Glossary

10.2 Gödel's constructible universe and consistency of CH

3 min read•august 7, 2024

is a groundbreaking concept in set theory. It's a special universe where every set can be built step-by-step using simple rules. This universe follows all the usual set theory rules, plus some extra ones.

In this universe, the is true. This was a big deal because it showed that the Continuum Hypothesis can't be disproved using standard set theory. It opened up new ways of thinking about sets and infinity.

Gödel's Constructible Universe

Overview of Gödel's Constructible Universe

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  • developed the concept of the constructible universe in the 1930s
  • The constructible universe, denoted by [L](https://www.fiveableKeyTerm:l)[L](https://www.fiveableKeyTerm:l), is a subclass of the von Neumann universe VV
  • LL is constructed by iteratively defining sets using formulas in the language of set theory
  • The construction of LL is done in stages, indexed by ordinals, where each stage LαL_\alpha contains all sets definable from elements of previous stages using formulas with parameters from LαL_\alpha

Properties and Axioms of the Constructible Universe

  • The , denoted by V=LV=L, states that every set is constructible
  • If V=LV=L holds, then the universe of sets is identical to the constructible universe
  • LL satisfies all the axioms of Zermelo-Fraenkel set theory (ZF) and the (AC)
  • In LL, the (GCH) holds, which states that for any infinite cardinal κ\kappa, there is no cardinal between κ\kappa and 2κ2^\kappa

Inner Models and Relative Consistency

  • An is a transitive class that contains all the ordinals and satisfies the axioms of ZF
  • LL is an inner model of ZF and serves as a canonical example of an inner model
  • The concept of inner models is used to study the of statements in set theory
  • If the existence of an inner model satisfying certain properties can be proved from the axioms of ZF, then those properties are consistent with ZF (relative )

Consistency of the Continuum Hypothesis

Gödel's Consistency Proof

  • In 1938, Gödel proved that the Continuum Hypothesis () is consistent with the axioms of (Zermelo-Fraenkel set theory with the Axiom of Choice)
  • Gödel's proof showed that if ZFC is consistent, then ZFC + CH is also consistent
  • The proof involves constructing the constructible universe LL and showing that LL satisfies both ZFC and CH
  • Gödel's result does not prove that CH is true, but rather that it cannot be disproved from the axioms of ZFC alone

Relative Consistency and Independence

  • Relative consistency means that if a theory TT is consistent, then adding a statement ϕ\phi to TT results in a consistent theory T+ϕT + \phi
  • Gödel's proof established the relative consistency of CH with ZFC
  • In 1963, proved the of CH from ZFC using the method of
  • The combination of Gödel's and Cohen's results shows that CH is independent of the axioms of ZFC, meaning it can neither be proved nor disproved from these axioms alone

Role of the Axiom of Choice

  • The Axiom of Choice (AC) is an axiom in set theory that states that for any collection of non-empty sets, it is possible to select an element from each set to form a new set
  • AC is used in the construction of the constructible universe LL and is essential for proving the relative consistency of CH with ZFC
  • The consistency of AC with the other axioms of ZF was proven by Gödel in his constructible universe
  • Some statements in set theory, such as the , are equivalent to AC, while others like CH are independent of AC

Key Terms to Review (26)

