Glossary

8.3 The Continuum Hypothesis and Generalized Continuum Hypothesis

4 min read•august 7, 2024

The (CH) and (GCH) are key ideas in set theory. They deal with the sizes of infinite sets, asking if there's a set bigger than the natural numbers but smaller than the real numbers.

These concepts are crucial for understanding the structure of infinite sets. Surprisingly, CH and GCH can't be proved or disproved using standard set theory axioms, showing the limits of our mathematical foundations.

The Continuum Hypothesis and Its Generalization

Cardinality of the Continuum and Aleph Numbers

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  • Continuum Hypothesis (CH) states there is no set with strictly between the cardinality of the natural numbers (ℵ0\aleph_0) and the cardinality of the continuum (2ℵ02^{\aleph_0}, or the cardinality of the real numbers)
  • In other words, CH asserts ℵ1=2ℵ0\aleph_1 = 2^{\aleph_0}, where ℵ1\aleph_1 is the smallest greater than ℵ0\aleph_0
  • Generalized Continuum Hypothesis (GCH) extends CH to all infinite sets, stating for every infinite cardinal κ\kappa, there is no cardinal between κ\kappa and 2κ2^\kappa
  • GCH implies the cardinality of the power set of any infinite set is the next larger cardinal number
  • (ℵ0,ℵ1,ℵ2,…\aleph_0, \aleph_1, \aleph_2, \ldots) represent the sequence of infinite cardinal numbers, with each aleph number being the smallest cardinal greater than the preceding one

Implications and Consequences of CH and GCH

  • If CH is true, there are no sets with cardinality between the natural numbers and the real numbers, implying a "gap" in the hierarchy of infinite sets
  • CH has implications for the structure of the real line and its subsets, such as the Borel hierarchy and the projective hierarchy
  • GCH, if true, would provide a complete description of the infinite cardinals and their power sets
  • However, both CH and GCH have been shown to be independent of the standard axioms of set theory (ZFC), meaning they can neither be proved nor disproved within ZFC

Independence and Consistency

Independence of CH and the Axiom of Choice

  • In 1938, showed that the Continuum Hypothesis (CH) is consistent with the axioms of (ZFC), meaning CH cannot be disproved from ZFC alone
  • However, in 1963, proved that the negation of CH is also consistent with ZFC, establishing the independence of CH from ZFC
  • The independence of CH means that both CH and its negation can be added as axioms to ZFC without introducing a contradiction
  • The (AC) is another statement independent of the other axioms of ZFC
  • AC states that given any collection of non-empty sets, it is possible to select an element from each set to form a new set
  • AC is equivalent to the , which states that every set can be well-ordered, and , a powerful tool in algebra and analysis

Consistency and Relative Consistency

  • A set of axioms is consistent if no contradiction can be derived from them
  • The consistency of ZFC, or any sufficiently complex axiom system, cannot be proved within the system itself due to Gödel's Second Incompleteness Theorem
  • However, relative consistency results can be established, showing that if one system (e.g., ZFC) is consistent, then another system (e.g., ZFC + CH) is also consistent
  • The independence results for CH and AC demonstrate the relative consistency of these statements with ZFC

Methods for Proving Independence

Constructible Universe and Forcing

  • (L) is a model of ZFC in which CH and GCH hold
  • L is constructed by iteratively defining sets using a restricted form of comprehension, starting from the empty set and iterating through the ordinals
  • The constructible universe satisfies all axioms of ZFC and additional statements like CH, demonstrating the consistency of ZFC + CH
  • , introduced by Cohen, is a technique for extending a model of set theory to create new models satisfying desired properties
  • Forcing adds new sets, called generic sets, to an existing model in a controlled manner, preserving the axioms of ZFC while altering the truth values of independent statements
  • Cohen used forcing to construct a model of ZFC in which CH fails, proving the independence of CH from ZFC

