Concept
Multiverse (set theory)
Overview
In mathematical set theory, the multiverse view is that there are many models of set theory, but no "absolute", "canonical" or "true" model. The various models are all equally valid or true, though some may be more useful or attractive than others. The opposite view is the "universe" view of set theory in which all sets are contained in some single ultimate model.
The collection of countable transitive models of ZFC (in some universe) is called the hyperverse and is very similar to the "multiverse".
A typical difference between the universe and multiverse views is the attitude to the continuum hypothesis. In the universe view the continuum hypothesis is a meaningful question that is either true or false though we have not yet been able to decide which. In the multiverse view it is meaningless to ask whether the continuum hypothesis is true or false before selecting a model of set theory. Another difference is that the statement "For every transitive model of ZFC there is a larger model of ZFC in which it is countable" is true in some versions of the multiverse view of mathematics but is false in the universe view.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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This leads us to define the number of a given collection as the class of all collections that are similar to it; that is to say, we set up the following formal definition: "The number of terms in a given class" is defined as meaning "the class of all classes that are similar to the given class." This definition, as Frege (expressing it in slightly different terms) showed, yields the usual arithmetical properties of…
Bertrand Russell, Our Knowledge of the External World as a Field for Scientific Method in Philosophy
States what Multiverse is.
In that case we say the manifold of these systems of impressions forms a physical continuum _C_. And each of these systems is called an _element_ of the continuum _C_. How many dimensions has this continuum?
Henri Poincaré, The Foundations of Science
States what Multiverse is.
As regards the distinction between the mathematically and the dynamically unconditioned which is the aim of the regression of the synthesis, I should call the two former, in a more limited signification, cosmical conceptions, the remaining two transcendent physical conceptions. This distinction does not at present seem to be of particular importance, but we shall afterwards find it to be of some value.
Immanuel Kant, The Critique of Pure Reason
States what Multiverse is.
Related ideas
Place in the tradition
Part of Modal logic.