arXiv subjects

Werner DePauli-Schimanovich

Publications and source records attributed to Werner DePauli-Schimanovich.

7 recordsLinked to original sources

arXiv2008

Paradigms-Shift in Set Theory

In this article the author claims that there is a paradigm shift from ZFC to NFUM and further to NACT - due to philosophical reasons, not mathematical ones. The goal is to construct systems where every "Not-Properclass" is a set! With help of Non-Monotonic Logic, the consistent systems NACT-MoonW, NACT*W, and NACT-SunW are producing "largest possible universes" of sets. Using self-evident philosophical principles, three approximations are suggested: NACT+NFUM-closed (to NACT*), NACT&ZFC4+(GCH) (also to NACT*) and NACT-NFUM (to NACT-Sun). Also the system NACT[&ZFC4-closed]+(FCA) is considered. NFUM = NFU with (AC) and measurable properclass Ord. NFUM-closed is NFUM where the set-constituting formulas Ai in the set operator need not only to be stratified but also to be made parameter-free (i.e., have only x as free variable over which the set is comprehended). In other formulas, free variables are allowed. Keywords: Set Theory, ZFC, Naive Set Theory, Predicate-Extension, Church Schema of Comprehension, NF, NFUM, Stratification, Universal Sets, Eliminative Class Theory, NBG, NACT (= Naive Axiomatic Class Theory), NACT-Sun, NACT-Moon, NACT-Star, NACT+NFUM-closed, ZFCK, NACT+NFUM, (GCH), (FCA) [= Finite or Countable Anti-Thesis], NACT+ZFC4+(GCH), NACT[+ZFC4-[closed]+(FCA).

math.LO↗

arXiv2008

ZFK := ZFC with a Complement, or: Hegel and the Synto-Set-Theory

What is the slightest modification of ZF to add a complement-axiom? The answer in my Ph.D. thesis 1971 was ZF'': Zermelo-Fraenkel with replacement for only well-founded domains and an omega-axiom. In 1974, Alonzo Church published a similar system, as did Urs Oswald in 1976. In his 1976 Ph.D. thesis, E. Mitchell also designed, a system very similar to ZF''. In this article we argue that ZF'' is the slightest modification of ZF among all these systems and that its successor ZFK is the simplest set theory with a universal set at all.

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arXiv2008

Higher Cardinals are only a Convention

Zermelo's Axiom of Separation is: Exist x: Forall y: (y in x <==> y in a & E(y)) with definite(E) and parameter a. Thoralf Skolem suggested to characterize the terminus "definite" by "the property E should be representable by a FOL formula". But that is trivial. "definite" must mean more. The author claims that "definite" means "in accordance with the theory of definitions of logic". In this case the theorem of Cantor is no longer a theorem, but a undecidable sentence, and has to be established explicitly as axiom. This is not done by the community, but it is made a silent assumption that we can drop the appendix "definite(E)" from the axiom of separation at all. But this is a convention (even when it is silent) and it is nothing else than an axiom.

math.LO↗

arXiv2008

A Brief History of Future Set Theory

Mathematicians still use Naive Set Theory when generating sets without danger of producing any contradiction. Therefore their working method can be considered as a consistent inference system with an experience of over 100 years. My conjecture is that this method works well because mathematicians use only those predicates to form sets, which yield closed hereditary consistent predicate extensions. And for every open formula they use in the process of constructing of a certain (special) set (bottom up), we can always find an "almost-closed" formula (i.e. a parameter-free formula with only the free variable "x") which yields the same certain (special) set as predicate extension as constructed in the bottom up process before. Therefore the use of predicates with free parameters in the Comprehension Scheme does not cause any difficulties and can be "lifted" by meta-mathematical considerations. KEYWORDS: naive set theory, Quine, new foundation, NF, universal sets, comprehension schema, predicate extension, philosophy of set theory, Zermelo, Fraenkel, ZF, complement. CT is the Class-Theoretical frame: logic + "=" + Church Schema + Extensionality Axiom.

math.LO↗

arXiv2008

The Notion "Pathology" in Set Theory

When we study the paradoxes of set theory we find out that there are mainly 2 types: the pathologies and the antinomies. These 2 notions are made precise and compared with the somehow inductively definable concept "abnormal". (See my paper "Naive Axiomatic Mengenlehre for Experiments" in arXiv.) In the following 5 Patho Theses are discussed in order to formalize this notion of pathology. This allows us to define formally the property "Hereditary-non-Pathological" for well-formed formulas. With this property the system NACT* of Naive Axiomatic Class Theory is constructed, which has a "unique maximal" universe (in a special sense).

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arXiv2008

Naive Axiomatic Class Theory: A Solution for the Antinomies of Naive Mengenlehre

Since the axioms in (Consi-CoS) are not recursively enumerable, NACT* is no axiom system in the classical sense . Therefore we construct a series of partial systems which form a recursive axiom system too. Starting with the "dichotomic" systems NACT# and its variant NACT#4, we are going on to the "disjunctive" systems NACT+ and NACT+4, and eventually to NACT+Strat. After that we discuss the medium classes of these systems. Finally we present the inconsistent NSA-systems based on Not-SelfApplicability and explain their help for computational set theory.

math.LO↗

arXiv2008

Naive Axiomatic Mengenlehre for Experiments

The main goal of "Naive Axiomatic Mengenlehre" (NAM) is to find a more or less adequately explicit criterion that precisely formalizes the intuitive notion of a "normal set". NAM is mainly a construction procedure for building several formal systems NAMix, each of which can turn out to be an adequate codification of the contentual naive set theory. ("i" is a natural number which enumerates the used "normality" condition, and "x" is a letter which points to the variants of the used axioms.) Parallel to NAM, the Naive Axiomatic Class Theory NACT is constructed as a system of systems too.

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