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Are We Closer to a Solution of the Continuum Problem? Carlos Augusto

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Are We Closer to a Solution of the Continuum Problem? Carlos Augusto CDD: 511.322 ARE WE CLOSER TO A SOLUTION OF THE CONTINUUM PROBLEM? CARLOS AUGUSTO DI PRISCO Instituto Venezolano de Investigaciones Cient´ıficas Departamento de Matem´aticas Apartado 21827 Caracas 1020-A, Venezuela [email protected] Abstract: The Continuum Hypothesis has motivated a considerable part of the development of axiomatic set theory for over a century. We present, in a very schematic way, some of the results that give information related to Cantor’s Continuum Problem. Key-words: Axiomatic Set Theory. Continuum Hypothesis. Inde- pendence of the continuum hypothesis. Large cardinals. Forcing. De- terminacy. 1. THE CONTINUUM HYPOTHESIS The notion of infinite appears in mathematics in many different ways. The notion of limit or endless processes of approximations have been considered since ancient times, but it was in the decade of 1870 that the systematic study of infinite collections as completed totalities was initiated by George Cantor, originating what is now known as set theory. Cantor’s original motivations, related to certain problems of mathematical analysis, led him to consider properties of sets of real numbers, in particular their size, or number of elements. Given two collections of objects, we say they have the same size, or the same number of elements, if we can establish a one to one correspon- dence between them, that is, if to each element of the first collection we can assign a unique element of the second, in such a way that every Manuscrito - Rev. Int. Fil., Campinas, v. 28, n. 2, p. 331-350, jul.-dez. 2005. 332 CARLOS AUGUSTO DI PRISCO element of the second collection gets assigned to exactly one element of the first collection. In mathematical terms, this is called a bijection between the two collections. We will thus say that two sets A and B are equipotent, or that they have the same cardinality, if there is a bijection from A to B. It is easy, to verify that the set N of natural numbers (0, 1, 2,... ) has the same cardinality as the set of even numbers (just set the correspondence sending the number n to the number 2n). Some interesting collections of real numbers have also the same cardinality, for example, the set of rational numbers, and the set of algebraic real numbers (those real numbers which are roots of a polynomial with integer coefficients) can be also put in a bijective correspondence with N. Cantor proved that there is no bijection between the points of the real line R and the set of natural numbers; there are, thus, at least two different kinds of infinite sets: those infinite sets which have the same cardinality as N, called countable, and the others, like the set R of real numbers, which are said to be uncountable. Cantor also noted that the k interval [0, 1] and R have the same cardinality, as well as the sets R , RN and {0, 1}N, the set of all sequences of zeroes and ones. All of these sets are uncountable. A very natural question to ask is if every uncountable set of real numbers has the same cardinality as R. Cantor’s continuum hypothesis is precisely that statement: every uncountable set of real numbers has the same cardinality as the whole set of real numbers. Cantor worked without success to prove this hypothesis, and, as we will see, still today the problem remains unsolved. Hilbert considered this problem so important that he put it first in the list of problems he presented to the International Congress of Mathematicians held in Paris in 1900. It continues to be one of the most important problems of mathematics, and it can be said that it has guided the development of set theory for over a century. We will try to present here, in a very schematic way, some of the most important developments related to the continuum hypothesis. The book (Jech 2003) is a good reference to consult for those interested in further reading on this subject and set theory in general; Kanamori (1994) is also an excellent source of information, specially on topics related to large cardinals. Manuscrito - Rev. Int. Fil., Campinas, v. 28, n. 2, p. 331-350, jul.-dez. 2005. A SOLUTION OF THE CONTINUUM PROBLEM? 