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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such ...
with the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
(ZFC), this is equivalent to the following equation in aleph numbers: 2^=\aleph_1, or even shorter with beth numbers: \beth_1 = \aleph_1. The continuum hypothesis was advanced by
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
in 1878, and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in 1900. The answer to this problem is
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independe ...
of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent. This independence was proved in 1963 by Paul Cohen, complementing earlier work by
Kurt Gödel Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imm ...
in 1940. The name of the hypothesis comes from the term '' the continuum'' for the real numbers.


History

Cantor believed the continuum hypothesis to be true and for many years tried in vain to prove it. It became the first on David Hilbert's list of important open questions that was presented at the
International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be rena ...
in the year 1900 in Paris.
Axiomatic set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
was at that point not yet formulated.
Kurt Gödel Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imm ...
proved in 1940 that the negation of the continuum hypothesis, i.e., the existence of a set with intermediate cardinality, could not be proved in standard set theory. The second half of the independence of the continuum hypothesis – i.e., unprovability of the nonexistence of an intermediate-sized set – was proved in 1963 by Paul Cohen.


Cardinality of infinite sets

Two sets are said to have the same ''
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
'' or ''
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. ...
'' if there exists a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
(a one-to-one correspondence) between them. Intuitively, for two sets ''S'' and ''T'' to have the same cardinality means that it is possible to "pair off" elements of ''S'' with elements of ''T'' in such a fashion that every element of ''S'' is paired off with exactly one element of ''T'' and vice versa. Hence, the set has the same cardinality as . With infinite sets such as the set of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s or
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s, the existence of a bijection between two sets becomes more difficult to demonstrate. The rational numbers seemingly form a counterexample to the continuum hypothesis: the integers form a proper subset of the rationals, which themselves form a proper subset of the reals, so intuitively, there are more rational numbers than integers and more real numbers than rational numbers. However, this intuitive analysis is flawed; it does not take proper account of the fact that all three sets are infinite. It turns out the rational numbers can actually be placed in one-to-one correspondence with the integers, and therefore the set of rational numbers is the same size (''cardinality'') as the set of integers: they are both
countable set In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numb ...
s. Cantor gave two proofs that the cardinality of the set of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s is strictly smaller than that of the set of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s (see
Cantor's first uncountability proof Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is his "revolutionary discovery" that the set of all real numbers is ...
and Cantor's diagonal argument). His proofs, however, give no indication of the extent to which the cardinality of the integers is less than that of the real numbers. Cantor proposed the continuum hypothesis as a possible solution to this question. The continuum hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers. That is, every set, ''S'', of real numbers can either be mapped one-to-one into the integers or the real numbers can be mapped one-to-one into ''S''. As the real numbers are equinumerous with the
powerset In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
of the integers, , \mathbb, =2^ and the continuum hypothesis says that there is no set S for which \aleph_0 < , S, < 2^. Assuming the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
, there is a unique smallest cardinal number \aleph_1 greater than \aleph_0, and the continuum hypothesis is in turn equivalent to the equality 2^ = \aleph_1.


