Large Cardinals and Set Theory
Contains material written by a large language model that may be incorrect. I am not an expert in this area and cannot assess the accuracy of the content.
References: [1], [2], [3], [4], [5].
1 Goodstein’s Theorem and a “Small” Large Cardinal
The proof of Goodstein’s theorem using transfinite ordinals, particularly involving \(\omega\), shares a conceptual resemblance with the use of large cardinals in set theory, though they operate at vastly different scales and contexts.
The introduction of a new “large” object, whether it’s an ordinal in arithmetic or a large cardinal in set theory, serves as a powerful tool to extend our mathematical capabilities and prove results that are unattainable within the original, more constrained framework.
In the case of Goodstein’s theorem, the introduction of transfinite ordinals like \(\omega\) allows us to encode and analyze the progression of sequences in a way that’s not possible within the confines of Peano arithmetic alone. This demonstrates how a higher-level perspective or a more extensive mathematical framework can unveil properties and results that are obscured or unreachable at a lower level.
Similarly, in set theory, large cardinals introduce new levels of infinity that extend the universe of discourse, enabling the proof of propositions that are otherwise unprovable. The use of a large cardinal in Solovay’s construction to create a model where all subsets of the reals are Lebesgue measurable is a profound illustration of how such “larger” objects can fundamentally alter our understanding and capabilities within mathematical structures.
This comparison underlines a recurring theme in mathematics: expanding our framework or introducing new, more powerful objects often leads to breakthroughs and deeper insights, showcasing the layered and hierarchical nature of mathematical understanding.
In Goodstein’s theorem, the use of \(\omega\) (the first transfinite ordinal, which is the set of all natural numbers) and other ordinal notations provides a higher-level perspective from which the behavior of Goodstein sequences can be understood and proven to converge to 0. The introduction of ordinals extends beyond the realm of Peano arithmetic, offering a more powerful framework for proving statements about natural numbers.
Similarly, in Solovay’s construction, a large cardinal is used to extend the universe of set theory to a model where additional desirable properties hold, such as every subset of the reals being Lebesgue measurable. Both scenarios involve stepping up to a more robust mathematical framework to prove something that cannot be established within the original, more restricted framework.
However, there are notable differences:
Scale and Complexity: \(\omega\) is a basic transfinite ordinal, foundational in the hierarchy of ordinals, whereas large cardinals are much more complex and exist at a much higher level in the set-theoretic universe.
Purpose and Application: In Goodstein’s theorem, ordinals provide a way to encode and compare the growth of sequences, leading to a proof of termination. In Solovay’s model, large cardinals are used to create a richer set-theoretic universe where certain properties of sets can be controlled or ensured.
Nature of Extensions: The extension from Peano arithmetic to include ordinals is more elementary than the leap to large cardinals in set theory. Large cardinals introduce a significant extension to the foundational axioms, while ordinals extend the number concept to describe order types and sizes of well-ordered sets.
While both are examples of how extending a mathematical framework can provide the tools to prove additional results, they serve different purposes and operate at different levels of mathematical abstraction. The analogy captures the essence of using higher-order structures to achieve proofs not possible within the original system, but it’s important to appreciate the differences in scale and context.
2 Large Cardinals
In Zermelo-Fraenkel (ZF) set theory, “large cardinals” refer to certain kinds of infinite cardinals that have strong properties, extending far beyond the standard arithmetic of infinite cardinals. These cardinals provide insights into the deep structure of the set-theoretic universe.
Large cardinals are of various kinds, with each larger type embedding stronger consistency assumptions and providing more structure to the universe of set theory. Some common types include inaccessible cardinals, Mahlo cardinals, measurable cardinals, supercompact cardinals, and many others. Each type comes with its specific definition and properties, forming a hierarchy that reflects their relative strengths.
For instance, a measurable cardinal is one for which there exists a non-trivial, σ-additive, 2-valued measure defined on all its subsets. This concept is stronger than that of inaccessible cardinals and has significant implications for the structure of sets.
Regarding Lebesgue measurability, it’s notable that the existence of large cardinals can influence the Solovay model, where assuming the existence of an inaccessible cardinal, one can show that ZF plus the axiom of dependent choice (DC) implies that all subsets of \(\mathbb{R}\) are Lebesgue measurable. This result contrasts sharply with the Zermelo-Fraenkel set theory with the axiom of choice (ZFC), where non-measurable sets can be constructed.
Large cardinals provide powerful tools for exploring fundamental questions in set theory, such as the nature of the continuum, the structure of the real line, and the hierarchy of sets. Their existence often yields consistency results, showing that if ZFC is consistent, then so is ZFC plus the existence of a large cardinal of a particular kind, though these cardinals themselves cannot be proven to exist within ZFC.
2.1 A “Smallest” Large Cardinal?
The concept of the “smallest” large cardinal is somewhat nuanced because the hierarchy of large cardinals is complex and not entirely linear. However, the most commonly accepted candidate for the “smallest” large cardinal is the inaccessible cardinal.
