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
DOI: https://doi.org/10.1090/S0002-9947-1975-0469695-0
MathSciNet review: 0469695
Full-text PDF

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] J. Barwise, Absolute logics and $ {L_{\infty \omega }}$, Ann. Math. Logic 4 (1972), 309-340. MR 0337483 (49:2252)
  • [FN] J. E. Fenstad and D. Normann, On absolutely measurable sets, Fund. Math. 81 (1974), 91-98. MR 0338299 (49:3065)
  • [Ga] H. Gaifman, Infinite Boolean polynomials.I, Fund. Math. 54 (1964), 229-250. MR 29 #5765. MR 0168503 (29:5765)
  • [Gr] J. Gregory, Incompleteness of a formal system for infinitary finite-quantifier formulas, J. Symbolic Logic 36 (1971), 445-455. MR 0332431 (48:10758)
  • [JK] R. B. Jensen and C. Karp, Primitive recursive set functions, Proc. Sympos. Pure Math., vol. 13, part I, Amer. Math. Soc., Providence, R. I., 1971, 143-167. MR 43 #7317. MR 0281602 (43:7317)
  • [Ka] C. 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, 1-32. MR 36 #1320. MR 0218232 (36:1320)
  • [Lé] A. Lévy, A hierarchy of formulas in set theory, Mem. Amer. Math. Soc. No. 57 (1965). MR 32 #7399.
  • [Na] M. Nadel, An application of set theory to model theory, Israel J. Math. 11 (1972), 386-393. MR 46 #3298. MR 0304163 (46:3298)
  • [Pa] K. R. Parthasarathy, Probability measures on metric spaces, Probability and Mathematical Statistics, no. 3, Academic Press, New York, 1967. MR 37 #2271. MR 0226684 (37:2271)
  • [So] R. 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 42 #64. MR 0265151 (42:64)
  • [St] J. Stavi, Extensions of Kripke's embedding theorem, Ann. Math. Logic (to appear). MR 0409175 (53:12937)
  • [We] E. Wesley, Extensions of the measurable choice theorem by means of forcing, Israel J. Math. 14 (1973), 104-114. MR 0322129 (48:493)

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

DOI: https://doi.org/10.1090/S0002-9947-1975-0469695-0
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society