Toy multiverses of set theory

This is a talk at the Philosophy of Set Theory and Foundations workshop, University of Konstanz, August 1, 2019.
Slides

In the early 20th century, optimism ran high that a formal development of mathematics through first-order logic within appropriate axiom systems would set up mathematical knowledge as incontrovertible truth. With the introdution of set theory, all mathematics could now be unified within a single field whose fundamental properties were codified by the Zermelo-Fraenkel axioms ${\rm ZFC}$. But already within a few decades, discoveries were made which suggested that the concept of set was prone to relativity. Löwenheim and Skolem showed that if there is a model of ${\rm ZFC}$, then there is a countable model of ${\rm ZFC}$. This countable model would, of course, think that its reals are uncountable, demonstrating the non-absoluteness of countability. Gödel proved the Completeness Theorem, an immediate consequences of which was the existence of nonstandard models of set theory, even with nonstandard natural numbers, demonstrating the non-absoluteness of well-foundedness. Soon afterwards, Gödel's First Incompleteness Theorem showed the general limitations inherent in formal method: no computable theory extending ${\rm ZFC}$ could decide the truth of all set theoretic assertions. In the next few decades came forcing, large cardinals, canonical inner models, and a model theoretic treatment of ${\rm ZFC}$. Set theory ever since has indisputably been the study of a multiverse of universes each instantiating its own concept of set.

Cohen's technique of forcing showed how to extend a given universe of sets to a larger universe satisfying a desired list of properties. Forcing could be used to construct universes in which the continuum was any cardinal of uncountable cofinality (including singular cardinals), indeed the continuum function on the regular cardinals could be made to satisfy any desired pattern (modulo a few necessary restrictions). There could be universes with or without a Suslin line, universes with non-constructible reals, universes whose cardinal characteristics had different values and different relationships.

At the same time, set theorists started exploring the consistency strength hierarchy of large cardinal axioms asserting existence of improbably large infinite objects. Other set theoretic assertions, no matter how unrelated to large cardinals, inevitably fell somewhere inside the hierarchy. Set theorists could now combine forcing with large cardinals to produce universes in which large cardinals were compatible with other desired properties. Many of the larger large cardinals implied that the universe had transitive sub-universes (inner models) into which it would elementarily embed. Such sub-universes, obtained from large cardinals or through other considerations, joined the menagerie of mathematical worlds populating the field.

In yet another direction, motivated by Godel's construction of $L$, set theorists began exploring how to build ever more sophisticated canonical inner models, universes built bottom-up from a recipe specifying what kind of sets should exists.

Finally, a model theoretic treatment of ${\rm ZFC}$ produced some unexpected results. Given two models $M$ and $N$ of ${\rm ZFC}$, let us say that $N$ is an end-extension of $M$ if $M$ is transitive in $N$, let us say that $N$ is a top-extension of $M$ if every element of $N\setminus M$ has higher rank than all ordinals of $M$. Keisler and Morley showed that every countable model of ${\rm ZFC}$ has a proper elementary top-extension (Keisler-Morley Extension Theorem [1]). Thus, any countable model could be grown in height without changing its theory. Barwise showed that every countable model of ${\rm ZFC}$ has an end-extension to a model of $V=L$ (Barwise Extension Theorem [2]). Thus, sets could become constructible in a larger universe.

What does set theoretic practice tell us about the nature of sets? Is there the one true universe of sets, the mathematical world in which all mathematics lives, but our epistemic limitations have so far prevented us from understanding its structure? Or is there simply no one true universe, and instead the multiverse of universes encountered by set theorists is the true mathematical reality?

The Universist position in the philosophy of set theory asserts that there is the one true universe of sets. Incompleteness is simply a by-product of formal methods we are forced to use to investigate mathematics, not an indication of some inherent relativity in the nature of sets. After all, in spite of a plethora of nonstandard models of arithmetic, most mathematicians would agree that the natural numbers is the only true model of arithmetic. The Peano Axioms, actually do a rather excellent job of capturing the fundamental properties of natural numbers, they are bound by the incompleteness phenomena but number theorists don't often run into statements independent of them. This certainty with regards to number theory comes from the deep intuition we believe we hold about the nature of natural numbers. The goal of set theory should then be to obtain the same deep intuition about the nature of sets so that ${\rm ZFC}$ can be extended to a theory deciding most of the more significant properties of sets. Woodin for instance believes that such a theory will come from a construction of a sufficiently sophisticated canonical inner model compatible with all known large cardinals.

The Multiversist position in the philosophy of set theory asserts that there is simply no one true universe of sets. All universes of sets discovered by set theorists: forcing extensions, canonical and non-canonical inner models, models with and without large cardinals, have equal ontological status. They all populate the multiverse of mathematical worlds, each instantiating its own equally valid concept of set. The Multiversist position splits along the lines of just how non-absolute should the notion of set be under this view. Can the multiverse contain models of different heights? Are ill-founded models allowed? The Radical Multiversist position, espoused most notably by Hamkins, asserts that all these models belong in the multiverse. Indeed, once you adapt the position that there is no absolute set theoretic background, no background aught to be better than any other and in fact each should be as bad as the next in the following sense. In the Hamkins perspective each universe in the multiverse will be seen to be countable and ill-founded from the perspective of a better universe. In particular, under this view the universist is mistaken even about the nature of the natural numbers. The Hamkins position is captured by his Multiverse Axioms.

