Multiverse vs Universe View

Background

The universe view of set theory fixes an interpretation of the $\in$-relation as ‘real’. The attitude towards nonstandard models w/in the universe view is that they are dismissed as ‘not real.’

The multiverse view of set theory does not fix an interpretation of the $\in$-relation as ‘real’, instead all interpretation of $\in$-relations are considered as equally real. Thus, there is no notion of ‘standard’ and ‘non-standard’ models.

The multiverse is a collection of all of a theory $T$. Thus, the multiverse can be defined as an elementary class $\mathcal{K}=\{\mathcal{M}|\mathcal{M}\models T\}$. $\to$ For a multiverse to be multiverse there must be at least two models which are not elementary equivalent to each other (which allows for different interpretations of statements).

Universe View: There is a single absolute universe of set $V$ w/ a fixed interpretation of the $\in$-relation which we take to be the objectively true '$\in$'-relation, and a true conception of set. Other interpretations from other structures are ‘nonstandard’ and are not viewed as the ‘correct’ interpretation.

Multiverse View: There is no single correct interpretation of the $\in$-relation, there are a variety of conceptions of sets each of which are equally valid. We work over a set of structures rather than w/in a single structure $V$ (i.e., by considering alternate universes such as generic extensions). - Horizontal Multiverse: The multiverse is a collection of all forcing extension of some ground model (usually taken to be the universe of set $V$). This multiverse conception realizes all possible ‘worlds’ of a model - Vertical Multiverse: The vertical multiverse is a collection of nested models $\{V_{\alpha }\models ZFC|\alpha \in ORD\}$ $\to$ i.e., the multiverse consists of sequentially taller universe of sets (stages of $V$).

E.g., If CH or $\neg$CH is true is relative to the model you are working in

Note: The notion of a ‘canonical’ model is a philosophical (and arguably subjective) one which can only apply to theories whose models are not unique up to isomorphism (i.e., non-categorical theories) such as $ZFC$. The Löwenheim-Skolem Theorem guarantees the existence of non-standard models given that the theory (in a countable language) has an infinite model.

Hyperuniverse: The collection of all countable transitive models of $ZFC$, can be treated as a multiverse (related to the toy-model approach).

Multiverse Conceptions

Full-Blooded Platonism (FBP): All conceivable mathematical universes exists. That is, there is a plurality of set-conceptions. $\to$ ‘Conceivable’, meaning consistent and set-theoretic, not necessarily model of just $ZFC$, but can also include models of weaker theories such as $ZF,Z{F}^{-},KP$. i.e., ‘fundamental’ axioms such as the axiom of existentiality and the axiom of foundation are kept.

Note: There is no complete formulation of FBP, only fragments of it via $2^{nd}$-order modal sentences.

FBP may imply unwarranted ontological inflation. Arithmetic statements are not changed via forcing, hence an object such as $\omega$ does not vary in models obtained through forcing $\to$ different conceptions of ’$\omega$.’ $\to$ Concerns about multiverse-membership; i.e., are ill-founded models of $ZFC$ considered as universes of set theory?

RFBP: A further extension of FBP which allows for models of inconsistent set-conceptions, essentially asserting that impossible objects can exist. This is related to extended modal realism and trivialism (see Philosophy and Trivialism).

Shelah’s Pluralism:

• There are different extensions of $ZFC$ each w/ their own collection of models, none of which are ‘canonical’ than the other.
• There are no preferred such extensions of $ZFC$, some axioms may be more useful in terms of their consequence, but this does not imply that they’re ‘better’ than other axioms (i.e., semi-axioms “As many possible set theories conforming to $ZFC$”

Issues:

• The notion of semi-axiom may be too vague
• The task of establishing new set-theoretic truths are ‘barred’
• Why did we pick up the $ZFC$ axioms and why do we stop w/ those (from the formalist conception)?

Woodin’s Position:

• Restrictive view
• The multiverse is a collection of all set generic extensions of a given model i.e.,
  • The multiverse is a collection of all $V_{\alpha }^{\mathbb{B}}$ (all Boolean-valued universes, which are generated through set-forcing).
  • Multiverse laws are fixed in an analogy w/ Tarski’s notion of truth
  • Through the $\Omega$-conjecture and the acceptance of a proper-class of Woodin cardinals, the multiverse laws are violated $\to$ As a consequence, the set-generic multiverse is flawed

Note: A set-generic extension will have the same cardinality as the ground model as seen externally from the universe of set $V$ (although, not necessarily internally as a model can be ‘forced’ to believe it is has a different cardinality, i.e., viewing itself as uncountable even though it is countable).

Toy-Model Approach: The multiverse view can be carried out w/in a family of models, we first select a countable, transitive model $\mathcal{M}$ of $ZFC$ and then construct a ‘horizontal’ multiverse (w/ in $V$) via taking the set of all forcing extensions $\mathcal{M}[G]$. Since the forcing extensions have the same cardinality, they will also be countable. i.e., $\mathcal{S}=\{\mathcal{M}[G]|\mathcal{M}[G]\text{ is a set-generic extensions of a countable, transitive model }\mathcal{M}\}$

Note: This family of sets is not a model of $ZFC$ but instead may satisfy certain multiverse axioms/rules.

This is in contrast to considering a multiverse of forcing extensions of the universe $V$, which works over proper-classes ,while the toy-model approach is done w/in $V$ by selecting a countable transitive element of it.

The toy-model approach can be used to gain insight about the universe $V$

Why must the ground model be countable (and transitive)?

Mirage Axiom: Every model of $ZF$ is seen as ill-founded (specifically, in there interpretation of the natural numbers, $\mathbb{N}$) by some other model.

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Modal Realism: All logically possible ‘worlds’ exist although are spatially and temporally isolated from each other (physical existence).

Extended Modal Realism: An extension of modal realism which admits the existence of impossible worlds (i.e., some which are paraconsistent and contradictions such as $p$ and $\neg p$ are both true, as well as completely trivial worlds where anything is true). Some may argue that impossible worlds do not even exists metaphysically (i.e., dictated on logic) rather than just physically.

Tegmark’s Ultimate Ensemble: There is a multiverse of universes, where each universe is described by (i.e., act as models for) an axiomatic mathematical theory (given that it is valid). Related to the notion of mathematics being ‘discovered’ not created. In this view, different systems of mathematics describe real physical universes which actually exists

note that set of all mathematical worlds is a subset of all logically possible worlds/

Full-Blooded Platonism: Each valid system of mathematics are a mathematical or platonic reality (not physical as described in Tegmark’s ultimate ensemble). Thus, w/in the context of set theory, we have a plurality of different set conceptions.

References

Philosophy of Sets & Proper-Classes Forcing Philosophy Trivialism Modal Logic