The Essence of Intuitive Set Theory

Intuitive Set Theory (IST) is defined as the theory we get, when we add Axiom of Monotonicity and Axiom of Fusion to Zermelo-Fraenkel set theory. In IST, Continuum Hypothesis is a theorem, Axiom of Choice is a theorem, Skolem paradox does not appear, nonLebesgue measurable sets are not possible, and the unit interval splits into a set of infinitesimals

Zermelo in the mirror of the Baer correspondence, 1930–1931

AbstractAround 1931 Zermelo had an extended correspondence with the young Reinhold Baer concerning the edition of Cantor's collected works. Some of the letters also deal with Skolem's paradox and Gödel's first incompleteness theorem. Whereas Zermelo's letters are lost, most of Baer's letters are contained in the Zermelo Nachlass. Besides giving insight into Zermelo's reaction to Skolem's and Gödel's results, the letters also demonstrate Baer's clear understanding of the behavior of models of set theory and of the relevance of Gödel's first incompleteness theorem

Ernst Zermelo's Project of Infinitary Logic

This paper is a summary of a more comprehensive work Infinitarna Logika Ernsta Zermela (The Infinitary Logic of Ernst Zermelo) being currently under preparation for the research grant KBN 2H01A 00725 Metody nieskończonościowe w teorii definicji (Infinitary methods in the theory of definitions) headed by Professor JANUSZ CZELAKOWSKI at the Institute of Mathematics and Information Science of the University of Opole, Poland. The presentation of Zermelo's ideas is accompanied with some remarks concerning the development of infinitary logic.This paper is a summary of a more comprehensive work Infinitarna Logika Ernsta Zermela (The Infinitary Logic of Ernst Zermelo) being currently under preparation for the research grant KBN 2H01A 00725 Metody nieskończonościowe w teorii definicji (Infinitary methods in the theory of definitions) headed by Professor JANUSZ CZELAKOWSKI at the Institute of Mathematics and Information Science of the University of Opole, Poland. The presentation of Zermelo's ideas is accompanied with some remarks concerning the development of infinitary logic.

Modal Set Theory

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

On the Axiom of Canonicity

The axiom of canonicity was introduced by the famous Polish logician Roman Suszko in 1951 as an explication of Skolem's Paradox (without reference to the L\"{o}wenheim-Skolem theorem) and a precise representation of the axiom of restriction in set theory proposed much earlier by Abraham Fraenkel. We discuss the main features of Suszko's contribution and hint at its possible further applications

To the Beat of Different Drumer....Freedom, Anarchy and Conformism in Research

In this paper I attempt to make a case for promoting the courage of rebels within the citadels of orthodoxy in academic research environments. Wicksell in Macroeconomics, Brouwer in the Foundations of Mathematics,Turing in Computability Theory, Sraffa in the Theories of Value and Distribution are, in my own fields of research, paradigmatic examples of rebels, adventurers and non-conformists of the highest calibre in scientific research within University environments. In what sense, and how, can such rebels, adventurers and nonconformists be fostered in the current University research environment dominated by the cult of picking winners? This is the motivational question lying behind the historical outlines of the work of Wicksell, Brouwer, Hilbert, Bishop, Veronese, Gödel, Turing and Sraffa that I describe in this paper. The debate between freedom in research and teaching and the naked imposition of correct thinking, on potential dissenters of the mind, is of serious concern in this age of austerity of material facilities. It is a debate that has occupied some the finest minds working at the deepest levels of foundational issues in mathematics, metamathematics and economic theory. By making some of the issues explicit, I hope it is possible to encourage dissenters to remain courageous in the face of current dogmas.Non-conformist research, macroeconomics, foundations of mathematics, intuitionism, constructivism, formalism, Hilbertís Dogma, Hilbertís Program, computability theory