An inaccessible cardinal \(\kappa\) is defined as follows:
- \(\kappa\) is uncountable.
- \(\kappa\) is a regular cardinal, meaning its cofinality is \(\kappa\) itself. This means there is no smaller cardinal whose limit points reach \(\kappa\).
- \(\kappa\) is a strong limit cardinal, which means that for any cardinal \(\lambda < \kappa\), \(2^\lambda < \kappa\).
These properties make inaccessible cardinals “large” in the sense that they are not reachable by the usual set-theoretic operations applied to smaller cardinals.
Inaccessible cardinals are significant because they mark the boundary where set theory starts to exhibit properties that cannot be proven from the Zermelo-Fraenkel set theory with the axiom of choice (ZFC). For example, if \(\kappa\) is an inaccessible cardinal, then \(V_\kappa\) (the set of all sets with rank less than \(\kappa\)) is a model of ZFC.
Despite being the “smallest” in the large cardinal hierarchy, inaccessible cardinals are still very powerful and cannot be proven to exist in ZFC due to Gödel’s incompleteness theorems. Their existence is considered a relatively mild large cardinal assumption compared to stronger ones like measurable or supercompact cardinals.
2.2 Rank vs Cardinalty
Rank and cardinality are fundamental concepts in set theory but serve different purposes:
Rank:
- The rank of a set is a measure of its complexity in terms of set-theoretic construction.
- Formally, the rank of a set is defined by the transfinite recursion:
- The rank of the empty set is 0.
- The rank of any other set \(S\) is \(\alpha + 1\), where \(\alpha\) is the supremum of the ranks of elements in \(S\) (if \(S\) is non-empty).
- This definition leads to every set having a well-defined rank, which is an ordinal number. It essentially tells you the “level” at which a set appears in the cumulative hierarchy of sets.
Cardinality:
- Cardinality measures the “size” or number of elements in a set, whether finite or infinite.
- Two sets have the same cardinality if there is a one-to-one correspondence (bijection) between them.
- Infinite sets can have different cardinalities, and these are measured by infinite cardinals like \(\aleph_0\) (the cardinality of the set of natural numbers), \(\aleph_1\), etc.
The distinction is that rank is about the depth or stage in the set-theoretic construction hierarchy at which a set appears, reflecting its structural complexity. In contrast, cardinality is about the quantity of elements in a set, regardless of how those elements are structured or the depth at which the set itself resides in the set-theoretic universe.
3 Inaccessible Cardinals
Inaccessible cardinals, while abstract, can be intuitively understood in the context of the set-theoretic universe. Here’s a more intuitive way to think about inaccessible cardinals:
Building Blocks of Sets: Think of the set-theoretic universe as built in layers, with each layer consisting of all sets that can be constructed from lower layers in a specific manner. Inaccessible cardinals represent levels in this hierarchy that are so large they cannot be reached by the usual processes of set formation (union, power set, etc.) from lower levels.
Beyond Reach: Inaccessible cardinals are “inaccessible” because they cannot be reached by any step-by-step process starting from smaller sets. While you can reach some large cardinals by operations like taking unions or power sets from below, an inaccessible cardinal is beyond these operations. It’s like a “new universe” of sets that starts beyond all the sets you could construct from below.
Universes Within Universes: If you consider the universe of sets up to a certain rank (like \(V_\kappa\) for an inaccessible cardinal \(\kappa\)), it behaves like a self-contained universe that mirrors the full set-theoretic universe. This “universe” has all the properties you would expect from set theory, making inaccessible cardinals foundational for certain types of “inner models” within set theory.
A New Kind of Infinity: While all infinite cardinals are “large” compared to finite numbers, inaccessible cardinals are a new level of large. They’re not just bigger than any finite number or the set of natural numbers; they’re so large that they mark a qualitative leap in size, where the usual operations that generate larger and larger sets (like taking power sets) can’t reach them.
Thinking about inaccessible cardinals in these ways can help provide some intuition, though the concept remains deeply rooted in the abstract framework of set theory.
Question: Is the inaccessible cardinal is bigger than P(P(…(P(omega))…)) where P is the power set operation and omega is all the finite ordinals?
Yes, exactly. An inaccessible cardinal \(\kappa\) is indeed larger than any set you could construct by repeatedly applying the power set operation starting from \(\omega\) (the set of all finite ordinals, which is equivalent to the set of natural numbers in terms of cardinality) a finite number of times.
The process of taking power sets increases cardinality. Starting with \(\omega\), each application of the power set operation moves you to a strictly larger cardinality:
- \(P(\omega)\) has cardinality \(2^{\aleph_0}\) (the cardinality of the continuum).
- \(P(P(\omega))\) has cardinality greater than \(2^{\aleph_0}\).
- And so on.
