Axioms of Set Theory

Object x being a member of itself or not and x being a member of R or not constitute two vastly different concepts. This paper attempts to locate the reflection of such an utter difference within the formal structure of the axiom of extensionality. -/- .

A collection of twenty-three essays examining the philosophical themes of the animated cartoon show about life in the year 3000, Futurama.

¿Qué ha pasado con el problema del cardinal del continuo después de Gödel (1938) y Cohen (1964)? Intentos de responder esta pregunta pueden encontrarse en los artículos de José Alfredo Amor (1946-2011), "El Problema del continuo después de Cohen (1964-2004)", de Carlos Di Prisco , "Are we closer to a solution of the continuum problem", y de Joan Bagaria, "Natural axioms of set and the continuum problem" , que se pueden encontrar en la biblioteca digital de mi blog de Lógica (...) Matemática y Fundamentos de la Matemática (ver). También en la entrada "The Continuum Hypothesis" de la web de Enciclopedia de Filosofía de la Universidad de Stanford existe información importante y actualizada al respecto. En esta breve nota se comenta sobre el tema de una manera divulgativa. (shrink)

ABSTRACT We shall defend three philosophical theses about the extent of intrinsic justification based on various technical results. We shall present a set of theorems which indicate intriguing structural similarities between a family of “weak” reflection principles roughly at the level of those considered by Tait and Koellner and a family of “strong” reflection principles roughly at the level of those of Welch and Roberts, which we claim to lend support to the view that the stronger reflection principles are intrinsically (...) justified as well as the weaker ones. We consider connections with earlier work of Marshall. (shrink)

This paper presents and defends an argument that the continuum hypothesis is false, based on considerations about objective chance and an old theorem due to Banach and Kuratowski. More specifically, I argue that the probabilistic inductive methods standardly used in science presuppose that every proposition about the outcome of a chancy process has a certain chance between 0 and 1. I also argue in favour of the standard view that chances are countably additive. Since it is possible to randomly pick (...) out a point on a continuum, for instance using a roulette wheel or by flipping a countable infinity of fair coins, it follows, given the axioms of ZFC, that there are many different cardinalities between countable infinity and the cardinality of the continuum. (shrink)

We present an extension of constructive Zermelo{Fraenkel set theory [2]. Constructive sets are endowed with an applicative structure, which allows us to express several set theoretic constructs uniformly and explicitly. From the proof theoretic point of view, the addition is shown to be conservative. In particular, we single out a theory of constructive sets with operations which has the same strength as Peano arithmetic.

In set theory without the Axiom of Choice, we investigate the set-theoretic strength of the principle NDS which states that there is no function f on the set ω of natural numbers such that for everyn ∈ ω, f ≺ f, where for sets x and y, x ≺ y means that there is a one-to-one map g : x → y, but no one-to-one map h : y → x. It is a long standing open problem whether NDS implies (...) AC. In this paper, among other results, we show that NDS is a strong axiom by establishing that ACLO ↛ NDS in ZFA set theory. The latter result provides a strongly negative answer to the question of whether “every Dedekind-finite set is finite” implies NDS addressed in G. H. Moore “Zermelo’s Axiom of Choice. Its Origins, Development, and Influence” and in P. Howard–J. E. Rubin “Consequences of the Axiom of Choice”. We also prove that ACWO ↛ NDS in ZF and that “for all infinite cardinals m, m + m = m” ↛ NDS in ZFA. (shrink)

This article presents an overview of the basic philosophical motivations for, and some recent work in, modal set theory.

1994.Επιστημολογία της Λογικής. Συγγραφέας Επαμεινώνδας Ξενόπουλος Μοναδική μελέτη και προσέγγιση της θεωρίας της γνώσης, για την παγκόσμια βιβλιογραφία, της διαλεκτικής πορείας της σκέψης από την λογική πλευρά της και της μελλοντικής μορφής που θα πάρουν οι διαλεκτικές δομές της, στην αδιαίρετη ενότητα γνωσιοθεωρίας, λογικής και διαλεκτικής, με την «μέθοδο του διαλεκτικού υλισμού». Έργο βαρύ με θέμα εξαιρετικά δύσκολο διακατέχεται από πρωτοτυπία και ζωντάνια που γοητεύει τον κάθε ανήσυχο στοχαστή από τις πρώτες γραμμές.

The Necessary Maximality Principle for c. c. c. forcing with real parameters is equiconsistent with the existence of a weakly compact cardinal. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim).

The results of this paper concern the effective cardinal structure of the subsets of [ω1]<ω1, the set of all countable subsets of ω1. The main results include dichotomy theorems and theorems which show that the effective cardinal structure is complicated.