Abstract. This is a non-standard paper, containing some problems in set theory I have in various degrees been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems are presented with a discussion of how I have tried to solve them, and sometimes with failed tries, anecdote and opinion. So the discussion is quite personal, in other words, egocentric and somewhat accidental. As we discuss many problems, history and side references are erratic, usually kept at a minimum (“see... ” means: see the references there and possibly the paper itself). The base were lectures in Rutgers Fall ’97 and reflect my knowledge then. The other half, [122], concentrating on model theory, will subsequently appear. I thank Andreas Blass and Andrzej Ros̷lanowski for many helpful comments. (666) revision:2001-11-12 modified:2003-11-18

Abstract. This paper is an introduction to forcing axioms and large cardinals. Specifically, we shall discuss the large cardinal strength of forcing axioms in the presence of regularity properties for projective sets of reals. The new result shown in this paper says that ZFC + the bounded proper forcing axiom (BPFA) + “every projective set of reals is Lebesgue measurable ” is equiconsistent with ZFC + “there is a Σ1 reflecting cardinal above a remarkable cardinal.” 1. Introduction. The current paper ∗ is in the tradition of the following result.

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The Ostrowski theorem in question is that an additive function bounded (above, say) on a set T of positive measure is continuous. In the converse direction, recall that a topological space T is pseudocompact if every function continuous on T is bounded. Thus theorems of ‘converse Ostrowski’ type relate to ‘additive (pseudo)compactness’. We give a different characterization of such sets, in terms of the property of ‘generic subuniversality’, arising from the Kestelman-Borwein-Ditor theorem and relate these to various new forms of compactness.

The aim of this paper is to present a detailed explanation of three models of Shelah. We show the rule of the amalgamation in the construction of models in which all definable sets of reals have Baire property or are Lebesgue measurable. Next we construct a model in which every projective set of reals has Baire property, a model with the Uniformization Property and a model of ZF + DC in which all subsets of the real line are Lebesgue measurable but there is a set without Baire property.

This article, based on an invited lecture at the Logic Colloquium '93 in Keele, is a sequel to van Lambalgen [1992]. Apart from presenting new results, it differs from its predecessor in the following respects: (i) the presentation of the axioms is simplified, following some suggestions of Wojciech Buszkowski, (ii) the axioms have been strengthened, and (iii) the philosophical discussion has (hopefully) been improved. The article has appeared in W. Hodges et al (eds.), Logic: from Foundations to Applications (European Logic Colloquium), Oxford University Press 1996

Abstract. We show that Σ 1 3-absoluteness under Sacks forcing is equivalent to the Sacks measurability of every ∆ 1 2 set of reals. We also show that Sacks forcing is the weakest forcing notion among all of the preorders which always add a new real with respect to Σ 1 3 forcing absoluteness. 1.

Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a far-away planet. Would their mathematics be set-based? What are the alternatives to the set-theoretic foundation of mathematics? Besides, set theory seems to play a significant role in computer science; is there a good justification for that? We discuss these and some related issues.

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Abstract. We continue investigations of forcing notions with strong ccc properties introducing new methods of building sweet forcing notions. We also show that quotients of topologically sweet forcing notions over Cohen reals are topologically sweet while the quotients over random reals do not have to be such. 856 revision:2005-10-27 modified:2005-10-27 One of the main ingredients of the construction of the model for all projective sets of reals have the Baire property presented in Shelah [7, §7] was a strong ccc property of forcing notions called sweetness. This property is preserved in amalgamations and also in compositions with the Hechler forcing notion D and the Universal Meager