Inner Models and Boolean-Valued Models

  • are subclasses of the von Neumann universe (V) that satisfy all the axioms of ZFC
  • Examples of inner models include Gödel's constructible universe (L), the core model (K), and the Dodd-Jensen core model (KDJ)
  • Inner models are used to establish consistency results and study the fine structure of the universe of sets
  • , introduced by Scott and Solovay, provide an alternative approach to forcing
  • In a Boolean-valued model, each statement is assigned a value from a complete Boolean algebra, representing its "truth value" in the model
  • Boolean-valued models can be used to prove independence results and study the relationships between different axioms and statements in set theory

Key Terms to Review (22)

Aleph Numbers: Aleph numbers are a way to describe the sizes of infinite sets, particularly in terms of cardinality. They start with aleph-null (ℵ₀), which represents the cardinality of the set of natural numbers, and increase to represent larger infinities, such as the cardinality of the set of real numbers. Understanding aleph numbers is essential when discussing the Continuum Hypothesis and its implications for the sizes of infinite sets.
Axiom of Choice: The Axiom of Choice states that for any collection of non-empty sets, there exists a way to select one element from each set, even if there is no explicit rule for making the selection. This concept is fundamental in set theory and connects various results and theorems across different areas of mathematics.
Boolean-valued models: Boolean-valued models are a type of mathematical structure used in set theory and logic where the truth values of propositions are not limited to just true or false but can take on values from a Boolean algebra. This approach allows for a richer exploration of set-theoretic concepts, particularly in relation to the Continuum Hypothesis and the Generalized Continuum Hypothesis. By utilizing Boolean algebras, these models provide a framework to analyze the consistency and independence of various mathematical statements, thus playing a crucial role in advanced set-theoretic studies.
Cantor's Theorem: Cantor's Theorem states that for any set, the power set of that set (the set of all its subsets) has a strictly greater cardinality than the set itself. This theorem highlights a fundamental aspect of the nature of infinity and implies that not all infinities are equal, leading to insights about the structure of different sizes of infinity.
Cardinal number: A cardinal number is a number that indicates quantity, representing the size of a set. They are used to compare the sizes of different sets, such as finite sets, infinite sets, and can even illustrate the concept of different 'sizes' of infinity. Understanding cardinal numbers is essential for grasping deeper concepts in set theory, such as infinite sets and their properties.
Cardinality: Cardinality refers to the measure of the 'number of elements' in a set, providing a way to compare the sizes of different sets. This concept allows us to classify sets as finite, countably infinite, or uncountably infinite, which is essential for understanding the structure of mathematical systems and their properties.
Continuum hypothesis: The continuum hypothesis posits that there is no set whose size is strictly between that of the integers and the real numbers, specifically stating that the cardinality of the continuum is equal to the cardinality of the first uncountable ordinal. This idea connects deeply with concepts of infinite sets, providing insights into the structure and properties of various infinities.
Countable infinity: Countable infinity refers to a type of infinity that can be put into a one-to-one correspondence with the set of natural numbers, meaning its elements can be listed in a sequence, even if that sequence goes on forever. This concept is crucial for understanding different sizes of infinity, as it helps distinguish between sets that can be counted and those that cannot. Countable infinity plays a significant role in comparing the sizes of sets, analyzing uncountable sets, and exploring hypotheses related to the continuum of real numbers.
Forcing: Forcing is a technique used in set theory to extend a given model of set theory to create a new model where certain statements hold true, particularly in proving the independence of various mathematical propositions. This method allows mathematicians to show that certain axioms, like the Continuum Hypothesis, can be independent of Zermelo-Fraenkel set theory, thus demonstrating their consistency or inconsistency.
Generalized continuum hypothesis: The generalized continuum hypothesis (GCH) extends the classic continuum hypothesis by asserting that for any infinite set, there is no set whose cardinality is strictly between that of the set and its power set. This concept challenges our understanding of cardinalities and the sizes of infinite sets, linking to deeper implications in set theory and model theory.
