Uniqueness Violations

Int. J. Contemp. Math. Sciences, 2011

A relatively complete and subjective attitude towards the axiomatic set theory which was achieved by mathematician, such as Cantor-Zermelo and Fraenkel resulted in the presentation of ten Zermelo-Fraenkel axioms and proposed Cantor and Russell , s paradoxes. In this paper we have tried to remove some proposed paradoxes by defining an exclusive set which we call universal set.

Social Science Research Network, 2003

If we apply an extension of the Deduction meta-Theorem to Gödel's meta-reasoning of "undecidability", we can conclude that Gödel's formal system of Arithmetic is not omega-consistent. If we then interpret [(Ax)F(x)] as "There is a general, xindependent, routine to establish that F(x) holds for all x", instead of as "F(x) holds for all x", it follows that a constructively interpreted omega-inconsistent system proves Hilbert's Entscheidungsproblem negatively. 1.1 Notation We generally follow the notation of Gödel [Go31a]. However, we use the notation "(Ax)", which classically interprets as "for all x", to denote Gödel's special symbolism for Generalisation. We use square brackets to indicate that the expression (including square brackets) only denotes the string 2 named within the brackets. Thus, "[(Ax)]" is not part of the formal system P, and would be replaced by Gödel's special symbolism for Generalisation in order to obtain the actual string in which it occurs. Following Gödel's definitions of well-formed formulas 3 , we note that juxtaposing the string "[(Ax)]" and the formula 4 "[F(x)]" is the formula "[(Ax)F(x)]", juxtaposing the 1 The author is an independent scholar.

This work is partially motivated by the recent articles related to the continuum hypothesis and the forcing axioms. Ever since Gödel's and Cohen's proofs that the ZFC cannot be used to disprove or approve continuum hypothesis, i.e., that it is independent of the ZFC axioms, there have been efforts to add at least one new axiom to ZFC that should illuminate the structure of infinite sets. The two main contenders for addition to ZFC are the inner-model axiom 'V-ultimate L' that builds a universe of sets from the ground up and Martin's axiom that allows to expand it outward in all directions. If Martin's axiom is correct, then the continuum hypothesis is false, and if the inner-model axiom is right, then the continuum hypothesis is true. There is also an orthogonal view assuming that there are myriad mathematical universes, some in which the continuum hypothesis is true and others in which it is false, but all equally worth of exploring. That model dependent view is underlined with the Löwenheim-Skolem theorem, no theory can have only nondenumerable models; if a theory has a nondenumerable model, it must have denumerably infinite ones as well. Accepting axiomatization in whatever shape means the notions 'denumerable' and 'nondenumerable' turn out to be relative, and it requires to accept the relativization of cardinals Skolem and Carnap first point it out. To show the impact of axiomatization, consider will be as an example a set created by a successive generation of algebraic pairs, each filled with a single transcendental number, and examined will be the impact of the axiom of choice on the determining the property of such set. It will be shown that the given example supports the Löwenheim-Skolem theorem. To underline the uncertainty implied by that theorem considered will be Wang's Σ model that generates a set in a constructive way, different from ZF. For the first time, provided will be an explicit example that satisfies Wang's model. In our example, a set of algebraic numbers will correspond to the 0-th Wang's layer, the 1-st layer will be a set of transcendental numbers generated by diagonal cuts applied to the 0 − th layers, and successive layers will be produced by diagonal cuts applied on the previously generated union of layers. It will be shown that this model also supports the Löwenheim-Skolem theorem.

2020

Those incompleteness theorems mean the relation of (Peano) arithmetic and (ZFC) set theory, or philosophically, the relation of arithmetical finiteness and actual infinity. The same is managed in the framework of set theory by the axiom of choice (respectively, by the equivalent well-ordering "theorem'). One may discuss that incompleteness form the viewpoint of set theory by the axiom of choice rather than the usual viewpoint meant in the proof of theorems. The logical corollaries from that "nonstandard" viewpoint the relation of set theory and arithmetic are demonstrated.

tribute to Professor Fernando Campello, PhD, 2018

Abstract: In this short note it is shown that Fernando Campello, PhD, does not obey the Peano axioms for arithmetic counting over the natural set. It is also shown that any infinite researcher is isomorphic to F. Campello. Résumé: Dans cette courte note, on montre que Fernando Campello, PhD, n'obéit guère à l'axiome du comptage. On montre également que tout chercheur infini est isomorphe à F. Campello. Zusammenfassung: In dieser kurzen Notiz wird gezeigt, dass Fernando Campello, PhD, dem Axiom des Grafen nicht gehorcht. Es wird auch gezeigt, dass jeder unendliche Forscher isomorph zu F. Campello ist. Sommario: In questa breve nota è dimostrato che Fernando Campello, PhD, non obbedisce all'assioma del conteggio. Viene anche mostrato che qualsiasi ricercatore infinito è isomorfo di F. Campello. Summarium: In hoc opusculum, Ferdinandus Campellus, PhD, axioma arithmeticam Peanesem non sufficere est demonstratus. Atque Campellus infinita copia peritorum isomorphicus est. Do not ask whether a statement is true until you know what it means. – Errett Bishop.

Journal of Philosophical Logic, 1996

In his new introduction to the 1925 second edition of Principia Mathematica, Russell maintained that by adopting Wittgenstein's idea that a logically perfect language should be extensional mathematical induction could be rectified for finite cardinals without the axiom of reducibility. In an Appendix B, Russell set forth a proof. Gödel caught a defect in the proof at *89.16, so that the matter of rectification remained open. Myhill later arrived at a negative result: Principia with extensionality principles and without reducibility cannot recover mathematical induction. The finite cardinals are indefinable in it. This paper shows that while Gödel and Myhill are correct, Russell was not wrong. The 1925 system employs a different grammar than the original Principia. A new proof for *89.16 is given and induction is recovered.

Axioms

The systems of arithmetic discussed in this work are non-elementary theories. In this paper, natural numbers are characterized axiomatically in two different ways. We begin by recalling the classical set P of axioms of Peano’s arithmetic of natural numbers proposed in 1889 (including such primitive notions as: set of natural numbers, zero, successor of natural number) and compare it with the set W of axioms of this arithmetic (including the primitive notions like: set of natural numbers and relation of inequality) proposed by Witold Wilkosz, a Polish logician, philosopher and mathematician, in 1932. The axioms W are those of ordered sets without largest element, in which every non-empty set has a least element, and every set bounded from above has a greatest element. We show that P and W are equivalent and also that the systems of arithmetic based on W or on P, are categorical and consistent. There follows a set of intuitive axioms PI of integers arithmetic, modelled on P and propos...

In this paper, we consider significant alternative systems of first order predicate calculus. We consider systems where the meta-assertion "PA proves: F(x)" translates under interpretation as "F(x) is satisfied for all values of x, in the domain of the interpretation, that can be formally represented in PA", whilst "PA proves:

One of the most important achievements of the last century is the knowledge of the existence of non-countable sets. The proof by Cantor's diagonal method requires the assumption of actual infinity. By two paradoxes we show that this method sometimes proves nothing because of it can involve self-referential definitions. To avoid this inconvenient, we introduce another proving method based upon the information in the involved object definitions. We also introduce the concepts of indiscernible mathematical construction and sub-cardinal. In addition, we show that the existence of indiscernible mathematical constructions is an unavoidable consequence of uncountability.

2007