The Relationship of Arithmetic As Two Twin Peano Arithmetic(s) and Set Theory: A New Glance From the Theory of Information

The paper introduces and utilizes a few new concepts: “nonstandard Peano arithmetic”, “complementary Peano arithmetic”, “Hilbert arithmetic”. They identify the foundations of both mathematics and physics demonstrating the equivalence of the newly introduced Hilbert arithmetic and the separable complex Hilbert space of quantum mechanics in turn underlying physics and all the world. That new both mathematical and physical ground can be recognized as information complemented and generalized by quantum information. A few fundamental mathematical problems of the present such as Fermat’s last theorem, four-color theorem as well as its new-formulated generalization as “four-letter theorem”, Poincaré’s conjecture, “P vs NP” are considered over again, from and within the new-founding conceptual reference frame of information, as illustrations. Simple or crucially simplifying solutions and proofs are demonstrated. The link between the consistent completeness of the system mathematics-physics on the ground of information and all the great mathematical problems of the present (rather than the enumerated ones) is suggested

Infinity and Continuum in the Alternative Set Theory

Alternative set theory was created by the Czech mathematician Petr Vop\v enka in 1979 as an alternative to Cantor's set theory. Vop\v enka criticised Cantor's approach for its loss of correspondence with the real world. Alternative set theory can be partially axiomatised and regarded as a nonstandard theory of natural numbers. However, its intention is much wider. It attempts to retain a correspondence between mathematical notions and phenomena of the natural world. Through infinity, Vop\v enka grasps the phenomena of vagueness. Infinite sets are defined as sets containing proper semisets, i.e. vague parts of sets limited by the horizon. The new interpretation extends the field of applicability of mathematics and simultaneously indicates its limits. This incidentally provides a natural solution to some classic philosophical problems such as the composition of a continuum, Zeno's paradoxes and sorites. Compared to strict finitism and other attempts at a reduction of the infinite to the finite Vop\v enka's theory reverses the process: he models the finite in the infinite.Comment: 25 page

Infinity and Continuum in the Alternative Set Theory

Alternative set theory was created by the Czech mathematician Petr Vopěnka in 1979 as an alternative to Cantor's set theory. Vopěnka criticised Cantor's approach for its loss of correspondence with the real world. Alternative set theory can be partially axiomatised and regarded as a nonstandard theory of natural numbers. However, its intention is much wider. It attempts to retain a correspondence between mathematical notions and phenomena of the natural world. Through infinity, Vopěnka grasps the phenomena of vagueness. Infinite sets are defined as sets containing proper semisets, i.e. vague parts of sets limited by the horizon. The new interpretation extends the field of applicability of mathematics and simultaneously indicates its limits. Compared to strict finitism and other attempts at a reduction of the infinite to the finite Vopenka's theory reverses the process: he models the finite in the infinite

