Independent sets of axioms for boolean algebras

Abstract

In this work, we review axiomatic systems and prove some of the equivalent axiomatizations of Boolean algebras.  Also we prove the independence of three axioms, proposed by Huntington and then by Robbins, which form a minimal set of axioms for Boolean algebras.

Keywords

• Classical logic,Boolean algebras,axiomatizability,consistence,independence

Boole cebirleri için bağımsız aksiyom kümeleri

Abstract

Bu çalışmada, aksiyomatik sistemler araştırıldı ve Boole cebirlerinin denk aksiyomlaştırmalarının bazıları ispatlandı.  Ayrıca, Huntington ve sonrasında Robbins tarafından ileri sürülen, Boole cebirleri için aksiyomların bir minimal kümesini oluşturan üç aksiyomun bağımsızlığını ispatlandı.

Keywords

• Klasik lojik,Boole cebirleri,aksiyomlaştırılabilirlik,tutarlılık,bağımsızlık

References

1. Coxeter, H.S.M., Non-Euclidean geometry, Mathematical Association of America, (1998).
2. Huntingtion, E. V., Sets of independent postulates for the algebra of logic, Transaction of the American Mathematical Society, 5, 208-309, (1904).
3. Huntingtion, E. V., New sets of independent postulates for the algebra of logic, with special reference to Whitehead and Russell’s principia mathematica, Transaction of the American Mathematical Society, 35, 274-304, (1933).
4. Tarski, A., Logic, Semantics, Mathematics, The Clarendron Press, Oxford, (1956).
5. Kreisel, G., Independent recursive axiomatization, Journal of Symbolic Logic, 22, 109, (1957).
6. Kreisel, G., Addition aux cours, corrections et renseignements bibliographiques, Polycopie, Paris, (1962).
7. Reznikoof, I., Tout ensemble de formules de la logique classique est equivaleut un ensemble independant, Comptes Rendus De L’Académie Des Sciences Mathematique, 2385-2388, (1965).
8. Oner, T. ve Terziler, M., Independence of countable set of formlulas of the propositional calculus, Ars Combinatoria, 112, 73-80, (2013).

9. Givant, S. and Halmos, P., Introduction to Boolean algebra, Springer-Verlag, (2007).
10. Winker, S., Absorption and idempotency criteria for a problem in near-boolean algebras, Journal of Algebra, 153, 414-423, (1992).
11. McCune, W., Solution of the Robbins problem, Journal of Automated Reasoning, 19, 277-318, (1997).
12. Mann, A. L., A case study in automated theorem proving: otter and EQP, Master Thesis, Universty of Colorado, Department of Mathematics, (2003).