Independence from ZFC

The independence of the continuum hypothesis (CH) from
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such ...
(ZF) follows from combined work of
Kurt Gödel Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imm ...
and Paul Cohen. Gödel showed that CH cannot be disproved from ZF, even if the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
(AC) is adopted (making ZFC). Gödel's proof shows that CH and AC both hold in the constructible universe L, an
inner model In set theory, a branch of mathematical logic, an inner model for a theory ''T'' is a substructure of a model ''M'' of a set theory that is both a model for ''T'' and contains all the ordinals of ''M''. Definition Let L = \langle \in \rangl ...
of ZF set theory, assuming only the axioms of ZF. The existence of an inner model of ZF in which additional axioms hold shows that the additional axioms are consistent with ZF, provided ZF itself is consistent. The latter condition cannot be proved in ZF itself, due to
Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the phil ...
, but is widely believed to be true and can be proved in stronger set theories. Cohen showed that CH cannot be proven from the ZFC axioms, completing the overall independence proof. To prove his result, Cohen developed the method of forcing, which has become a standard tool in set theory. Essentially, this method begins with a model of ZF in which CH holds, and constructs another model which contains more sets than the original, in a way that CH does not hold in the new model. Cohen was awarded the
Fields Medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award h ...
in 1966 for his proof. The independence proof just described shows that CH is independent of ZFC. Further research has shown that CH is independent of all known '' large cardinal axioms'' in the context of ZFC. Moreover, it has been shown that the cardinality of the continuum can be any cardinal consistent with König's theorem. A result of Solovay, proved shortly after Cohen's result on the independence of the continuum hypothesis, shows that in any model of ZFC, if \kappa is a cardinal of uncountable
cofinality In mathematics, especially in order theory, the cofinality cf(''A'') of a partially ordered set ''A'' is the least of the cardinalities of the cofinal subsets of ''A''. This definition of cofinality relies on the axiom of choice, as it uses t ...
, then there is a forcing extension in which 2^ = \kappa. However, per König's theorem, it is not consistent to assume 2^ is \aleph_\omega or \aleph_ or any cardinal with cofinality \omega. The continuum hypothesis is closely related to many statements in
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
, point set
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
. As a result of its independence, many substantial
conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in ...
s in those fields have subsequently been shown to be independent as well. The independence from ZFC means that proving or disproving the CH within ZFC is impossible. However, Gödel and Cohen's negative results are not universally accepted as disposing of all interest in the continuum hypothesis. Hilbert's problem remains an active topic of research; see Woodin and
Peter Koellner Peter Koellner is Professor of Philosophy at Harvard University. He received his Ph.D from MIT in 2003. His main areas of research are mathematical logic, specifically set theory, and philosophy of mathematics, philosophy of physics, analytic philo ...
for an overview of the current research status. The continuum hypothesis was not the first statement shown to be independent of ZFC. An immediate consequence of Gödel's incompleteness theorem, which was published in 1931, is that there is a formal statement (one for each appropriate
Gödel numbering In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number. The concept was developed by Kurt Gödel for the proof of h ...
scheme) expressing the consistency of ZFC that is independent of ZFC, assuming that ZFC is consistent. The continuum hypothesis and the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
were among the first mathematical statements shown to be independent of ZF set theory.