However, no matter how many times you apply the power set operation in this manner, you’ll always end up with a cardinality that is less than an inaccessible cardinal. This is because an inaccessible cardinal is not just uncountable but also a strong limit cardinal, meaning that its size exceeds any cardinality you can reach through such operations starting from smaller cardinals.
4 Existence
The existence of inaccessible cardinals cannot be proven within Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC); it is introduced as an additional axiom. The axiom asserting the existence of an inaccessible cardinal is independent of ZFC, meaning that both ZFC plus the existence of an inaccessible cardinal and ZFC plus the non-existence of such cardinals are consistent, assuming ZFC itself is consistent.
The concept of an inaccessible cardinal extends beyond the reach of what can be constructed or proven to exist using the standard axioms of set theory. Therefore, to discuss or assume the existence of inaccessible cardinals, one must extend the axiomatic framework of ZFC by adding a new axiom that specifically asserts such cardinals exist.
This approach is similar to how other large cardinal axioms are introduced, each asserting the existence of increasingly stronger types of large cardinals, none of which can be proven to exist within the standard axiomatic system.
5 Model of the Reals Where All Subsets Are Lebesgue Measurable
To construct a model of the set of real numbers in which all subsets are Lebesgue measurable, we typically refer to the Solovay model. The construction of this model relies on the existence of an inaccessible cardinal.
Here’s a brief overview of how the existence of an inaccessible cardinal \(\kappa\) enables the construction of the Solovay model:
Starting Point: Assume there is an inaccessible cardinal \(\kappa\). The universe of sets is then built up to this level, creating a model \(V_\kappa\) that reflects the entire set-theoretic universe up to \(\kappa\).
Collapsing: Next, we perform a forcing process to collapse \(\kappa\) to \(\aleph_1\) (the smallest uncountable cardinal), while preserving the structure of the sets below \(\kappa\). This creates a new model where the cardinality structure is altered but the relative constructibility of sets is preserved.
Solovay Model: In this new model, constructed by Solovay, every set of real numbers is Lebesgue measurable, has the property of Baire, and is determinate. This contrasts with the standard ZFC universe, where the existence of non-measurable sets can be proven.
The critical point here is that the inaccessible cardinal provides a “high enough” starting point to build a model of set theory that includes all sets we’re interested in (like the reals and their subsets) and then apply a forcing technique to change the cardinality structure without affecting the constructibility properties of sets. This process ensures that in the resulting model, all subsets of reals have the desired measurability properties.
It’s important to note that this construction doesn’t imply that all subsets of reals are Lebesgue measurable in ZFC. Instead, it shows that within a specific model constructed using an inaccessible cardinal, this property holds, illustrating the profound impact of large cardinal axioms on the structure of set-theoretic universes.
5.1 Disallowing Vitale?
The Solovay model does not explicitly disallow sets like the Vitali set; rather, it changes the context in such a way that every set of real numbers in the model is Lebesgue measurable. In standard set theory, the Vitali set is a classic example of a non-measurable set, constructed using the axiom of choice to select one representative from each equivalence class of \(\mathbb{R}\) under the equivalence relation \(x \sim y\) if \(x - y\) is rational.
In the Solovay model, the construction uses a form of the axiom of choice that is weakened (dependent choice, DC) but still sufficient for most analysis. By collapsing an inaccessible cardinal to \(\aleph_1\) and carefully constructing the model, Solovay was able to ensure that all sets of reals in this model are measurable. This doesn’t mean sets like the Vitali set are disallowed; instead, the model is constructed in such a way that what we identify as the “Vitali set” within it becomes measurable.
The key is that in the Solovay model, the construction ensures that all subsets of real numbers can be well-ordered in a way that is compatible with their Lebesgue measurability. In essence, the model circumvents the usual construction of non-measurable sets by ensuring that any set you can define within the model is measurable, without needing to explicitly ban or alter sets like the Vitali set. This is a subtle but profound shift in the foundational framework that governs the properties of sets in this model.
5.2 More on Measurability
Assuming the existence of an inaccessible cardinal, you can construct a model of set theory (the Solovay model) in which all subsets of the reals are Lebesgue measurable. This model behaves like the “usual” real numbers in many ways, supporting the same algebraic structure (it’s a complete ordered field isomorphic to the real numbers), but it has different set-theoretic properties.
In this model, the kind of construction that produces the Vitali set in ZFC (using the axiom of choice to select representatives from each equivalence class of \(\mathbb{R}\) modulo the rational numbers) cannot be carried out in the same way. The Solovay model uses a weaker version of the choice axiom (the Axiom of Dependent Choice, DC), which is sufficient for most mathematical analysis but not strong enough to support the construction of a non-measurable set like the Vitali set.
So, while algebraically the field of real numbers in the Solovay model is identical to the “normal” real numbers, set-theoretically, it’s quite different: every set of reals in this model is Lebesgue measurable, avoiding the paradoxical constructions possible in ZFC with the full Axiom of Choice. This demonstrates how changing set-theoretic assumptions can lead to different mathematical universes, each with its own version of “real numbers” and their properties.
