A Reconstruction of Steel’s Multiverse Project

Maddy, Penelope; Meadows, Toby

doi:10.1017/bsl.2020.5

Cited by 15 publications

(23 citation statements)

References 23 publications

(20 reference statements)

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“…The collection of all models M G provides a complete semantics, and the axiom which guarantees the completeness of MV T is, as has been shown by [Maddy and Meadows, 2020], p. 25, Amalgamation. But, as stressed by the authors, a consequence of this fact is that, in any model of the form M G , not all generic filters for posets in M may be taken into account to produce forcing extensions which act, as required by the Meta-Theoretic Constraint of section 2.1, as the 'worlds' of MV, but only those which are produced by the Coll(ω, Ord M )-generic filter G over M .…”

Section: Models For MVmentioning

confidence: 87%

“…Further discussion of Steel's MV axioms may be found in [Maddy and Meadows, 2020] (for the discussion of the 'core hypothesis', see, in particular, sections 5-6).…”

Section: Steel's Programmementioning

confidence: 99%

“…16 Both [Maddy, 2017], pp. 310ff., and [Maddy and Meadows, 2020], pp. 7-8 stress the importance of this fact, by contrasting Steel's conception with those of Hamkins and Woodin, which would, on the contrary, openly commit to significantly more ontological forms of multiversism, and to a 'metaphysics of universes' from the beginning.…”

Section: Natural Theories Large Cardinals and Worldsmentioning

confidence: 99%

“…28 [Steel, 2014], p. 166. For further details on (Transl), see [Maddy and Meadows, 2020], section 5, pp. 30-38.…”

Section: G and The Translation Functionmentioning

confidence: 99%

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Steel's Programme: Evidential Framework, the Core and Ultimate-L

Bagaria¹,

Ternullo²

2020

Preprint

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We address Steel's Programme to identify a 'preferred' universe of set theory (and the best theory extending ZFC), in the context of the multiverse axioms MV and of the 'core hypothesis'. In the first part, we examine the evidential framework for MV, in particular the use of large cardinals, and of 'worlds' obtained through forcing, to represent and bind together alternative extensions of ZFC. In the second part, we address the existence and the possible features of the core of MVT (where T is ZFC+Large Cardinals). In the last part, we discuss the hypothesis that the core is Ultimate-L, and examine whether and how the Core Universist might use salient facts about the core provable in MVT to settle on V =Ultimate-L as the best (and ultimate) extension of ZFC. To this end, we take into account several strategies, and assess their prospects in the light of MV's evidential framework.We may re-state Weak Absolutism in a slightly different, but maybe more perspicuous, way, as follows:Core Universism. Set theory is the theory of multiple (set-theoretic) universes, each of which contains a unique core universe, which has a better claim to be seen as the 'preferred' universe of sets than any other universe.To state very quickly the main differences between the two positions: the Classic Universist may be standardly characterised as someone believing that our intuition of sets, or the 'concept of set' itself, will provide us with a unique, consistent extension of the ZFC axioms which will uniquely fix the truth-value of the undecidable statements. 11 By contrast, the Core Universist may be characterised as someone who takes all alternative 'universes' as equally legitimate; however, such a Universist will also hold that each universe (or, if you wish, theory) contains 'traces' of a single, 'preferred' universe, and much of the value of the position consists in showing that the claim is true, that is, that a core universe is really detectable within the multiverse itself. Of course, the Core Universist also expects to be able to describe the properties of the core in a satisfactory way.In order to attain a reduction of set-theoretic incompleteness, the Classic Universist will suggest further exploration of our intuitions about sets, or sharpening of the concept of set, whereas the Core Universist will suggest further exploration of the properties of the core through the multiverse axioms. Now, it is clear that Steel's Programme advocates the Core Universist's standpoint, and, therefore, has deep implications on the preferability of Core over Classic Universism: as already said, the Programme, if successful, would lend support to Core Universism. Indeed, the Core Universist's construal of the Programme's goals and results could be condensed as follows: 'Non-pluralism about set-theoretic ontology cannot be correct, as we are aware of the existence of many alternative universes (as well as of alternative theories extending ZFC). However, given a suitable version of the multiverse, one resting upon significant bits of current s...

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Section: Models For MVmentioning

confidence: 87%

“…Further discussion of Steel's MV axioms may be found in [Maddy and Meadows, 2020] (for the discussion of the 'core hypothesis', see, in particular, sections 5-6).…”

Section: Steel's Programmementioning

confidence: 99%

Section: Natural Theories Large Cardinals and Worldsmentioning

confidence: 99%

“…28 [Steel, 2014], p. 166. For further details on (Transl), see [Maddy and Meadows, 2020], section 5, pp. 30-38.…”

Section: G and The Translation Functionmentioning

confidence: 99%

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Steel's Programme: Evidential Framework, the Core and Ultimate-L

Bagaria¹,

Ternullo²

2020

Preprint

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“…What we are told by Steel, at the very outset, is that a set-theoretic statement does qualify as 'natural' if it is consistent with ZFC, asserts some 'facts' about sets, and is not of a metamathematical or proof-theoretic nature, but this is really not much. 21 Moreover, MV's scope for 'naturalness' is too restrictive from the beginning, as it leaves out theories, such as ZF+AD, which unquestionably express deep set-theoretic facts.…”

Section: Natural Theories Large Cardinals and Worldsmentioning

confidence: 99%

STEEL’S PROGRAMME: EVIDENTIAL FRAMEWORK, THE CORE AND ULTIMATE-L

Bagaria¹,

Ternullo²

2021

The Review of Symbolic Logic

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We address Steel’s Programme to identify a ‘preferred’ universe of set theory and the best axioms extending $\mathsf {ZFC}$ by using his multiverse axioms $\mathsf {MV}$ and the ‘core hypothesis’. In the first part, we examine the evidential framework for $\mathsf {MV}$ , in particular the use of large cardinals and of ‘worlds’ obtained through forcing to ‘represent’ alternative extensions of $\mathsf {ZFC}$ . In the second part, we address the existence and the possible features of the core of $\mathsf {MV}_T$ (where T is $\mathsf {ZFC}$ +Large Cardinals). In the last part, we discuss the hypothesis that the core is Ultimate-L, and examine whether and how, based on this fact, the Core Universist can justify V=Ultimate-L as the best (and ultimate) extension of $\mathsf {ZFC}$ . To this end, we take into account several strategies, and assess their prospects in the light of $\mathsf {MV}$ ’s evidential framework.

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Intellectual Autobiography

Maddy

2024

Outstanding Contributions to Logic

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A Reconstruction of Steel’s Multiverse Project

Cited by 15 publications

References 23 publications

Steel's Programme: Evidential Framework, the Core and Ultimate-L

Steel's Programme: Evidential Framework, the Core and Ultimate-L

STEEL’S PROGRAMME: EVIDENTIAL FRAMEWORK, THE CORE AND ULTIMATE-L

Intellectual Autobiography

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