Bolzano-Weierstrass principle of choice extended towards ordinals
Abstract
The Bolzano-Weierstrass principle of choice is the oldest method of the set theory, traditionally used in mathematical analysis. We are extending it towards transfinite sequences of steps indexed by ordinals. We are introducing the notions: hiker's tracks, hiker's maps and principles Pn(X,Y,m); which are used similarly in finite, countable and uncountable cases. New proofs of Ramsey's theorem and Erdős-Rado theorem are presented as some applications
References
2. P. Erdős, R. Rado, A partition calculus in set theory, Bull. Amer. Math. Soc. 62 (1956), 427-489.
3. J.D. Monk, Appendix on set theory, in: Handbook of Boolean algebras, Elsevier Science Publishers 1989, 1213-1233.
4. H.J. Prömel, B. Voigt, Aspects of Ramsey-theory I: sets, Forschungsinstitut für Diskrete Mathematik Institut für Ökonometrie und Operations Research Rheinische Friedrich-Wilhelms-Universität Bonn, report No 87 495-OR (1989).
5. F.P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 2 (1930), 264-286.
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
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