Maximality Principles in the Hyperuniverse Programme
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Maximality Principles in the Hyperuniverse Programme Sy‑David Friedman1 · Claudio Ternullo2 Accepted: 27 September 2020 © Springer Nature B.V. 2020
Abstract In recent years, one of the main thrusts of set-theoretic research has been the investigation of maximality principles for V, the universe of sets. The Hyperuniverse Programme (HP) has formulated several maximality principles, which express the maximality of V both in height and width. The paper provides an overview of the principles which have been investigated so far in the programme, as well as of the logical and model-theoretic tools which are needed to formulate them mathematically, and also briefly shows how optimal principles, among those available, may be selected in a justifiable way. Keywords Set theory · Universe of sets · Maximality principles · Hyperuniverse · New axioms
1 The Question of Maximality One of the main goals of current set-theoretic research is to identify and justify maximality principles for the universe of sets V.1 Although the standard axioms of set theory, that is, ZFC, provide a nice characterisation of all sets as elements of V, they are not able to establish the truth or falsity of fundamental set-theoretic statements. Therefore, one may legitimately expect that isolating further fundamental properties of V will provide us with crucial insights concerning these statements. 1
As is known, V is recursively defined as follows: V0 = �, V𝛼+1 = P(V𝛼 ) , where 𝛼 is a successor ordinal, ⋃ and V𝜆 = 𝛼 𝜅 is the least inaccessible after 𝜅 . The iterated procedure yields a ‘tower’ of nested models of ZFC2 , and for any two
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The topic is fully explored by one of the authors and Neil Barton in Barton and Friedman (2020). Zermelo’s proof is discussed (and defended) at length in Martin (2001). A philosophical discussion of Zermelo’s conception is also in Tait (1998). Also see the more recent Linnebo (2017), pp. 179–182. 15 Recall that an uncountable cardinal 𝜅 is inaccessible if and only if: (1) 𝜅 is regular (that is, its cofinality is 𝜅 ) and (2) 𝜅 is a limit cardinal (that is, it is not the successor of any cardinal). 𝜅 is strongly inaccessible if, in addition, is: (3) strong limit, that is, if for all cardinals 𝜆 < 𝜅 , 2𝜆 < 𝜅. 14
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Maximality Principles in the Hyperuniverse Programme
models V and V ∗ such that V ⊂ V ∗ , one has that the limit-number of V ∗ is greater than the limit-number of V. The collection of all V’s may also be described as a vertical multiverse, although Zermelo’s collection of stacked ‘normal domains’ is not, as is clear, a multiverse in the proper sense, nor is there any evidence that he construed the set-theoretic realm in a pluralist way: for him, set-theoretic reality was a fully determinate realm of objects, consisting of all sets in V; only, the concept ‘universe of all sets’ had, for him, an indefinite extension, insofar as what one might potentially take to be the collection of all sets at some level V𝜅 (where 𝜅 is, as said, at least inaccessible) can always automatically be extended
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