Reconstructions of science

'In vier Rekonstruktionen wird versucht, die natur- und sozialwissenschaftliche Diskussion der Basiskonzepte von Raum und Zeit zu vereinen. Dazu bedarf es einer neuen Diskursform, die bereits in Alfred North Whiteheads Naturphilosophie anklingt. Auf sozial-wissenschaftlicher Seite besinnen wir uns grundlegender Themen von Sozialphänomenologie, Strukturalismus und Interaktionismus. Fragestellungen von PrähistorikerInnen, ÄgyptologInnen und EthnomathematikerInnen werden wichtig, wo wir zeigen, daß unsere Konzepte von Raum und Zeit kulturelle Institutionen der Bedeutung sind, die ihrerseits Gesellschaft konstituieren und konstanter Rekonstruktion bedürfen. Die vierte Rekonstruktion greift die Frage der theoretischen Physik auf und stützt sich auf das integrative Instrument der Theorie der geometrischen Clifford Algebren. Wir leiten ab, daß und wie die inneren Symmetrien der Materie mit den äußeren Symmetrien der Raum-Zeit verbunden sind und daß die Metapher vom 'achtfachen Pfad', die Gell-Mann für einen Teil des Standardmodells verwendete, entgegen seiner Auffassung nicht als Witz zu verstehen ist. Der Faktor (D4)m in der Dirac-Gruppe jeder geometrischen Clifford Algebra C/p,q bildet eine Grundstruktur von Orientierung und Logik ab und korrespondiert daher mit einem Interface zwischen Geist und Materie.' (Autorenreferat)'In four reconstructions it is attempted to lead the natural and social science debate of the basis concepts of space and time in common. For this we need a new mode of science discourse which has already been initiated in Alfred North Whitehead's philosophy of nature. In social science we reconsider the basis themes of social phenomenology, structuralism and interactionism as far as those contribute to a space-time topic. Investigations of prehistorians, egyptologists and ethno-mathematicians are of importance where we demonstrate that our concepts of space and time represent cultural institutions of meaning which on their part constitute society and require that we constantly reconstruct them. The fourth reconstruction deals with the space-time of postmodern theoretical physics and is founded on the integrative instrument of the theory of geometric Clifford algebras. We show that and how the inner symmetrics of matter are connected with the outer symmetries of space-time and that Gell-Mann's metaphor of the 'eightfold path' that he used to denote part of the standard model of physics cannot be interpreted as quirk, in opposition to his own intention. The factor (D4)m in the Dirac group of any geometric Clifford Algebra C/p,q represents a ground template (or archetypal structure) for both orientation and logic and corresponds therefore with an interface between matter and mind.' (author's abstract)

Countable Short Recursively Saturated Models of Arithmetic

Short recursively saturated models of arithmetic are exactly the elementary initial segments of recursively saturated models of arithmetic. Since any countable recursively saturated model of arithmetic has continuum many elementary initial segments which are already recursively saturated, we turn our attention to the (countably many) initial segments which are not recursively saturated. We first look at properties of countable short recursively saturated models of arithmetic and show that although these models cannot be cofinally resplendent (an expandability property slightly weaker than resplendency), these models have non-definable expansions which are still short recursively saturated

On the Categoricity of Quantum Mechanics

The paper offers an argument against an intuitive reading of the Stone-von Neumann theorem as a categoricity result, thereby pointing out that, against what is usually taken to be the case, this theorem does not entail any model-theoretical difference between the theories that validate it and those that don't.Comment: 20 page

Gödel mathematics versus Hilbert mathematics. I. The Gödel incompleteness (1931) statement: axiom or theorem?

The present first part about the eventual completeness of mathematics (called “Hilbert mathematics”) is concentrated on the Gödel incompleteness (1931) statement: if it is an axiom rather than a theorem inferable from the axioms of (Peano) arithmetic, (ZFC) set theory, and propositional logic, this would pioneer the pathway to Hilbert mathematics. One of the main arguments that it is an axiom consists in the direct contradiction of the axiom of induction in arithmetic and the axiom of infinity in set theory. Thus, the pair of arithmetic and set are to be similar to Euclidean and non-Euclidean geometries distinguishably only by the Fifth postulate now, i.e. after replacing it and its negation correspondingly by the axiom of finiteness (induction) versus that of finiteness being idempotent negations to each other. Indeed, the axiom of choice, as far as it is equivalent to the well-ordering “theorem”, transforms any set in a well-ordering either necessarily finite according to the axiom of induction or also optionally infinite according to the axiom of infinity. So, the Gödel incompleteness statement relies on the logical contradiction of the axiom of induction and the axiom of infinity in the final analysis. Nonetheless, both can be considered as two idempotent versions of the same axiom (analogically to the Fifth postulate) and then unified after logicism and its inherent intensionality since the opposition of finiteness and infinity can be only extensional (i.e., relevant to the elements of any set rather than to the set by itself or its characteristic property being a proposition). So, the pathway for interpreting the Gödel incompleteness statement as an axiom and the originating from that assumption for “Hilbert mathematics” accepting its negation is pioneered. A much wider context relevant to realizing the Gödel incompleteness statement as a metamathematical axiom is consistently built step by step. The horizon of Hilbert mathematics is the proper subject in the third part of the paper, and a reinterpretation of Gödel’s papers (1930; 1931) as an apology of logicism as the only consistent foundations of mathematics is the topic of the next second part