Arguments for and against the continuum hypothesis

Gödel believed that CH is false, and that his proof that CH is consistent with ZFC only shows that the Zermelo–Fraenkel axioms do not adequately characterize the universe of sets. Gödel was a platonist and therefore had no problems with asserting the truth and falsehood of statements independent of their provability. Cohen, though a formalist, also tended towards rejecting CH. Historically, mathematicians who favored a "rich" and "large"
universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the univers ...
of sets were against CH, while those favoring a "neat" and "controllable" universe favored CH. Parallel arguments were made for and against the axiom of constructibility, which implies CH. More recently,
Matthew Foreman Matthew Dean Foreman is an American mathematician at University of California, Irvine. He has made notable contributions in set theory and in ergodic theory. Biography Born in Los Alamos, New Mexico, Foreman earned his Ph.D. from the Univers ...
has pointed out that ontological maximalism can actually be used to argue in favor of CH, because among models that have the same reals, models with "more" sets of reals have a better chance of satisfying CH. Another viewpoint is that the conception of set is not specific enough to determine whether CH is true or false. This viewpoint was advanced as early as 1923 by Skolem, even before Gödel's first incompleteness theorem. Skolem argued on the basis of what is now known as Skolem's paradox, and it was later supported by the independence of CH from the axioms of ZFC since these axioms are enough to establish the elementary properties of sets and cardinalities. In order to argue against this viewpoint, it would be sufficient to demonstrate new axioms that are supported by intuition and resolve CH in one direction or another. Although the axiom of constructibility does resolve CH, it is not generally considered to be intuitively true any more than CH is generally considered to be false. At least two other axioms have been proposed that have implications for the continuum hypothesis, although these axioms have not currently found wide acceptance in the mathematical community. In 1986, Chris Freiling presented an argument against CH by showing that the negation of CH is equivalent to Freiling's axiom of symmetry, a statement derived by arguing from particular intuitions about
probabilities Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
. Freiling believes this axiom is "intuitively true" but others have disagreed. A difficult argument against CH developed by
W. Hugh Woodin William Hugh Woodin (born April 23, 1955) is an American mathematician and set theorist at Harvard University. He has made many notable contributions to the theory of inner models and determinacy. A type of large cardinals, the Woodin cardinals, ...
has attracted considerable attention since the year 2000.
Foreman __NOTOC__ A foreman, forewoman or foreperson is a supervisor, often in a manual trade or industry. Foreman may specifically refer to: *Construction foreman, the worker or tradesman who is in charge of a construction crew * Jury foreman, a head ju ...
does not reject Woodin's argument outright but urges caution. Woodin proposed a new hypothesis that he labeled the , or "Star axiom". The Star axiom would imply that 2^ is \aleph_2, thus falsifying CH. The Star axiom was bolstered by an independent May 2021 proof showing the Star axiom can be derived from a variation of
Martin's maximum In set theory, a branch of mathematical logic, Martin's maximum, introduced by and named after Donald Martin, is a generalization of the proper forcing axiom, itself a generalization of Martin's axiom. It represents the broadest class of forcings ...
. However, Woodin stated in the 2010s that he now instead believes CH to be true, based on his belief in his new "ultimate L" conjecture.
Solomon Feferman Solomon Feferman (December 13, 1928 – July 26, 2016) was an American philosopher and mathematician who worked in mathematical logic. Life Solomon Feferman was born in The Bronx in New York City to working-class parents who had immigrated to th ...
has argued that CH is not a definite mathematical problem. He proposes a theory of "definiteness" using a semi-intuitionistic subsystem of ZF that accepts
classical logic Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this class ...
for bounded quantifiers but uses
intuitionistic logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
for unbounded ones, and suggests that a proposition \phi is mathematically "definite" if the semi-intuitionistic theory can prove (\phi \lor \neg\phi). He conjectures that CH is not definite according to this notion, and proposes that CH should, therefore, be considered not to have a truth value.
Peter Koellner Peter Koellner is Professor of Philosophy at Harvard University. He received his Ph.D from MIT in 2003. His main areas of research are mathematical logic, specifically set theory, and philosophy of mathematics, philosophy of physics, analytic philo ...
wrote a critical commentary on Feferman's article.
Joel David Hamkins Joel David Hamkins is an American mathematician and philosopher who is O'Hara Professor of Philosophy and Mathematics at the University of Notre Dame. He has made contributions in mathematical and philosophical logic, set theory and philosophy o ...
proposes a
multiverse The multiverse is a hypothetical group of multiple universes. Together, these universes comprise everything that exists: the entirety of space, time, matter, energy, information, and the physical laws and constants that describe them. The dif ...
approach to set theory and argues that "the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and, as a result, it can no longer be settled in the manner formerly hoped for". In a related vein,
Saharon Shelah Saharon Shelah ( he, שהרן שלח; born July 3, 1945) is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey. Biography Shelah was born in Jerusalem on July 3, ...
wrote that he does "not agree with the pure Platonic view that the interesting problems in set theory can be decided, that we just have to discover the additional axiom. My mental picture is that we have many possible set theories, all conforming to ZFC".