In the Solovay model, you can define subsets, perform operations like unions, intersections, and complements, and work with power sets, just as you would in the standard model of the real numbers. The operations you’re accustomed to in analysis and basic set theory remain valid.
The real numbers in the Solovay model maintain their usual properties and algebraic structure, meaning they form a complete ordered field, just like the real numbers in the standard model of set theory. They have the same cardinality as the “usual” real numbers, \(2^{\aleph_0}\), preserving the familiar continuum.
What changes in the Solovay model is not the algebraic or topological structure of the reals but rather the set-theoretic properties concerning measurability and choice. The model is constructed to ensure that while you can define sets and perform standard operations, the Axiom of Choice is weakened so that you cannot construct non-measurable sets as you can in ZFC.
This unique feature illustrates how altering set-theoretic foundations (like modifying the Axiom of Choice or assuming large cardinals) can impact the nature of sets in the universe, particularly in terms of their measurability, without affecting the underlying algebraic and topological structure of the real number system.
5.3 More than Just ZF + weaker AC?
The Solovay model isn’t just a matter of Zermelo-Fraenkel set theory (ZF) plus a weaker version of the Axiom of Choice (AC); it’s a bit more nuanced. The model does indeed utilize a weaker form of choice, the Axiom of Dependent Choice (DC), but the construction of the model also critically involves the use of an inaccessible cardinal. Here’s a breakdown of its structure:
ZF: The foundation is ZF, the standard framework for set theory without the full Axiom of Choice.
Axiom of Dependent Choice (DC): The Solovay model incorporates DC, which is strong enough to support most of mathematical analysis but does not allow for the arbitrary selection of elements from disjoint sets, a key feature in constructing non-measurable sets like the Vitali set.
Inaccessible Cardinal: The construction of the Solovay model begins by assuming the existence of an inaccessible cardinal. This assumption is significantly stronger than DC and is not provable within ZF or ZFC.
Forcing: The model uses a technique called “forcing” to collapse the inaccessible cardinal to \(\aleph_1\) while maintaining the structure of the sets below it. This technique is central to creating a model where all subsets of the real numbers are Lebesgue measurable.
Lebesgue Measurability: In the resulting model, every subset of the real numbers is Lebesgue measurable, a direct consequence of the model’s construction and the limitations on set selection imposed by using DC instead of AC.
So, while DC plays a critical role in the model, the use of an inaccessible cardinal and the forcing technique are equally crucial. The Solovay model is a sophisticated construction that demonstrates how altering foundational set-theoretic assumptions can lead to a universe where all sets of real numbers have properties like Lebesgue measurability.
5.4 Relative Constructibility
The term “relative constructibility” of a set refers to its definability or the manner in which it can be constructed within a certain set-theoretic framework. In the context of set theory, particularly when discussing models like the Solovay model, this concept is crucial for understanding how sets are formed and identified in different models.
Constructibility in Set Theory: In set theory, a set is considered constructible if it can be built from simpler sets using a sequence of defined operations. The universe of constructible sets, denoted by \(L\), includes all sets that can be constructed in a well-defined manner from previously constructed sets, using operations like union, pairing, subset formation, and so on.
Relative to a Model: When discussing “relative constructibility,” we’re considering how a set is constructed within the context of a specific model of set theory. A set’s constructibility might be clear in one model but not in another, depending on the axioms and operations allowed in each model.
Impact on Properties: The relative constructibility of sets influences their properties within a model. For example, in the Solovay model, the constructibility is framed in such a way that all sets of real numbers are Lebesgue measurable. This contrasts with the broader universe of ZFC set theory, where some sets (like the Vitali set) are not constructible in a way that ensures measurability.
In essence, when we talk about the relative constructibility of a set, we are referring to whether and how that set can be defined or constructed within the confines of a particular model of set theory, which influences its properties and the theorems that hold within that model.
5.5 …and AC
The Solovay model does modify the use of the Axiom of Choice (AC) in a significant way. In the Solovay model, the full Axiom of Choice is not assumed; instead, a weaker form known as the Axiom of Dependent Choice (DC) is used. This adjustment is crucial for achieving the property that all subsets of the real numbers are Lebesgue measurable within the model.
Axiom of Dependent Choice (DC): DC is a weaker version of AC that allows for sequences to be constructed where each choice depends on the previous one, but it does not permit arbitrary selection from disjoint nonempty sets without a specified rule, which is allowed under AC. DC is strong enough to support most of the analysis and is particularly relevant in the context of real analysis and measurable sets.
Impact on Set Construction: In the Solovay model, because the full AC is not assumed, one cannot construct non-measurable sets in the same way as in ZFC, where AC is fully available. For example, the construction of a Vitali set relies on choosing a representative from each equivalence class of real numbers modulo the rational numbers, with no specific rule for selection. This kind of arbitrary selection is not available in the Solovay model, which helps in ensuring that all sets are measurable.
