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The set-theoretic multiverse
Joel David Hamkins
TL;DR
The paper addresses whether set theory has one absolute universe or many distinct set concepts and universes. It defends the multiverse view by interpreting forcing and related constructions as access to alternative universes, and argues that the continuum hypothesis is settled through extensive knowledge of its behavior across those worlds. The paper also notes limitations in some forcing formalizations, especially their reliance on assumptions or metamathematical arguments beyond ZFC.
Problem
The paper addresses the conflict between a single absolute set-theoretic universe and the diverse alternative universes revealed by set-theoretic practice.
Method
It defends the multiverse view by analyzing forcing, inner and outer models, and the mathematical access these constructions provide to alternative set-theoretic universes.
Results
The paper concludes that extensive knowledge of CH and ¬CH worlds settles the continuum hypothesis on the multiverse view and prevents settling it through a new natural deciding axiom.
Takeaways & Limitations
The multiverse view treats set-theoretic diversity as knowledge of multiple real universes rather than a phenomenon requiring reduction to one absolute universe.
Takeaways & Limitations
Some forcing approaches are limited because countable transitive models are not provably available in ZFC, while alternative consistency arguments do not themselves produce models and shift work into the metatheory.
Abstract
from arXiv · showhide
The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous diversity of set-theoretic possibilities, a phenomenon that challenges the universe view. In particular, I argue 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.
1. Introduction
The article contrasts the universe view’s single absolute set universe with a multiverse of distinct set concepts and corresponding universes. It argues that this multiverse perspective explains the diversity encountered in set theory and changes how the continuum hypothesis can be settled.
- The universe view posits one absolute concept of set and a universe in which every set-theoretic assertion has a definite truth value.
- The multiverse view posits diverse concepts of set, each instantiated in a corresponding universe with its own set-theoretic truths.The article identifies familiar models and forcing extensions as examples of such universes.
- The article argues that extensive knowledge of CH and ¬CH worlds settles the continuum hypothesis on the multiverse view, but rules out settling it through a new natural deciding axiom.The argument rests on experience with forcing both CH and its negation in diverse worlds.
- The multiverse is defended as a realist view of genuinely existing alternative universes, while allowing preferences among them.
- Set theory can investigate multiverse features mathematically through inner models, outer models, forcing extensions, modal logic, and geology.These investigations provide mathematical footholds for philosophical questions about set concepts.
2. The challenge of diverse set-theoretic possibilities
Set theory’s history has revealed an extensive diversity of alternative models, challenging the universe view’s claim that there is only one absolute set-theoretic universe. The multiverse view explains this experience by treating those alternative universes as real mathematical possibilities while retaining set theory’s foundational role.
- Set theory’s most prominent development has been the discovery and construction of a shocking diversity of set-theoretic possibilities.Forcing, ultrapowers, and canonical inner models are presented as tools for constructing and relating alternative universes.
- This diversity challenges the universe view because it must explain away alternative universes that appear fully set-theoretic to mathematicians.
- The multiverse view explains mathematicians’ experience by treating the alternative worlds revealed by their tools as real set-theoretic universes.
- Definable inner models are understood through their behavior inside different outer models, including forcing extensions.The example of HOD shows that definability can change across successive extensions.
- Forcing extensions most directly challenge the universe view because names and the forcing relation provide access to their objects and truths from a ground model.
- The multiverse view preserves set theory’s role as a foundation by locating familiar mathematical objects inside each universe.
3. The Ontology of Forcing
The dispute centers on whether forcing extensions are genuinely existing universes outside V or merely formal constructions within an absolute universe. Forcing provides controlled access to alternative ZFC models, while different implementations carry distinct metamathematical costs.
- The ontology of forcing: The central question is whether forcing extensions V[G] exist beyond V, which the universe view denies and the multiverse view treats as fully real.The multiverse view regards V as a relative notion and allows each universe to have corresponding forcing extensions.
- Constructing extensions: Forcing begins with a ground model V and a partial order P, producing extensions that can exhibit precisely controlled alternative set-theoretic truths.The method was initially used to prove independence results, including for choice and the continuum hypothesis.
- Constructing extensions: Every forcing extension V[G] satisfies ZFC, and its statements are connected to definable forcing conditions in the ground model.The forcing relation is definable in V for fixed statements or fixed complexity.
- Methodological limitations: The countable transitive ground-model method restricts forcing to certain models and cannot establish the existence of the transitive ZFC models it requires.This creates a stronger-hypothesis tax: the method may need assumptions whose consistency strength exceeds that of the theory being studied.
- Methodological limitations: A consistency-based alternative avoids assuming countable transitive models but does not itself produce a model and shifts important parts of the argument into the meta-theory.The approach relies on finite fragments and reflection, mixing internal finiteness with metatheoretic reasoning.
- The multiverse interpretation: The naturalist account effectively permits moving from any current universe V to a forcing extension V[G], while the multiverse view interprets this practice as evidence that such extensions actually exist.Set theorists access these extensions through names and the forcing relation even though the access is imperfect.