Realizability: If $M$ is a universe and $N$ is a definable class in $M$ such that $M$ believes that $N\models{\rm ZFC}$, then $N$ is a universe. (This includes set models, inner models, will-founded models, etc.)

Forcing Extension: If $M$ is a universe and $\mathbb P\in M$ is a forcing notion, then there is a universe $M[G]$ where $G\subseteq\mathbb P$ is $M$-generic.

Class Forcing Extension: If $M$ is a universe and $\mathbb P$ is a definable ${\rm ZFC}$-preserving forcing notion in $M$, then there is a universe $M[G]$ where $G\subseteq \mathbb P$ is $V$-generic.

Reflection: If $M$ is a universe, then there is a universe $N$ such that $M\prec V^N_\theta\prec N$ for some rank initial segment $V^N_\theta$ of $N$. (Recall the Keisler-Morley Extension Theorem.)

Countability: Every universe $M$ is a countable set in another universe $N$.

Well-founded Mirage: Every universe $M$ is a set in another universe $N$ which thinks that $\mathbb N^M$ is ill-founded.

Reverse Embedding: If $M$ is a universe and $j:M\to N$ is an elementary embedding definable in $M$, then there is a universe $M^*$ and an elementary embedding $j^*:M^*\to M$ definable in $M^*$ such that $j=j^*(j^*)$. (Every elementary embedding has already been iterated many times.)

Absorbtion into $L$: Every universe $M$ is a transitive set in another universe $N$ satisfying $V=L$. (This is stronger than the Barwise Extension Theorem.)

A way to investigate the viability of a collection of multiverse axioms, without coming up with a formal background beyond ${\rm ZFC}$, is by considering toy multiverses of ${\rm ZFC}$ models. Assuming that ${\rm ZFC}$ is consistent, a toy multiverse is some set collection of models of ${\rm ZFC}$. Does a given collection of multiverse axioms have a toy multiverse, does it have a natural toy multiverse? With Hamkins, we showed that his Multiverse Axioms, radical as their assertions may seem, have a very natural toy multiverse, namely the collection of all countable computably saturated models [3]. In fact, every model in any toy multiverse satisfying the Hamkins Multiverse Axioms must be computably saturated because every model of ${\rm ZFC}$ that is an element of a model of ${\rm ZFC}$ with an ill-founded $\omega$ is computably saturated (see the previous post).

In a recent joint work with Godziszewski, Meadows, and Williams, we consider various weakenings of the Hamkins Multiverse Axioms that have toy multiverses not consisting entirely of computably saturated models. First, we consider the Hamkins Multiverse Axioms with a weak version of well-founded mirage which asserts that every universe in the multiverse is merely ill-founded, not necessarily at $\omega$, from the perspective of a better universe.

Weak well-founded mirage: Every universe $M$ is a set in another universe which thinks that its membership relation is ill-founded.

Recall that if there is a transitive model of ${\rm ZFC}$, then the Cohen-Shepherdson model $L_\alpha$ is the minimum transitive model of ${\rm ZFC}$.

Theorem: (G., Godziszewki, Meadows, Williams) The collection in $L_\alpha$ of all models $M$ such that $L_\alpha$ thinks that $M$ is a countable model of ${\rm ZFC}$ satisfies:

  1. Realizability
  2. Set Forcing and Class Forcing Extension
  3. Countability
  4. Weak Well-founded Mirage

Next we consider weakening the axioms to the so-called covering versions, where instead of demanding that a universe be a set inside another universe, we ask that each universe has, a cover, an end-extension that is a set in a better universe.

Covering Countability: For every universe $M$, there is a universe $N$ and a countable set $A\in N$ with $M\subseteq A$.

Covering Well-founded Mirage: For every universe $M$, there is a universe $N$ and a model $M^*\in N$ end-extending $M$ such that $N$ thinks that $\mathbb N^{M^*}$ is ill-founded.

Covering Absorbtion into $L$: For every universe $M$, there is a universe $N\models{\rm ZFC}+V=L$ and a model $M^*\in N$ end-extending $M$.

Theorem: (G., Godziszewki, Meadows, Williams) There is a toy multiverse, not all of whose models are computably saturated, satisfying:

  1. Realizability
  2. Forcing Extension
  3. Class Forcing Extension for ${\rm Ord}$-cc forcing
  4. Covering Countability
  5. Covering Well-founded Mirage
  6. Covering Absorbtion into $L$

In this post, I purposely didn't tell you what my philosophical position is on all of this. I will leave it up to the reader to guess.

References

  1. H. J. Keisler and M. Morley, “Elementary extensions of models of set theory,” Israel J. Math., vol. 6, pp. 49–65, 1968.
  2. J. Barwise, “Infinitary methods in the model theory of set theory,” in Logic Colloquium ’69 (Proc. Summer School and Colloq., Manchester, 1969), North-Holland, Amsterdam, 1971, pp. 53–66.
  3. V. Gitman and J. D. Hamkins, “A natural model of the multiverse axioms,” Notre Dame J. Form. Log., vol. 51, no. 4, pp. 475–484, 2010. Available at: http://dx.doi.org/10.1215/00294527-2010-030