333 2. AXIOMS OF SET THEORY Cantor introduced transfinite ordinals and cardinals to deal with infinite collections. When we count the elements in a collection, we assign natural numbers to the elements of the collection. If we count real numbers, for example, the elements of the unit interval [0.1], we run out of natural numbers before we end the counting, so we need new numbers to continue, the infinite ordinal numbers. The ordinals, thus, start with the natural numbers and go on indefinitely: 0, 1, 2, . , ω, ω + 1, ω + 2, . , ω + ω, ω + ω + 1,... When we count the elements of a set, we order these elements, the first one, the second, etc. ω is the first infinite ordinal, it is the number assigned to the first element counted after we run out of natural numbers, then comes ω + 1, obtained adding one new element, adding another we obtain ω + 2, and so on. In the limit we have ω + ω, The ordinals obtained starting from ω and applying the operations of adding one more element, and passing to the limit are the countable ordinals. Any set counted with a countable ordinal has the same cardinality as N, which was called ℵ0 by Cantor. It turns out that the set of all countable ordinals is not countable, and its cardinality is the next infinite cardinal, called ℵ1. The process can be continued, if ω1 is the ordinal that comes after all the countable ordinals; then, adding one more element we get ω1 +1, then ω1 +2, and so on. The limit is ω1 +ω, and in this way we get the ordinals that count sets which have the same cardinality as ω1. The cardinality of this set of ordinals is ℵ2, the next cardinal after ℵ1. The transfinite cardinals are ℵ0, ℵ1, ℵ2,..., ℵω, ℵω+1,..., ℵω1 ,... For each ordinal α there is a cardinal ℵα, and these are all different “sizes” of infinite sets. An infinite cardinal ℵβ is a successor cardinal if β = α + 1 for some ordinal α, otherwise it is a limit cardinal. Cantor conjectured that every set can be put in one to one corre- spondence with an ordinal. In other words, that every set can be well ordered, which means totally ordered in such a way that every non- empty subset has a minimal element. This statement is called the well Manuscrito - Rev. Int. Fil., Campinas, v. 28, n. 2, p. 331-350, jul.-dez. 2005. 334 CARLOS AUGUSTO DI PRISCO ordering principle. If it holds, then the real numbers, in particular, can be well ordered and there is a unique cardinal c equipotent with the set of real numbers; c is the cardinality of the set of real numbers. The continuum hypothesis just means that c is ℵ1, the least uncountable cardinal. Since the set of real numbers is equipotent with the set of sequences of sequences of zeroes and ones, c, the cardinality of the set of real numbers is the same as 2ℵ0 . Thus, the continuum hypothesis ℵ is sometimes expressed as 2 0 = ℵ1. Soon after, around the turn of the century, several paradoxes were discovered in the theory of sets, the most famous of which is Russell’s paradox. Bertrand Russell showed that the intuitive idea that every property determines a set (the collection of objects having the property) leads to contradictions. Consider the collection B of all sets x with the property that x is not an element of x. If B is not an element of B then, by its own definition, B is an element of B; and vice-versa, if B is an element of B it is because B is not an element of B. Therefore this collection B cannot be a set. The collection of all sets is another collection which cannot be a set, as well as the collection of all ordinals. Ernst Zermelo, in 1908, proposed an axiomatization of set theory to avoid these paradoxes. This axiomatization, modified later by Abra- ham Fraenkel, is the theory ZF , and together with the axiom of choice it is called the theory of sets ZFC (Zermelo-Fraenkel with Choice). The axioms of ZFC are the following. 1. Axiom of extensionality. If X and Y have the same elements, they are equal. 2. Axiom of pairs. Given X and Y , there is a set {X, Y } whose elements are exactly X and Y . 3. Axiom of union. For every set X, there is a set Y = ∪X which is the union of the elements of X. 4. Axiom of power set. For every set X, there is a set Y = P(X), the set of all subsets of X. 5. Axiom of infinity. There is an infinite set. Manuscrito - Rev. Int. Fil., Campinas, v. 28, n. 2, p. 331-350, jul.-dez. 2005. A SOLUTION OF THE CONTINUUM PROBLEM? 335 6. Axiom of replacement. If F is a definable function, for every set X there is a set Y = {F (x): x ∈ X}. 7. Axiom of regularity. Every non-empty set has an ∈-minimal ele- ment. 8. Axiom of choice. Every family of non-empty sets has a choice function. The well ordering principle, which states that every set can be well ordered, is equivalent to the axiom of choice. The ordinals are identified in the theory ZFC with some specific sets, in such a way that each ordinal is the set of its predecessors.
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