15 found
  1. Analysis without actual infinity.Jan Mycielski - 1981 - Journal of Symbolic Logic 46 (3):625-633.
    We define a first-order theory FIN which has a recursive axiomatization and has the following two properties. Each finite part of FIN has finite models. FIN is strong enough to develop that part of mathematics which is used or has potential applications in natural science. This work can also be regarded as a consistency proof of this hitherto informal part of mathematics. In FIN one can count every set; this permits one to prove some new probabilistic theorems.
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  2. A Lattice of Chapters of Mathematics.Jan Mycielski, Pavel Pudlák, Alan S. Stern & American Mathematical Society - 1990 - American Mathematical Society.
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  3.  59
    Locally finite theories.Jan Mycielski - 1986 - Journal of Symbolic Logic 51 (1):59-62.
    We say that a first order theoryTislocally finiteif every finite part ofThas a finite model. It is the purpose of this paper to construct in a uniform way for any consistent theoryTa locally finite theory FIN which is syntactically isomorphic toT.Our construction draws upon the main idea of Paris and Harrington [6] and generalizes the syntactic aspect of their result from arithmetic to arbitrary theories. The first mathematically strong locally finite theory, called FIN, was defined in [1]. Now we get (...)
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  4. A Mathematical Axiom Contradicting the Axiom of Choice.Jan Mycielski, H. Steinhaus & S. Swierczkowski - 1971 - Journal of Symbolic Logic 36 (1):164-166.
  5.  32
    A lattice of interpretability types of theories.Jan Mycielski - 1977 - Journal of Symbolic Logic 42 (2):297-305.
  6.  23
    New set-theoretic axioms derived from a lean metamathematics.Jan Mycielski - 1995 - Journal of Symbolic Logic 60 (1):191-198.
  7.  34
    Quantifier-free versions of first order logic and their psychological significance.Jan Mycielski - 1992 - Journal of Philosophical Logic 21 (2):125 - 147.
  8.  18
    Annual Meeting of the Association for Symbolic Logic Denver, 1983.Carl G. Jockusch, Richard Laver, Donald Monk, Jan Mycielski & Jon Pearce - 1984 - Journal of Symbolic Logic 49 (2):674 - 682.
  9.  32
    Canonical universes and intuitions about probabilities.Randall Dougherty & Jan Mycielski - 2006 - Dialectica 60 (4):357–368.
  10.  15
    Strong measure zero and infinite games.Fred Galvin, Jan Mycielski & Robert M. Solovay - 2017 - Archive for Mathematical Logic 56 (7-8):725-732.
    We show that strong measure zero sets -totally bounded metric space) can be characterized by the nonexistence of a winning strategy in a certain infinite game. We use this characterization to give a proof of the well known fact, originally conjectured by K. Prikry, that every dense \ subset of the real line contains a translate of every strong measure zero set. We also derive a related result which answers a question of J. Fickett.
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    Meeting of the association for symbolic logic, Dallas 1973.J. Donald Monk, Jan Mycielski & Jürgen Schmidt - 1973 - Journal of Symbolic Logic 38 (3):541-549.
  12.  39
    On the tension between Tarski's nominalism and his model theory (definitions for a mathematical model of knowledge).Jan Mycielski - 2004 - Annals of Pure and Applied Logic 126 (1-3):215-224.
    The nominalistic ontology of Kotarbinski, Slupecki and Tarski does not provide any direct interpretations of the sets of higher types which play important roles in type theory and in set theory. For this and other reasons I will interpret those theories as descriptions of some finite structures which are actually constructed in human imaginations and stored in their memories. Those structures will be described in this lecture. They are hinted by the idea of Skolem functions and Hilbert's -symbols, and they (...)
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    Shadows of the axiom of choice in the universe $$L$$.Jan Mycielski & Grzegorz Tomkowicz - 2018 - Archive for Mathematical Logic 57 (5-6):607-616.
    We show that several theorems about Polish spaces, which depend on the axiom of choice ), have interesting corollaries that are theorems of the theory \, where \ is the axiom of dependent choices. Surprisingly it is natural to use the full \ to prove the existence of these proofs; in fact we do not even know the proofs in \. Let \ denote the axiom of determinacy. We show also, in the theory \\), a theorem which strenghtens and generalizes (...)
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    The meaning of pure mathematics.Jan Mycielski - 1989 - Journal of Philosophical Logic 18 (3):315 - 320.
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  15.  4
    Canonical Universes and Intuitions About Probabilities.Randall Dougherty & Jan Mycielski - 2006 - Dialectica 60 (4):357-368.
    This paper consists of three parts supplementing the papers of K. Hauser 2002 and D. Mumford 2000: There exist regular open sets of points in with paradoxical properties, which are constructed without using the axiom of choice or the continuum hypothesis. There exist canonical universes of sets in which one can define essentially all objects of mathematical analysis and in which all our intuitions about probabilities are true. Models satisfying the full axiom of choice cannot satisfy all those intuitions and (...)
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