Ensuring Measurability: By restricting the axiom of choice, the Solovay model avoids the creation of non-measurable sets like the Vitali set, as the model does not support the kind of “free” selection AC would allow. This leads to a universe where every subset of the real numbers is Lebesgue measurable, which is a stark contrast to the standard ZFC universe where the existence of non-measurable sets is provable.
In summary, the Solovay model’s approach to the Axiom of Choice is a key factor in its distinctive property where all subsets of the real numbers are Lebesgue measurable, highlighting how foundational set-theoretic axioms influence the properties of the mathematical universe.
5.6 DC and Well-Ordering
The Axiom of Dependent Choice (DC) does not allow for the construction of a well-ordering of an arbitrary set in the same comprehensive way that the Axiom of Choice (AC) does. DC is strong enough to construct sequences where each element is chosen based on the previous one, but it does not provide the unrestricted selection ability that AC offers, which is necessary for well-ordering any set.
With AC, you can well-order any set by essentially the process you described: selecting “one element from each set” in a certain way to construct a well-order. This process relies on the ability to make arbitrary selections from an infinite number of sets simultaneously, something DC does not permit.
DC can help construct well-orders in a sequential and dependent manner, but it falls short of enabling the well-ordering of arbitrary sets, especially in the absence of any other form of ordering or selection criteria. This limitation is particularly evident with sets like \(\mathbb{R}\), where constructing a well-ordering requires choices that are not sequentially dependent in a straightforward manner, illustrating a clear distinction between the capabilities of AC and DC.
6 Ordering of Large Cardinals
Large cardinals are typically organized in a hierarchy of consistency strength, and this ordering is a central aspect of their study in set theory. The hierarchy suggests that if a certain type of large cardinal is consistent with the axioms of set theory (like ZFC), then all smaller types of large cardinals in the hierarchy are also consistent. However, it’s important to note that the consistency of large cardinals cannot be proven within ZFC itself due to Gödel’s incompleteness theorems. Here’s a basic overview of how they are ordered and their consistency relations:
Consistency Hierarchy: Starting from weaker notions like inaccessible cardinals, the hierarchy moves up to stronger and stronger types, such as Mahlo cardinals, measurable cardinals, supercompact cardinals, and so on. Each step up in this hierarchy represents a cardinal with strictly stronger properties and greater consistency strength.
Implications of Existence: If a stronger large cardinal exists (consistent with ZFC), it implies the existence (consistency) of the weaker ones in the hierarchy. For example, if a supercompact cardinal is consistent with ZFC, then so are measurable cardinals, Mahlo cardinals, and inaccessible cardinals.
Non-Implication for Larger Cardinals: The existence of a weaker large cardinal does not imply the existence of stronger ones. For instance, the consistency of inaccessible cardinals with ZFC does not imply the consistency of measurable cardinals with ZFC.
Relative Consistency: The relationships are typically framed in terms of relative consistency: if ZFC is consistent, then ZFC plus the existence of a certain type of large cardinal is consistent. These relationships help mathematicians understand the potential impact of assuming one or another type of large cardinal in their theoretical frameworks.
Open Questions and Research: There are ongoing research and discussions in set theory regarding the exact nature of the hierarchy, especially at higher levels where the distinctions become more subtle and complex.
In summary, while large cardinals are organized in a hierarchy where stronger cardinals imply the consistency of weaker ones, the reverse is not true. Each type of large cardinal introduces additional strength to the set-theoretic universe, allowing for more complex and rich structures to be studied and understood.
6.1 Tree of Consistency
The hierarchy of large cardinals can be somewhat visualized as a tree of consistency, although it’s more often presented as a linear hierarchy due to the direct implication relationships. However, when considering different types of large cardinals and their relative strengths, the structure can indeed branch out, especially when considering different extensions or nuances of large cardinal properties.
Here are some of the branches and key points in this tree-like structure:
Inaccessible Cardinals: These are often considered the starting point for large cardinals. Every larger type of large cardinal implies the existence of inaccessible cardinals.
Mahlo Cardinals: A step up from inaccessible cardinals, Mahlo cardinals introduce a new level of complexity. A Mahlo cardinal is inaccessible, and it has the property that any normal function defined below it has stationary points below it.
Measurable Cardinals: These are stronger than Mahlo cardinals and introduce the concept of a non-trivial, σ-additive, 2-valued measure on a set. Measurable cardinals imply the existence of Mahlo cardinals and inaccessible cardinals.
Strong and Superstrong Cardinals: These introduce different notions of “lifting” embeddings from one model of set theory to another, reflecting even higher consistency strength.
Woodin Cardinals: These cardinals introduce properties related to determinacy and the structure of the real line, branching into areas concerning the foundations of descriptive set theory.
