the new predictions is also attempted.

Any two infinite-dimensional (separable) Hilbert spaces are unitarily isomorphic. The sets of all their self-adjoint operators are also therefore unitarily equivalent. Thus if all self-adjoint operators can be observed, and if there is no further major axiom in quantum physics than those formulated for example in Dirac’s ‘Quantum Mechanics’, then a quantum physicist would not be able to tell a ...

Ex. 17 Let us call the new system L, i.e. its axioms are all propositional tautologies (Axiom 1) plus the axioms 2> (Axiom 2) and 2φ∧2ψ → 2(φ∧ψ) (Axiom 3), and the rules modus ponens and φ→ ψ 2φ→ 2ψ We have to show that for all formulas φ `K φ ⇔ `L φ. ⇒: For this direction we have to show that L derives all axioms of K and all its rules. Axiom 1 of K is the same as Axiom 1 in L, thus we have no...

KEY WORDS Contextual spaces, context relation, contextual rough sets, elements of a contextual rough set, the counterpart of the axiom of extensionality. ABSTRACT This paper originates from the conceptions of rough sets presented by Pawlak (1982, 1992). It also refers to Ziarko's conception (1993) and the conceptions of Blizard's multisets (1989a, 1989b), Zadeh's fuzzy sets (1965), and the auth...

§9.1 The Axiom of Choice We come now to the most important part of set theory for other branches of mathematics. Although infinite set theory is technically the foundation for all mathematics, in practice it is perfectly valid for a mathematician to ignore it – with two exceptions. Of course most of the basic set constructions as outlined in chapter 2 (unions, intersections, cartesian products,...

Any two infinite-dimensional (separable) Hilbert spaces are unitarily isomorphic. The sets of all their self-adjoint operators are also therefore unitarily equivalent. Thus if all self-adjoint operators can be observed, and if there is no further major axiom in quantum physics than those formulated for example in Dirac’s ‘Quantum Mechanics’, then a quantum physicist would not be able to tell a ...