4. The analogy between set theory and geometry
The paper compares the multiverse view with the acceptance of non-Euclidean geometries. Alternative set-theoretic universes, initially treated as simulations useful for independence results, are presented as worlds that set theorists have learned to study as fully real.
- Historical analogy: Geometry shifted from treating non-Euclidean systems as curiosities or simulations to accepting them as genuine alternative geometrical worlds.The paper uses this historical transition as the model for understanding the multiverse in set theory.
- Historical analogy: Set theory underwent a parallel shift as L and forcing made alternative set-theoretic universes known and increasingly central to research.The universe view still regards these alternatives as not fully real, whereas the multiverse view accepts them as fully real.
- Modes of access: The analogy concerns not only ontology but also method: mathematicians study alternative geometries from another geometry, just as set theorists study alternative universes from a given universe.This provides a model for reasoning about worlds that are accessed through another mathematical setting.
- Modes of acceptance: The paper presents acceptance of irrational, negative, imaginary, and complex numbers, followed by non-Euclidean geometries, as precedents for accepting V-generic filters.The conclusion frames V-generic filters as the next initially puzzling mathematical objects to be accepted as real.
5. Multiverse response to the categoricity arguments
The multiverse view challenges categoricity arguments by questioning whether distinct set concepts can be compared from a coherent common background. It also emphasizes that even arithmetic categoricity depends on a prior concept of sets and subsets.
- Categoricity arguments: Categoricity arguments support the universe view by claiming that set-theoretic concepts can be compared level-by-level through the ordinals.Martin’s comparison proceeds by agreement at successive stages of the cumulative hierarchy.
- Multiverse response: The multiversist rejects the presumption that two set concepts can be compared coherently without a shared background concept of set or property.Using either set concept as the comparison context does not establish that it can internally refer to the other.
- Multiverse response: Within a fixed background, agreement between any set concept claiming to contain all sets and that background does not establish categoricity across incompatible backgrounds.The argument changes substantially once different set-theoretic backgrounds are admitted.
- Multiverse response: The transfinite comparison also requires the compared concepts to agree sufficiently on ordinals and well-ordering, which the multiverse view does not assume.Each concept may be internally coherent while appearing inadequate from a common background.
- Related perspective: The categoricity tradition’s apparent convergence with Grothendieck universes illustrates how foundational universe concepts can arise in related mathematical practice.The cited comparison notes that the concepts nearly coincide, apart from treating Vω as a Grothendieck universe.
- Natural numbers: Peano’s second-order categoricity proof depends on a fixed concept of subsets of the natural numbers, so the apparent absoluteness of finite number carries set-theoretic baggage.Different set concepts may therefore yield different or incomparable conceptions of natural number.
6. The multiverse provides a context for universe adjudication
The multiverse can emulate any selected universe view by restricting attention to one universe and its lower cone. It also supplies the broader setting needed to compare competing candidates for the absolute universe.
- Simulating the universe view: A multiverse set-theorist can simulate the universe view by selecting a particular universe V and temporarily ignoring worlds outside its lower cone.Within that cone, V functions as the absolute background universe.
- Adjudicating universe proposals: Disagreement over which universe V should be treated as absolute is itself a question about choosing among set concepts within the multiverse.The dispute includes which universe’s features and truths should count as those of the real universe.
- Adjudicating universe proposals: The multiverse provides the natural forum for comparing competing proposals for a unique absolute background universe.This comparison is presented as a specifically multiverse task rather than an internal commitment to one universe view.
7. Case study: multiverse view on the continuum hypothesis
The continuum hypothesis is used as a case study for the multiverse view: CH and ¬CH are both extensively understood and forceable, so CH is settled as a multiverse question rather than by a single new decisive axiom. The paper argues that this experience rules out the traditional dream solution of finding an obviously true principle deciding CH.
- The CH case study: CH is a major set-theoretic question that is neither provable nor refutable from the usual ZFC axioms.The paper uses this independence as the starting point for comparing universe and multiverse treatments.
- Forcing both alternatives: Both CH and ¬CH are forceable over any model of set theory, showing that the two alternatives remain broadly available throughout the multiverse.This goes beyond merely knowing that CH is independent.
- Forcing both alternatives: Every universe V has forcing extensions with ¬CH that collapse no cardinals and with CH that add no new reals.The theorem gives two contrasting extensions with these stated preservation properties.
- Forcing both alternatives: CH behaves like a lightswitch because relatively mild forcing extensions can turn it on or off while preserving the original universe’s large cardinals.The paper presents this as the result of decades of detailed study of CH and ¬CH worlds.
- Multiverse conclusion: On the multiverse view, CH is settled by detailed knowledge of where it holds or fails and how either outcome combines with other set-theoretic properties.The paper therefore rejects describing CH as an open problem, while allowing that particular combinations may remain unresolved.
- The dream solution: The traditional dream solution sought an obviously true principle Φ whose consequences would decide CH or ¬CH.Its first step requires producing such a natural principle before proving that it determines CH.