Supercompact Cardinals: Supercompactness introduces a strong form of compactness over all higher cardinals and has significant implications for the structure of the universe of set theory.
Branches for Specific Properties: The tree can branch out for specific types of large cardinals that are not directly comparable or are variations with unique properties, like indescribable cardinals, Ramsey cardinals, etc.
Interactions and Implications: Some branches might converge or influence each other, especially when properties of one type of large cardinal have implications for another type or when one type subsumes another.
This tree-like structure is an abstraction to help understand the relationships and relative strengths of large cardinals. While it’s a simplification (the actual relationships can be more nuanced and complex), it provides a useful framework for understanding how different types of large cardinals relate to each other and to the foundational aspects of set theory.
7 Other Results Relying on Large Cardinals
Large cardinals have profound implications across various areas of mathematics, influencing both the foundational aspects and specific results. Here are some key areas and results where large cardinals play a crucial role:
Determinacy of Games: Large cardinals, especially Woodin cardinals and stronger notions, have significant implications for the determinacy of games played on real numbers. For instance, the axiom of determinacy (AD), which is inconsistent with the axiom of choice but interesting in models where choice is limited, can be shown to hold in the presence of certain large cardinals, affecting the outcome of infinite games.
Descriptive Set Theory: In descriptive set theory, large cardinals can impact the structure and properties of sets, particularly subsets of the real line and Baire space. Results about projective sets, for example, can be influenced by the existence of large cardinals, affecting the regularity properties like measurability, perfect set property, and the property of Baire.
Inner Model Theory: Large cardinals are central to the development of inner model theory, which seeks to construct and understand “canonical” models of set theory containing large cardinals. This area provides deep insights into the structure of the set-theoretic universe and the nature of large cardinals themselves.
Set-Theoretic Topology: In set-theoretic topology, the existence of large cardinals can influence the properties of topological spaces, such as compactness, connectedness, and various separation properties, altering the landscape of what’s provable in the field.
Category Theory and Homotopy Theory: Large cardinals can influence results in category theory and homotopy theory, where they can impact the existence of certain kinds of categories or the structure of higher-dimensional algebraic objects.
Combinatorics: In the realm of combinatorics, large cardinals have implications for partition properties, cardinal characteristics of the continuum, and other combinatorial principles, influencing the kind of combinatorial objects that can exist and their properties.
Model Theory: Large cardinals can impact model theory by influencing the compactness properties and the existence of elementary embeddings, which are crucial for understanding models of various theories.
These examples illustrate the broad influence of large cardinals across mathematics, showcasing their role not just as objects of intrinsic set-theoretic interest but as foundational tools that can reshape our understanding of mathematical structures and theorems.
8 Large Cardinals as “Extra Axioms”
Large cardinals can be viewed as extra axioms added on top of Zermelo-Fraenkel set theory (ZF) or Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). When we talk about assuming the existence of a large cardinal, we’re essentially talking about extending ZF or ZFC with additional axioms that assert the existence of these highly powerful and abstract mathematical objects.
These large cardinal axioms are not provable within ZF or ZFC if these systems are consistent—this follows from Gödel’s incompleteness theorems. Instead, they are taken as additional assumptions that can yield richer models of set theory with more robust structures and properties. Each large cardinal axiom strengthens the theory, enabling new theorems and insights that are not achievable in the absence of such axioms.
The hierarchy of large cardinals reflects a hierarchy of increasingly strong axioms, where the assumption of higher large cardinals encompasses the existence of all lower ones in the hierarchy, thereby expanding the foundational framework of set theory. This approach is similar to how one might consider the parallel postulate in geometry: by adopting or not adopting it, one can explore Euclidean or non-Euclidean geometries, respectively. In a similar vein, by adopting different large cardinal axioms, mathematicians explore various “universes” of set theory, each with its own unique properties and theoretical possibilities.
9 Large Cardinals and Descriptive Set Theory
Descriptive set theory studies the properties and classifications of sets, particularly subsets of the real line and other Polish spaces, using the tools and techniques of logic and topology. Large cardinals have a significant impact on descriptive set theory, particularly in the context of projective sets and their properties. Here are some key implications:
Projective Determinacy: Large cardinals provide the consistency strength necessary to prove the determinacy of projective sets. Projective determinacy is a principle stating that games whose winning conditions are defined by projective sets (a hierarchy of sets built from open and closed sets using operations of complementation and projection) are determined: for every game of this sort, one of the two players has a winning strategy. The existence of certain large cardinals (like Woodin cardinals) implies the determinacy of various levels of the projective hierarchy, which in turn has profound implications for the regularity properties of sets in this hierarchy.
Regularity Properties: In descriptive set theory, regularity properties like Lebesgue measurability, the Baire property, and the perfect set property are crucial. Large cardinals can ensure that sets, especially within the projective hierarchy, exhibit these properties. For instance, under the assumption of projective determinacy, which can be supported by the existence of large cardinals, all projective sets are Lebesgue measurable, have the Baire property, and if they are uncountable, contain perfect subsets.
