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Propositions Equivalent to the Continuum Hypothesis

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Propositions Equivalent to the Continuum Hypothesis Utah State University DigitalCommons@USU All Graduate Plan B and other Reports Graduate Studies 5-1968 Propositions Equivalent to the Continuum Hypothesis Bernard L. Bohl Follow this and additional works at: https://digitalcommons.usu.edu/gradreports Part of the Mathematics Commons Recommended Citation Bohl, Bernard L., "Propositions Equivalent to the Continuum Hypothesis" (1968). All Graduate Plan B and other Reports. 1118. https://digitalcommons.usu.edu/gradreports/1118 This Report is brought to you for free and open access by the Graduate Studies at DigitalCommons@USU. It has been accepted for inclusion in All Graduate Plan B and other Reports by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. PROPOSITIONS EQUIVALENT TO THE CONTINUUM HYPOTHESIS by Bernard L. Bohl A report submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in Mathematics Plan B Approved: UTAH STATE UNIVERSITY Logan, Utah 1968 To my wife Dorothy AC KNO\-JLEDGMENT I sincerely thank Professor E. E. Underwood for his assistance, encouragement, and confidence. Bernard L. Bohl TABLE OF CONTENTS Page INTRODUCTION 1 PRELIMINARY RESULTS 3 PROPOSITIONS EQUIVALENT TO THE CONTINUUM HYPOTHESIS 5 BIBLIOGRAPHY 28 VITA 29 INTRODUCTION Two sets A and B are said to have the same power if there exists a one-to-one correspondence between them. All sets which have the same power as the natural numbers are called countable and have power 5V0 . All sets which have the same po1,11er as the real numbers are said to have the power of the continuum, which is usually denoted by c. In this paper, the power of the continuum will be denoted by N 2 ° , since 2�0 can be shown to be equal to c as will be indicated in the preliminary results. Given an element a of a well-ordered set B, the set of all elements of B which precede a is called a segment of B. Every uncount­ able well-ordered set, all of whose segments are either finite or countable, is said to have powerJ½'. The Continuum Hypothesis is the N % o = hypothesis that the po1,11er of the continuum is 1 that is 2 %1. In the sequel, this equality will be called hypothesis H. This paper presents a number of interesting propositions equivalent to hypothesis H. It is, in the most part, a translation from the first chapter of W. Sierpinski's book, Hypothese Du Continu. These propositions were first developed by W. Sierpinski in an attempt to prove hypothesis H. Even though hypothesis H has been shown to be independent of existin� axioms by Paul J. Cohen in 1963, these pro­ positions are still interesting in themselves. In this paper. a function of a real variable need not be single valued. A curve will mean any set of points (x,y) of the plane which satisfies an equation of the form 2 y = f(x) or x = f(y) where f is a sin�le valued function of a real variable. A family of increasing sets is defined as a family F, such that for any two sets of the family, one will always be a subset of the other. A linear set will mean any subset of real numbers. The lettere will represent the set of all real numbers and D will represent the set of all natural numbers. All Greek letters will represent ordinal numbers. The symbol 0 will represent the empty set. It is assumed throughout this paper that the reader is familiar with the basic concepts of cardinal and ordinal numbers and with the topological concepts of nowhere dense set, perfect set, first category of Baire, countable base, measure zero, and G0 set. Some theorems that are used in the support of this paper, but not directly a part of it, will be stated without nroof in the preliminary results. 3 PRELIMINARY RESULTS 1. Every set M can be well ordered. 2. The set of all ordinals less than i has ordinal number f. 3. Any well-ordered set is similar to a set of ordinal numbers. 4. Any two ordinal numbers a,S satisfy precisely one of the three relations a < S, a = S, a > S. 5. The power of the continuum, c = 2 nfo. .l1fo 6. Jl.f1 . � 2 7. A countable union of countable sets is countable. 