12.4 Contemporary research directions in set theory

Set theory research is pushing the boundaries of mathematical knowledge. Large cardinal axioms and inner models are helping us understand the structure of the set-theoretic universe. These concepts are key to exploring consistency and independence results.

Forcing and set-theoretic geology are powerful tools for building new models and studying their relationships. Meanwhile, descriptive set theory and determinacy are shedding light on the nature of "definable" sets and their properties. These areas are at the forefront of set theory today.

Large Cardinals and Inner Models

Large Cardinal Axioms and Their Implications

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• Large cardinal axioms extend ZFC by positing the existence of cardinals with strong properties
• These axioms form a hierarchy, with stronger axioms implying the existence of larger cardinals
• Examples of large cardinal axioms include inaccessible, measurable, and supercompact cardinals
• Large cardinal axioms have consequences for the structure of the set-theoretic universe ($V$)
• They also have implications for the consistency strength of various mathematical theories

Inner Model Theory and the Search for Canonical Models

• Inner model theory studies transitive class models of set theory that contain all the ordinals
• The goal is to construct canonical inner models that satisfy certain large cardinal axioms
• These models provide a way to measure the consistency strength of large cardinal axioms
• Examples of inner models include $L$ (the constructible universe) and $L[U]$ (the universe constructible from a measurable cardinal $U$)
• Inner models can be used to prove consistency results and independence results in set theory

Woodin Cardinals and the Ultimate $L$

• Woodin cardinals are a type of large cardinal that play a central role in modern set theory
• They are defined using the notion of a "Woodin filter" and have strong reflection properties
• The existence of Woodin cardinals has many consequences, such as the determinacy of certain games
• Woodin's Ultimate $L$ program aims to construct a canonical inner model that satisfies all large cardinal axioms not known to be inconsistent
• This would provide a unified framework for studying the set-theoretic universe and its properties

Forcing and Set-Theoretic Geology

Forcing and the Independence of the Continuum Hypothesis

• Forcing is a technique for extending models of set theory by adding new sets
• It was introduced by Paul Cohen to prove the independence of the Continuum Hypothesis (CH) from ZFC
• Forcing allows for the construction of models where CH holds and models where CH fails
• The basic idea is to start with a model $M$ and a partial order $\mathbb{P} \in M$, then construct a generic extension $M[G]$ by adding a generic filter $G \subseteq \mathbb{P}$
• Properties of the generic extension depend on the choice of the partial order $\mathbb{P}$

Set-Theoretic Geology and the Modal Logic of Forcing

• Set-theoretic geology studies the collection of all models that can be obtained by forcing over a given model
• It uses modal logic to express properties of this collection, with the modal operators interpreted as quantifiers over forcing extensions
• The Maximality Principle (MP) states that any sentence that holds in some forcing extension already holds in the ground model
• The Inner Model Hypothesis (IMH) asserts that any sentence that holds in an inner model of the universe also holds in an inner model of the ground model
• These principles have consequences for the structure of the set-theoretic universe and the existence of certain large cardinals

Descriptive Set Theory and Determinacy

Descriptive Set Theory and the Projective Hierarchy

• Descriptive set theory studies the properties of "definable" subsets of Polish spaces (complete, separable metric spaces)
• The projective hierarchy classifies subsets of Polish spaces based on their complexity
• The first level of this hierarchy consists of the analytic sets ($\Sigma^1_1$) and the coanalytic sets ($\Pi^1_1$)
• Higher levels are defined by alternating existential and universal quantification over the reals
• The study of the projective hierarchy has led to important results in set theory, such as the consistency of the Axiom of Determinacy (AD) with ZF

Determinacy and Large Cardinals

• The Axiom of Determinacy (AD) states that for any two-player game with perfect information and a subset $A$ of the reals as the payoff set, one of the players has a winning strategy
• AD contradicts the Axiom of Choice (AC) but is consistent with ZF
• Determinacy for certain classes of sets (e.g., the analytic sets) follows from the existence of large cardinals
• For example, the existence of infinitely many Woodin cardinals implies that all projective sets are determined
• The study of determinacy and its relationship to large cardinals has been a major focus of modern set theory research