Absoluteness: Descriptive set theory often deals with statements or properties that are absolute between different models of set theory, meaning they remain true regardless of the choice of a particular model. Large cardinals can enhance absoluteness by ensuring that certain properties or theorems in descriptive set theory hold across a broader range of models, providing a more stable foundation for the theory.
Structural Insights: Large cardinals can reveal deep structural insights about the universe of set theory that have direct implications for descriptive set theory. For example, the existence of large cardinals can influence the complexity and structure of definable sets, shedding light on the intricacies of how sets can be constructed and classified.
By influencing the determinacy of games, the regularity properties of sets, and the absoluteness of certain principles, large cardinals enrich the landscape of descriptive set theory, allowing mathematicians to resolve questions that would otherwise be intractable.
10 Forcing
Forcing is a powerful technique in set theory, developed by Paul Cohen in the 1960s, to prove the independence of the Continuum Hypothesis and the Axiom of Choice from Zermelo-Fraenkel set theory (ZF). It has since become a central method in set theory for constructing new models from given ones, particularly to demonstrate the relative consistency of various mathematical statements.
10.1 Basic Idea:
Extensions of Models: Forcing is a method to extend a model of set theory to a larger model in which certain desired properties hold. This larger model contains all the sets of the original model plus new sets that ensure the addition of a specific feature or property.
Forcing Conditions: Forcing involves defining a set of conditions (a forcing notion or poset) that describe how new sets should behave. These conditions help build the new sets that will extend the original model.
Generic Filter: A central concept in forcing is a generic filter, a collection of conditions that is consistent (no two conditions contradict each other) and meets certain criteria to ensure it captures the desired properties in the extended model. The generic filter is chosen so that it intersects with every dense subset of the poset in the original model, ensuring that the new sets added to the model are “generic” or “independent” of the original model’s sets.
10.2 Applications and Outcomes:
Independence Proofs: Forcing can demonstrate the independence of certain mathematical statements from ZF or ZFC. If you can force a statement to be true in one model and false in another, then that statement is independent of the axioms.
Real Numbers and Continuum Hypothesis: Forcing can create models where the Continuum Hypothesis holds and others where it does not, showcasing its independence from ZFC.
Large Cardinals and Consistency: Forcing can be used to show that if a certain large cardinal exists, then other mathematical statements are consistent. For example, if there is a model of ZFC with a large cardinal, forcing can be used to construct a model where additional set-theoretic properties hold without contradicting the existence of that large cardinal.
10.3 Technique:
Construction: The actual technique involves constructing a generic extension of a model. The new elements (or sets) are added according to the generic filter, ensuring they align with the desired properties.
Names and Interpretation: Forcing introduces the concept of “names” for sets in the extended model, with rules on how these names are interpreted in the generic extension. This interpretation helps define how new sets are constructed within the extended model.
Forcing is a sophisticated tool in set theory that requires a deep understanding of models, posets, and genericity. Its development has significantly expanded the scope and depth of set theory, allowing set theorists to explore a vast landscape of mathematical possibilities and understand the implications of different axioms and hypotheses in set theory.
11 Forcing and Smaller Models
You’re correct, and I appreciate the opportunity to clarify this. can be used to extend a model of set theory to a larger model, but the process can also be conceptualized in a way that seems to “reduce” the size or alter the characteristics of certain sets or cardinalities, particularly when viewed from a certain perspective.
In the context of the Solovay model and Lebesgue measurable sets, here’s how the process works more specifically:
Starting with a Large Cardinal: You begin with a model of set theory that includes a large cardinal, an inaccessible cardinal \(\kappa\). This model is “large” in the sense that it contains a rich and extensive hierarchy of sets up to \(\kappa\).
Collapsing the Cardinal: Using forcing, you collapse \(\kappa\) to become \(\aleph_1\), the cardinality of the set of all countable ordinals. This process doesn’t exactly reduce the size of the universe but changes the cardinality of certain sets within the model. The “largeness” of \(\kappa\) is essential because it ensures that there are enough sets below \(\kappa\) to build a rich structure that can mimic the real line and its power set in the resulting model.
Resulting Model: The end result is a model where the real numbers (and their subsets) behave as if they were constructed within a universe where \(\kappa\) is now countable. In this new model, every subset of the real numbers is Lebesgue measurable, despite the model still containing a version of the real numbers that is isomorphic to the “usual” real numbers.
Conceptualizing the “Reduction”: While forcing is used to extend models, the effect of collapsing a cardinal can be seen as “reducing” the complexity or the “size” of the universe concerning certain properties or cardinalities. It’s like creating a scaled model of the universe where the relative sizes of sets and their properties are altered, even though the model itself is a larger or extended version of the original universe up to \(\kappa\).
This process showcases the versatility of forcing: it can be used to construct models with a wide variety of properties by altering the relationships between sets and their cardinalities, which can sometimes be perceived as a “reduction” or simplification of certain aspects of the universe.
