The set-theoretic multiverse (1108.4223v1)

Abstract: 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.

The Set-Theoretic Multiverse: Philosophical Insights and Implications

The paper discussed here presents the multiverse view as a philosophical framework for understanding set theory, posited by Joel David Hamkins. This view stands in contrast to the universe view, which suggests that there is an ultimate, singular set-theoretic universe where every mathematical proposition has a definitive truth value. The multiverse perspective, however, suggests that there are multiple set-theoretic universes, each housing distinct concepts of sets and exhibiting various mathematical truths.

Key Concepts and Arguments

• Multiverse vs. Universe View: The universe view holds on to the notion of a unique set-theory background, whereas the multiverse view embraces diverse concepts of sets, each forming different universes with unique properties. This concept is philosophical yet deeply rooted in mathematical reasoning, challenging the idea of absolute truths within a single universe.
• Continuum Hypothesis (CH): An essential argument within the multiverse framework concerns the Continuum Hypothesis. On the universe view, the CH is a definitively open problem, seeking settlement through additional axioms. On the multiverse view, however, the CH is "settled" given the vast understanding of its range—the hypothesis can be toggled on or off via forcing across the multiverse, illustrating a comprehensive set-theoretic experience that does not necessitate a unique resolution.

Implications and Future Directions

The multiverse view has substantial implications for both the philosophy of mathematics and practical set theory:

• Independence Phenomena: By framing set theory within the multiverse context, one acknowledges the pervasive independence phenomena, such as CH and others, not as deficiencies in current theories but as reflections of the multiverse's non-absolute nature. It suggests that independence is a vital characteristic of the multiverse rather than a hindrance.
• Set-Theoretic Realism: This view promotes higher-order realism about universes, solidifying the position that any consistent set-theoretic construction is meaningful. Such realism maintains that these universes genuinely exist, shaping how mathematicians interpret truth across different models without commitments to singular foundational principles.
• Potential for New Research: The multiverse view opens avenues for exploring further into modal logic and set-theoretic geology. The paper of modal logic within set theory—how forcing affects potential truths—has been deeply influenced by viewing the universe as a multiverse. Similarly, set-theoretic geology explores the universe's "bedrock," considering the layers of different mathematical truths like examining geological strata.

Conclusion

The multiverse view champions a philosophical approach that embraces the rich diversity seen in set-theoretical practice. It provokes rethinking not just particular mathematical problems like CH but the broader ontological makeup of mathematical reality itself. Understanding set theory as an expansive multiverse rather than a static universe fosters innovation and deeper comprehension of foundational questions, urging the field towards bold philosophical and mathematical exploration. As set theory evolves, the multiverse perspective may provide compelling insights and resolve longstanding challenges in the discipline.