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 be regular, . A -complete Boolean algebra on generators, which are restricted by just one countably long equation, is either atomic with atoms or isomorphic to the free -complete Boolean algebra on 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.

**[Ba]**J. Barwise,*Absolute logics and*, 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)**

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