- The dream solution: The paper argues that rich experience with both CH and ¬CH worlds makes any principle implying either outcome unacceptable as obviously true.Accepting Φ would conflict with the corresponding worlds that set theorists have extensively studied and constructed.
- Examples against the dream solution: Freiling’s axiom of symmetry illustrates the problem: it is equivalent to ¬CH but is not generally accepted as a solution because of concerns about non-measurable sets and functions.The episode instead warns against naive treatment of measure concepts.
8. Case study: multiverse view on the axiom of constructibility
The multiverse view treats V = L as one among many set-theoretic universes rather than an absolute limiting axiom. Results about transitive and constructible models show that substantial set-theoretic strength and arbitrary reals can appear within models satisfying V = L.
- V = L is commonly viewed as restrictive because it permits only constructible sets and often settles questions in counterintuitive ways.Its adoption is criticized for limiting interpretative power and producing counterexamples or obstacles rather than a unifying theory.
- On the multiverse view, “the” constructible universe denotes a rough grouping of different models satisfying V = L.Different universes can have different ordinals, reals, large cardinals, and arithmetic truths.
- V and L have transitive models of exactly the same constructible theories, including theories asserting extremely strong large cardinals.The equivalence follows from the complexity of having a transitive model and Shoenfield absoluteness.
- Every real can occur in a model of a chosen finite ZFC fragment plus V = L that is well-founded beyond any chosen countable ordinal.Under additional conditions, such models can satisfy full ZFC and be pointwise definable.
- Even a generic real produced by collapsing ω1 can belong to a well-founded model of ZFC + V = L that regards the real as constructible.The model may place the real at a necessarily nonstandard stage beyond the ambient universe’s ω1.
- Every transitive model can in principle be continued upward to a model of V = L, supporting the view that constructibility need not permanently restrict a universe.The article describes this as V = L becoming a possible “rewarder” of sufficiently patient extension.
9. Multiverse Axioms
The multiverse axioms articulate closure principles for treating constructed, extended, taller, and otherwise related models as genuine universes. They also expose formal and foundational limits: some principles are philosophically broader than standard first-order set theory, while others are redundant in toy formulations.
- Formalism and scope: The multiverse vision exceeds standard formalism for universe-existence principles, although questions about forcing and geology can still be formalized in first-order set theory.The article therefore presents the subject as requiring both technical mathematics and philosophical analysis.
- Basic universe existence principles: The Realizability Principle treats any model of set theory definable or interpretable in a universe as a universe in its own right.This expresses the multiverse capacity to regard constructed universes as real mathematical structures.
- Outer-model principles: The Forcing Extension Principle asserts that every universe has a forcing extension V[G] for every forcing notion P it contains.The article presents this as a minimal foundational commitment engaging the central universe–multiverse disagreement.
- Reflection: The Reflection Axiom says every universe is elementarily represented as an initial segment of a much taller universe with the same truths.Its formal statement requires a taller W and an ordinal θ with V ≾ Wθ ≺ W.
- Countability: The Countability Principle makes countability relative by asserting that every universe is countable from the perspective of another universe.This follows the observation that suitable forcing extensions can make any set countable.
- Absorption: The Absorption into L principle asserts that every universe is a countable transitive model inside another universe satisfying V = L.It generalizes the earlier observation that countable transitive models can be continued to models of constructibility.
10. Appendix: multiverse-Inspired Mathematics
Multiverse-inspired mathematics treats forcing extensions, grounds, and related inner models as alternative set-theoretic worlds whose structural relationships can be studied systematically. Results show that forcing has a common provable modal core, while particular models and geological structures can exhibit substantial variation.
- Modal logic of forcing: Forcing extensions form a robust Kripke model of possible mathematical worlds, with universes accessible through forcing.This modal perspective expresses general principles relating forcing and truth.
- Maximality Principle: The Maximality Principle asserts that every possibly necessary assertion is already necessary and is relatively consistent with ZFC.It expresses the idea that the universe is complete with respect to what forcing can make permanently true.
- Modal logic of forcing: The ZFC-provably valid principles of forcing are exactly S4.2, assuming ZFC is consistent.The proof establishes both validity of S4.2 and failure of every stronger candidate in an appropriate model.
- Modal logic of forcing: Different models of set theory can have forcing-validity theories ranging from S4.2 to S5.S4.2 occurs in models of V = L, while S5 occurs in models satisfying the Maximality Principle.
- Set-theoretic geology: Set-theoretic geology studies inner models by reversing the usual forcing perspective, treating a universe as a forcing extension of a ground.A ground W satisfies V = W[G] for some W-generic G, and a bedrock is a minimal ground satisfying the ground axiom.
- Set-theoretic geology: Every model of ZFC is the mantle of another model, so the mantle cannot be assumed to be a uniquely regular ancient core.Under directedness hypotheses, the mantle has better structural behavior, but the strong downward-directed grounds hypothesis is not known to be universal.
- Set-theoretic geology: Grounds are uniformly definable and parameterized, while questions about bedrocks and bottomless models remain open or model-dependent.Every ground is W_r for some parameter r, and the relation x ∈ W_r is first-order; some models have no bedrock.