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.
Axiom of Constructibility: The Axiom of Constructibility (V = L) states that every set is constructible, meaning that every set can be built up in a systematic way from simpler sets. This axiom has significant implications for the foundations of set theory and directly relates to the independence of the Continuum Hypothesis and Gödel's constructible universe, which show how certain mathematical truths can depend on the acceptance of this axiom.
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.
CH: CH, or the Continuum Hypothesis, is a proposition concerning the possible sizes of infinite sets, specifically stating that there is no set whose cardinality is strictly between that of the integers and the real numbers. This concept plays a crucial role in understanding the structure of infinite sets and their relationships to each other, linking directly to fundamental questions in set theory about the nature of infinity.
Consistency: Consistency refers to a property of a set of statements or axioms where no contradictions can be derived from them. In other words, a consistent system allows for the existence of at least one model that satisfies all its axioms, which is crucial in mathematical logic and set theory for establishing the reliability of the framework.
Constructible Sets: Constructible sets are collections of sets that can be explicitly defined and built up using a process that involves the cumulative hierarchy of sets. This idea is central to understanding Gödel's constructible universe, where every set can be constructed in a systematic way, leading to the exploration of consistency and independence results, particularly regarding the Continuum Hypothesis (CH). Constructible sets serve as a framework to analyze what can be known and established within set theory.
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.
Definability: Definability refers to the property of a mathematical object being able to be uniquely characterized or described by a specific set of properties or conditions. This concept is crucial in understanding how sets and structures can be interpreted and understood within a given framework, especially in relation to the limitations and possibilities of formal systems.
Elementary Substructure: An elementary substructure is a type of structure in model theory where a smaller structure can be seen as a model of the same theory as a larger structure, preserving all properties and relationships defined by the language of that theory. This concept helps in understanding how different models can relate to each other while maintaining consistency in terms of logical relations and truths within set theory.
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.
Gödel's Completeness Theorem: Gödel's Completeness Theorem states that for any consistent set of first-order sentences, there exists a model in which all those sentences are true. This theorem bridges the gap between syntactic provability and semantic truth, emphasizing that if something can be proven using the axioms and rules of a logical system, it is also true in some model. This connection lays the groundwork for understanding issues related to consistency, independence of axioms, and provides insights into the structure of mathematical theories.
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.
Gödel's incompleteness theorems: Gödel's incompleteness theorems are two fundamental results in mathematical logic that demonstrate inherent limitations in formal systems capable of expressing basic arithmetic. The first theorem states that any consistent formal system strong enough to encompass arithmetic cannot be both complete and consistent, meaning there are true statements that cannot be proven within the system. The second theorem shows that such a system cannot prove its own consistency. These results have profound implications for understanding the consistency and independence of axioms, as well as for the constructible universe and the consistency of set theory, particularly with respect to the Continuum Hypothesis.
Independence: Independence refers to a property of mathematical statements or axioms where a statement cannot be proven or disproven using a given set of axioms. This concept is essential in understanding the limits of formal systems, as it illustrates that certain truths exist beyond provability within those systems. Recognizing independence helps clarify relationships between axioms and the statements they encompass, which is pivotal in exploring consistency and the foundations of set theory.
Inner model: An inner model is a transitive set that satisfies the axioms of set theory and is contained within a larger model of set theory. Inner models are crucial in studying the foundations of mathematics, especially in analyzing properties like consistency and the existence of certain sets. They provide a way to explore how sets behave under different conditions and play an important role in understanding the implications of various axioms, including the Continuum Hypothesis.
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.
L: In the context of set theory and Gödel's constructible universe, the symbol 'l' often refers to a specific class of sets that are used to construct models of set theory. It represents a certain level or hierarchy within the constructible universe, denoted as L, which contains sets that can be explicitly defined through a definable process. This concept plays a critical role in understanding the consistency of the Continuum Hypothesis (CH) and the nature of set-theoretic truth.
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.
Ordinal: An ordinal is a type of number used to describe the position or order of elements within a well-ordered set. Unlike cardinal numbers that denote quantity, ordinals specifically indicate the rank of each element, such as first, second, or third. This concept is crucial in set theory as it helps in understanding how sets can be arranged in a sequence and how they relate to concepts like countability and hierarchy.
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.
Relative consistency: Relative consistency is a concept in mathematical logic that refers to the idea that the consistency of one theory can be established relative to another theory. This means that if one theory is consistent, then another theory can be shown to be consistent under the assumption that the first theory is true. It often involves comparing the axioms of different systems and understanding how they interact, particularly in relation to set theory and the Continuum Hypothesis.
Uncountable: In set theory, a set is described as uncountable if it has more elements than the set of natural numbers, meaning it cannot be put into a one-to-one correspondence with the natural numbers. This concept highlights the existence of different sizes of infinity, showcasing that some infinite sets are larger than others, particularly in discussions involving cardinal numbers and foundational aspects of mathematics.
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.
ZFC: ZFC, or Zermelo-Fraenkel set theory with the Axiom of Choice, is a foundational system for mathematics that provides a rigorous framework for understanding sets and their properties. This system is crucial in modern mathematical logic, allowing for the formulation of statements about sets and functions, and it serves as a basis for much of contemporary set theory. ZFC helps mathematicians navigate complex questions about the nature of infinity, cardinality, and the existence of various mathematical objects.
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