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Volume 3, Issue 3, June 2014, Page: 49-65
Stairs of Natural Set Theories
Maydim A. Malkov, Russian Research Center for Artificial Intelligence, Moscow, Russia
Received: May 15, 2014;       Accepted: Jun. 3, 2014;       Published: Jun. 10, 2014
DOI: 10.11648/j.pamj.20140303.11      View  3545      Downloads  161
All contemporary set theories have intersected classes. We have built the stairs of set theories with disjoint classes. We call such theories natural. We numerate these theories by ordinals. The first set theory is T_0. We build the theory from the set N of natural numbers by using the operations of direct products and of power set by finite times. The theory contains all results of Cantor’s theory. We argue that the theory can satisfy all needs of applied mathematics. We build theory T_1 by using the universe set of all sets of T_0 and by using the operations of direct products and of power set by finite times. We build theory T_α+1 from the set of previous by using the operations of direct products and of power set by finite times, too. We build theory T_ω from the set of all sets of T_α with α < ω again by using the operations of direct products and of power set by finite times. And so on for every theory T_α, if theory T_α-1 does not exists. We use the join of all these sets to build theory T_On without operation of power set. We call members of T_On families, members of families sets, families, which are not members of families, up-sets. Families are an analog of classes of the MK set theory and up-sets are an analog of proper classes of MK theory. The theory T_On is more strong than MK theory because we use more strong axiom of comprehension. The last theory T_On+1 is external to T_On. We use T_On+1 to prove those theorems of T_On that are unproved in T_On.
Set theory, Classifications, Disjoint classes of sets, Cantor ordinals, Cantor cardinals, Comprehension axiom
To cite this article
Maydim A. Malkov, Stairs of Natural Set Theories, Pure and Applied Mathematics Journal. Vol. 3, No. 3, 2014, pp. 49-65. doi: 10.11648/j.pamj.20140303.11
G. Cantor, 1874, Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen, J. reine and angew. Math. 77, 258-262.
B. Russell, 1903, The Principles of Mathematics, Cambridge University Press.
B. Russell, 1908, Mathematical logic as based on the theory of types, Amer. J. of Math. 30, 222-262.
E. Zermelo, 1908, Untersuchungen über die Grundlagen der Mengenlehre, Math. Ann., 65, 261-281.
K. Gödel, 1931, Über formal untentschedibare Saetze der Principia Mathematica und verwandter Systeme I, Monatsh Math. Phys., 38, 349-360.
A. Tarski, 1931, Sur les ensembles definissables de nombres reels I, Fund. Math., 17, 210-239.
W. Quine, 1937, New foundations for mathematical logic, Am. Math. Monthly, 44, 70-80.
W. Quine, 1940, Mathematical logic, Cambridge Mass.
A. Church, 1940, A formulation of the simple theory of types, J. of Symb. Log. 5(2), 56-68.
Hao Wang, 1954, The formalization of mathematics, J. of Symb. Log., 19, 241-266.
A. Fraenkel, 1922, Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre, Math. Annalen, 86, 230-237.
J. Neumann, 1925-1928, Eine Axiomatizierung der Mengenlehre, J. für Math., 154, 219-240 (1925); Die Axiomatizierung der Mengenlehre, Math. Z., 27, 669-752 (1928).
R.M. Robinson, 1937, The Theory of the Classes. A modification of von Neumann’s system, J. of Symb. Log., 2, 69-72.
P. Bernays, 1937-1954, A system of axiomatic set theory, J. of Symb. Logic, I: 2, 65-77 (1937), II: 6, 1-17 (1941), (1942), III: 7, 65-89 (1942), IV: 7, 133-145 (1942), V: 8, 89-106 (1943), VI: 13, 65-79 (1948), VII: 19, 81-96 (1954).
K. Gödel, 1940, The consistency of the axiom of choice and of the generalized continuum hypothesis with axioms of set theory, Princeton.
J. Kelley, 1955, General topology, Van Nostrand.
A. Morse, 1965, A teory of sets, Acadimiv Press.
G. Boolos, 1971, The iterative conception of set, J. of phil. 68 215-231.
P. Maddy, 1988, Believing the Axioms, J. of Symb. Log. I: 53(2), 481-511; II: 53(3), 736-764.
W. Woodin, 2001, The continuum hypothesis, part II, Notices of Amer. Math. Soc. 48(7), 681-690.
A. Kanamori, 2003, The Higher Infinite: Large Cardinals in Set Theory from their beginnings (2nd ed.), Springer.
J.R. Steel, 2000, Mathematics needs new axioms, Bull. Symb. Log., 6 422-433.
M. Foreman, A. Kanamory, ed., 2010, Handbook of set theory, v. 1-3, Springer.
E. Mendelson, 2013, Introduction to mathematical logic, 5th edition, Chapman and Hall.
D. Scott, 1965, Logic with denumerably long formulas and finite strings of quantifiers, in Proceedings of the 1963 International Symposium, Theory of Models, Berkeley, 329-341.
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