8. The set of all infinite sequences of real numbers has the power of the continuum. 9. The set of all infinite sequences of natural numbers has the power of the continuum. 10. There is no ca rdi na 1 number between _N and Slf . 0 1 2 = 11. For any ordi na 1 number a,% a SUa . 12. The equationJU1 ·.su1 =%1 is equivalent to the statement that the union of a family of powerN1 of disjoint sets of powerSU1 is a set of power .ru1 . 13. Any countable linear set is of the first category of Baire. 14. The family F of all linear nowhere dense perfect sets has power 1 ess than or equa 1 to 2N° . 15. Every linear set that is a countable union of nowhere dense sets is contained in a countable union of nowhere dense linear perfect sets. 4 16. The family of all linear G sets has power 2Afo. 0 17. A countably infinite union of sets of measure zero has measure zero. 18. Every linear set of measure zero is contained in a linear G set 0 of measure zero. 19. If E has powerJ)J1, then the family of all infinite subsets of E has power greater than N1. 51f0 20. If E has power R(1 = 2 , then the set of a11 countab1 e subsets of E has power ru1No N 21. For every ordina 1 number a, there exists a cardi na1 number a . 22. Any two cardinal numbers a,S satisfy precisely one of the three relations a <s , a= s, a> s. 23. The set of all infinite increasing sequences of natural numbers has power 21Uo . 5 PROPOSITIONS EQUIVALENT TO THE CONTINUUM HYPOTHESIS Proposition P . The set of all the points of the plane is a 1 union of two sets, one of which intersects every line parallel to the y-axis in at most a countable number of points and the other intersects every line parallel to the x-axis in at most a countable number of points. ° 1 H � P1. First the following lemma will be proven without making use of hypothesis H. Lemma. The set of all the points of the plane is a union of tv-1O sets, one of which intersects every line parallel to the y-axis in a set of power less than 2iU0 and the other intersects every line parallel to the x-axis in a set of power less than 2 .A.lo Proof. Since the real numbers can be well ordered, there exists a transfinite sequence (1 ) t , t , . , t , t t < 1 2 w w+ 1 , . , a, . ( a f) formed of all the real numbers, where the order type 'I' of this sequence is the smallest ordinal number of the power of the continuum. Let P be the set of all the points (x,y) of the plane and designate by A the set of all the points (t , t ) where B �a < 'f a 6 and 1et B = P - A where a < B < f . Let a be a real number. It is then a term of sequence (1), = that is a ta where a is an ordinal number less than f. The points of the set A on the line x = a are the points (t , t ) where B �a < 'f. a 6 M < Since 'f is the smallest ordinal number such that f :c: 2 o , a 2 No and therefore the line x = a contains less than 2 nfo roints of the set A. 6 Let b be a rea 1 number and as above, 1 et b = ts where S < f. y The points of the set Bon the line =b are the points (ta, ts) where a < S <f. Si nee f = 2Afo , S < 2 Afo and therefore the 1i ne y = b contains 1ess than 2ft.!o points of B. The 1 emma is then proven. Proposition P1 follows immediately from this lemma and hypothesis H. That H +P is then shown. 1 ° 2 P1 + H. Assume proposition P1 and let AUB be the decom­ position of the plane as given in P1. Suppose Eis a set of .ru1 lines parallel to the y-axis and let N be the set of all the points of A which are on the parallel lines forming the set E. Since E contains Qf1 lines and every line of E contains a set of at most a countable number of points of A, Nhas power less than or equal to %1. It follows that the orthogonal projection TI of the set Non the y-axis has power less than or equal to %1. Now show that TI contains all the points of the y-axis. Let (o,b) be an arbitrary point of this axis. The line y = b contains a set of at most a countable number of points of B; but, this line intersects the parallel lines of E in a set of points of power 5l[1. Among these points, there is then an uncountably infinite number of points which belong to A (since they do not belong to B) and their projection on the y-axis, notably, the point (o,b) belonqs to TI, according to the definition of TI.
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