The generalized continuum hypothesis

The generalized continuum hypothesis (GCH) states that if an infinite set's cardinality lies between that of an infinite set ''S'' and that of the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
\mathcal(S) of ''S'', then it has the same cardinality as either ''S'' or \mathcal(S). That is, for any infinite cardinal \lambda there is no cardinal \kappa such that \lambda <\kappa <2^. GCH is equivalent to: :\aleph_=2^ for every ordinal \alpha (occasionally called Cantor's aleph hypothesis). The beth numbers provide an alternate notation for this condition: \aleph_\alpha=\beth_\alpha for every ordinal \alpha. The continuum hypothesis is the special case for the ordinal \alpha=1. GCH was first suggested by Philip Jourdain. For the early history of GCH, see Moore. Like CH, GCH is also independent of ZFC, but Sierpiński proved that ZF + GCH implies the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
(AC) (and therefore the negation of the axiom of determinacy, AD), so choice and GCH are not independent in ZF; there are no models of ZF in which GCH holds and AC fails. To prove this, Sierpiński showed GCH implies that every cardinality n is smaller than some aleph number, and thus can be ordered. This is done by showing that n is smaller than 2^ which is smaller than its own Hartogs number—this uses the equality 2^\, = \,2\cdot\,2^ ; for the full proof, see Gillman.
Kurt Gödel Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imm ...
showed that GCH is a consequence of ZF + V=L (the axiom that every set is constructible relative to the ordinals), and is therefore consistent with ZFC. As GCH implies CH, Cohen's model in which CH fails is a model in which GCH fails, and thus GCH is not provable from ZFC. W. B. Easton used the method of forcing developed by Cohen to prove
Easton's theorem In set theory, Easton's theorem is a result on the possible cardinal numbers of powersets. (extending a result of Robert M. Solovay) showed via forcing that the only constraints on permissible values for 2''κ'' when ''κ'' is a regular cardinal ...
, which shows it is consistent with ZFC for arbitrarily large cardinals \aleph_\alpha to fail to satisfy 2^ = \aleph_. Much later,
Foreman __NOTOC__ A foreman, forewoman or foreperson is a supervisor, often in a manual trade or industry. Foreman may specifically refer to: *Construction foreman, the worker or tradesman who is in charge of a construction crew * Jury foreman, a head ju ...
and Woodin proved that (assuming the consistency of very large cardinals) it is consistent that 2^\kappa>\kappa^+ holds for every infinite cardinal \kappa. Later Woodin extended this by showing the consistency of 2^\kappa=\kappa^ for every \kappa. Carmi Merimovich showed that, for each ''n'' ≥ 1, it is consistent with ZFC that for each κ, 2κ is the ''n''th successor of κ. On the other hand, László Patai proved that if γ is an ordinal and for each infinite cardinal κ, 2κ is the γth successor of κ, then γ is finite. For any infinite sets A and B, if there is an injection from A to B then there is an injection from subsets of A to subsets of B. Thus for any infinite cardinals A and B, A < B \to 2^A \le 2^B . If A and B are finite, the stronger inequality A < B \to 2^A < 2^B holds. GCH implies that this strict, stronger inequality holds for infinite cardinals as well as finite cardinals.


Implications of GCH for cardinal exponentiation

Although the generalized continuum hypothesis refers directly only to cardinal exponentiation with 2 as the base, one can deduce from it the values of cardinal exponentiation \aleph_^ in all cases. GCH implies that: :\aleph_^ = \aleph_ when ''α'' ≤ ''β''+1; :\aleph_^ = \aleph_ when ''β''+1 < ''α'' and \aleph_ < \operatorname (\aleph_), where cf is the
cofinality In mathematics, especially in order theory, the cofinality cf(''A'') of a partially ordered set ''A'' is the least of the cardinalities of the cofinal subsets of ''A''. This definition of cofinality relies on the axiom of choice, as it uses t ...
operation; and :\aleph_^ = \aleph_ when ''β''+1 < ''α'' and \aleph_ \ge \operatorname (\aleph_). The first equality (when ''α'' ≤ ''β''+1) follows from: :\aleph_^ \le \aleph_^ =(2^)^ = 2^ = 2^ = \aleph_ , while: :\aleph_ = 2^ \le \aleph_^ ; The third equality (when ''β''+1 < ''α'' and \aleph_ \ge \operatorname(\aleph_)) follows from: :\aleph_^ \ge \aleph_^ > \aleph_ , by König's theorem, while: :\aleph_^ \le \aleph_^ \le (2^)^ = 2^ = 2^ = \aleph_ Where, for every γ, GCH is used for equating 2^ and \aleph_; \aleph_^2 = \aleph_ is used as it is equivalent to the axiom of choice.


See also

* Beth number *
Cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
*
Ω-logic In set theory, Ω-logic is an infinitary logic and deductive system proposed by as part of an attempt to generalize the theory of determinacy of pointclasses to cover the structure H_. Just as the axiom of projective determinacy yields a canonic ...
* Wetzel's problem


References

*


Sources

*


Further reading

* * * * Gödel, K.: ''What is Cantor's Continuum Problem?'', reprinted in Benacerraf and Putnam's collection ''Philosophy of Mathematics'', 2nd ed., Cambridge University Press, 1983. An outline of Gödel's arguments against CH. * Martin, D. (1976). "Hilbert's first problem: the continuum hypothesis," in ''Mathematical Developments Arising from Hilbert's Problems,'' Proceedings of Symposia in Pure Mathematics XXVIII, F. Browder, editor. American Mathematical Society, 1976, pp. 81–92. * *


External links

* {{DEFAULTSORT:Continuum Hypothesis Forcing (mathematics) Independence results Basic concepts in infinite set theory Hilbert's problems Infinity Hypotheses Cardinal numbers