Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Applications of a theorem of Lévy to Boolean terms and algebras

Author: Jonathan Stavi
Journal: Trans. Amer. Math. Soc. 205 (1975), 1-36
MSC: Primary 02B25; Secondary 02J05, 02K30
MathSciNet review: 0469695
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The paper begins with a short proof of the Gaifman-Hales theorem and the solution of a problem of Gaifman about the depth and length of Boolean terms. The main results are refinements of the following theorem: Let $ \kappa $ be regular, $ {\aleph _1} \leq \kappa \leq \infty $. A $ < \kappa $-complete Boolean algebra on $ {\aleph _0}$ generators, which are restricted by just one countably long equation, is either atomic with $ \leq {\aleph _0}$ atoms or isomorphic to the free $ < \kappa $-complete Boolean algebra on $ {\aleph _0}$ generators. The main tools are a Skolem-Löwenheim type theorem of Azriel Lévy and a coding of Borel sets and Borel-measurable functions by Boolean terms.

References [Enhancements On Off] (What's this?)

  • [Ba] K. Jon Barwise, Absolute logics and 𝐿_{∞𝜔}, Ann. Math. Logic 4 (1972), 309–340. MR 0337483
  • [FN] Jens Erik Fenstad and Dag Normann, On absolutely measurable sets, Fund. Math. 81 (1973/74), no. 2, 91–98. Collection of articles dedicated to Andrzej Mostowski on the occasion of his sixtieth birthday, II. MR 0338299
  • [Ga] H. Gaifman, Infinite Boolean polynomials. I, Fund. Math. 54 (1964), 229–250. MR 0168503
  • [Gr] John Gregory, Incompleteness of a formal system for infinitary finite-quantifier formulas, J. Symbolic Logic 36 (1971), 445–455. MR 0332431
  • [JK] Ronald B. Jensen and Carol Karp, Primitive recursive set functions, Axiomatic Set Thoory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 143–176. MR 0281602
  • [Ka] Carol Karp, A proof of the relative consistency of the continuum hypothesis, Sets, Models and Recursion Theory (Proc. Summer School Math. Logic and Tenth Logic Colloq., Leicester, 1965) North-Holland, Amsterdam, 1967, pp. 1–32. MR 0218232
  • [Lé] A. Lévy, A hierarchy of formulas in set theory, Mem. Amer. Math. Soc. No. 57 (1965). MR 32 #7399.
  • [Na] Mark Nadel, An application of set theory to model theory, Israel J. Math. 11 (1972), 386–393. MR 0304163
  • [Pa] K. R. Parthasarathy, Probability measures on metric spaces, Probability and Mathematical Statistics, No. 3, Academic Press, Inc., New York-London, 1967. MR 0226684
  • [So] Robert M. Solovay, A model of set-theory in which every set of reals is Lebesgue measurable, Ann. of Math. (2) 92 (1970), 1–56. MR 0265151
  • [St] Jonathan Stavi, Extensions of Kripke’s embedding theorem, Ann. Math. Logic 8 (1975), no. 4, 345–428. MR 0409175
  • [We] Eugene Wesley, Extensions of the measurable choice theorem by means of forcing, Israel J. Math. 14 (1973), 104–114. MR 0322129

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 02B25, 02J05, 02K30

Retrieve articles in all journals with MSC: 02B25, 02J05, 02K30

Additional Information

Article copyright: © Copyright 1975 American Mathematical Society