2 research outputs found
Hilbert's epsilon as an Operator of Indefinite Committed Choice
Paul Bernays and David Hilbert carefully avoided overspecification of
Hilbert's epsilon-operator and axiomatized only what was relevant for their
proof-theoretic investigations. Semantically, this left the epsilon-operator
underspecified. In the meanwhile, there have been several suggestions for
semantics of the epsilon as a choice operator. After reviewing the literature
on semantics of Hilbert's epsilon operator, we propose a new semantics with the
following features: We avoid overspecification (such as right-uniqueness), but
admit indefinite choice, committed choice, and classical logics. Moreover, our
semantics for the epsilon supports proof search optimally and is natural in the
sense that it does not only mirror some cases of referential interpretation of
indefinite articles in natural language, but may also contribute to philosophy
of language. Finally, we ask the question whether our epsilon within our
free-variable framework can serve as a paradigm useful in the specification and
computation of semantics of discourses in natural language.Comment: ii + 73 pages. arXiv admin note: substantial text overlap with
arXiv:1104.244
Fermat’s last theorem proved in Hilbert arithmetic. I. From the proof by induction to the viewpoint of Hilbert arithmetic
In a previous paper, an elementary and thoroughly arithmetical proof of Fermat’s last theorem by induction has been demonstrated if the case for “n = 3” is granted as proved only arithmetically (which is a fact a long time ago), furthermore in a way accessible to Fermat himself though without being absolutely and precisely correct. The present paper elucidates the contemporary mathematical background, from which an inductive proof of FLT can be inferred since its proof for the case for “n = 3” has been known for a long time. It needs “Hilbert mathematics”, which is inherently complete unlike the usual “Gödel mathematics”, and based on “Hilbert arithmetic” to generalize Peano arithmetic in a way to unify it with the qubit Hilbert space of quantum information. An “epoché to infinity” (similar to Husserl’s “epoché to reality”) is necessary to map Hilbert arithmetic into Peano arithmetic in order to be relevant to Fermat’s age. Furthermore, the two linked semigroups originating from addition and multiplication and from the Peano axioms in the final analysis can be postulated algebraically as independent of each other in a “Hamilton” modification of arithmetic supposedly equivalent to Peano arithmetic. The inductive proof of FLT can be deduced absolutely precisely in that Hamilton arithmetic and the pransfered as a corollary in the standard Peano arithmetic furthermore in a way accessible in Fermat’s epoch and thus, to himself in principle. A future, second part of the paper is outlined, getting directed to an eventual proof of the case “n=3” based on the qubit Hilbert space and the Kochen-Specker theorem inferable from it