12 Algebraic Analogs of Forcing
An algebraic analogy to forcing, while not a perfect parallel, can be drawn from the concept of field extensions in algebra, particularly when you think about constructing larger fields from a given field by adding roots of polynomials that are not solvable within the original field.
12.1 Field Extensions and Forcing: A Conceptual Parallel
Base Field and Model of Set Theory: Consider a base field in algebra, like \(\mathbb{Q}\) (the rationals). In set theory, this is analogous to your base model \(M\) of ZFC, where \(M\) satisfies certain axioms and properties.
Extension Field and Generic Extension: Just as you might extend \(\mathbb{Q}\) to \(\mathbb{Q}(\sqrt{2})\) by adding a solution to the equation \(x^2 - 2 = 0\) that is not solvable in \(\mathbb{Q}\), in set theory, you extend \(M\) to \(M[G]\) by adding a set \(G\) (a generic filter) that is not in \(M\) but interacts with \(M\) in a well-defined way to create new sets and properties.
Solving Equations and Determining Properties: In field extensions, you’re often interested in solving polynomials or exploring properties (like constructibility with a compass and straightedge) that weren’t possible in the base field. In forcing, you’re adding sets to solve problems or realize properties (like every set of reals being Lebesgue measurable) that aren’t achievable in the base model.
Independence and Algebraic Closure: Extending a field can be seen as making certain algebraic statements true, like “there exists a number whose square is 2” when extending \(\mathbb{Q}\) to \(\mathbb{Q}(\sqrt{2})\). Forcing can make set-theoretic statements true, such as “there exists a model where all sets of reals are measurable.” Just as algebraic closure can make a field “complete” in a sense by solving all its polynomial equations, forcing can create a model where certain set-theoretic statements hold.
12.2 Limitations of the Analogy
While this analogy provides a conceptual bridge, there are limitations:
Complexity and Interaction: Forcing involves complex interactions between sets, conditions, and the generic filter, which have no direct counterparts in the relatively straightforward process of extending fields.
Role of Axioms: Forcing deals with models of axiomatic systems where the notions of truth and provability are central, which doesn’t have a direct analogue in field extensions.
Structural Impact: Field extensions typically don’t change the underlying structure of the existing elements, while forcing can introduce sets that fundamentally alter the nature of the extended model.
Despite these differences, the analogy provides a way to conceptualize forcing in a familiar algebraic context, offering a bridge to understand how new elements (whether algebraic or set-theoretic) can be introduced to expand a structure and explore new properties.
13 Generic Filters
A generic filter is a fundamental concept in the technique of forcing in set theory. It’s used to construct new models of set theory from given ones. Here’s a breakdown of what a generic filter is and its role in forcing:
13.1 Definition and Properties:
Partial Order: Forcing starts with a partially ordered set (poset) \((P, \leq)\), often called a forcing notion. The elements of \(P\) are considered as conditions, and the order \(\leq\) represents strengthening of conditions (with \(p \leq q\) meaning that \(p\) is stronger or more informative than \(q\)).
Filter: A filter \(G\) on \(P\) is a subset of \(P\) that is upward closed (if \(p \in G\) and \(p \leq q\), then \(q \in G\)) and downward directed (for any \(p, q \in G\), there exists \(r \in G\) such that \(r \leq p\) and \(r \leq q\)). Essentially, a filter chooses consistent conditions within \(P\).
Genericity: A filter \(G\) is generic over a model \(M\) of set theory if it intersects every dense subset of \(P\) that is in \(M\). A subset \(D\) of \(P\) is dense if for every condition \(p \in P\), there is a stronger condition \(q \in D\) such that \(q \leq p\). Genericity ensures that \(G\) is “large” or “sufficiently generic” in the sense that it captures enough conditions from \(P\) relative to the model \(M\).
13.2 Role in Forcing:
Extending Models: The generic filter \(G\) is used to extend the model \(M\) to a new model \(M[G]\), where \(G\) adds new sets to \(M\). The construction of \(M[G]\) involves interpreting “names” or “terms” in \(M\) under \(G\), effectively translating potential objects defined in \(M\) into actual objects in \(M[G]\).
Adding New Sets: The generic filter can add new subsets to existing sets (like adding a new subset of \(\omega\)) or ensure certain properties (like making a certain real number or subset of reals in \(M[G]\) have specific characteristics not predetermined in \(M\)).
Preserving Models’ Properties: While \(G\) adds new sets, the construction ensures that \(M[G]\) remains a model of set theory. The genericity of \(G\) helps preserve the consistency of the extended model and ensures that the axioms of set theory are still satisfied.
In summary, a generic filter in forcing is a carefully selected set of conditions from a poset that intersects with every dense subset in the model. It’s used to construct a new model where new sets are added, or certain properties are realized, while maintaining the consistency and axiomatic structure of the original model.