Georg Cantor: Georg Cantor was a German mathematician known for founding set theory and introducing concepts such as different sizes of infinity and cardinality. His work laid the groundwork for much of modern mathematics, influencing theories about infinite sets, real numbers, and their properties.
Gödel's constructible universe: Gödel's constructible universe, often denoted as $L$, is a class of sets that are built up in a specific way, where every set is definable from earlier sets using certain operations. It plays a crucial role in set theory, especially regarding the Continuum Hypothesis (CH) and its consistency. By analyzing this universe, we gain insight into the nature of mathematical truth and the limitations of set theory, particularly in relation to infinite sets and cardinalities.
Inner models: Inner models are specific types of set-theoretic universes that provide a way to study the properties of the universe of all sets by looking at a smaller, contained model. They help in examining the consistency and implications of various set-theoretical statements, particularly in relation to large cardinals and the continuum hypothesis. Inner models serve as essential tools in understanding the structure of set theory and exploring the relationships between different axioms.
Kurt Gödel: Kurt Gödel was a renowned mathematician and logician, best known for his incompleteness theorems which revealed limitations in formal mathematical systems. His work established critical insights into the consistency and independence of axioms, influencing foundational aspects of mathematics and set theory.
Model theory: Model theory is a branch of mathematical logic that deals with the relationship between formal languages and their interpretations, or models. It studies the ways in which mathematical structures can represent various theories and explores the properties of these models, particularly in relation to consistency, completeness, and categoricity. Understanding model theory is crucial for analyzing the foundations of mathematics and examining the independence of axioms.
Paul Cohen: Paul Cohen was a prominent American mathematician known for his groundbreaking work in set theory and logic, particularly in demonstrating the independence of the Continuum Hypothesis from Zermelo-Fraenkel set theory with the Axiom of Choice. His innovative method of forcing transformed how mathematicians approached the foundations of set theory and significantly influenced subsequent research directions in the field.
Schroeder-Bernstein Theorem: The Schroeder-Bernstein Theorem states that if there are injective functions from set A to set B and from set B to set A, then there exists a bijection between the two sets A and B. This theorem connects the concept of cardinality with the idea of countable and uncountable sets, making it essential for understanding relationships between different sizes of infinity, as well as implications for cardinal arithmetic and hypotheses regarding the continuum.
Set-theoretic universe: The set-theoretic universe refers to the totality of all sets that can be considered within a particular set theory framework. It serves as the foundational backdrop for discussing concepts like the Continuum Hypothesis, where one examines the sizes of infinite sets and their relationships to one another. Understanding this universe is crucial for analyzing the properties of different cardinalities, especially when exploring questions about the existence of certain sets and the nature of infinities.
Uncountable infinity: Uncountable infinity refers to a type of infinity that is larger than countable infinity, meaning that its elements cannot be put into a one-to-one correspondence with the natural numbers. This concept plays a crucial role in understanding the different sizes of infinite sets, distinguishing between sets like the natural numbers and the real numbers, and leading to significant implications in set theory, particularly concerning the nature of infinite sets and their cardinalities.
Well-Ordering Theorem: The Well-Ordering Theorem states that every set can be well-ordered, meaning that every non-empty set has a least element under a specified order. This concept plays a crucial role in various areas of mathematics, providing the foundation for the properties of ordered sets, particularly in understanding how infinite sets can be structured.
Zermelo-Fraenkel Set Theory: Zermelo-Fraenkel Set Theory (ZF) is a foundational system for mathematics that uses sets as the basic building blocks, formalized by a collection of axioms that dictate how sets behave and interact. This theory serves as a framework for discussing concepts such as infinity, ordinals, and the continuum hypothesis, while also addressing paradoxes in set theory and providing a rigorous basis for mathematical reasoning.
Zorn's Lemma: Zorn's Lemma is a principle in set theory that states if every chain (totally ordered subset) in a partially ordered set has an upper bound, then the entire set contains at least one maximal element. This concept is crucial in many areas of mathematics and is closely related to the structure of partially ordered sets, the axioms of set theory, and various forms of the Axiom of Choice.
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