Here I refer to Hamkins' slides:
http://lumiere.ens.fr/~dbonnay/files/talks/hamkins.pdf
particularly, to the "Universe view simulated inside Multiverse", p. 22.
My question is: is it very unsound to ask if the Multiverse view could be simulated (in a similar sense) inside Universe?
If it is, why is it? If it is not, why should one prefer one view to the other?
Update.(Sep 6, 2011) My paper on the multiverse is now available at the math arxiv at The set-theoretic multiverse, and gives a fuller account of the ideas in the slides mentioned in the question. The particular issue of the question arises in the discussion of the toy-model approach to formalization, discussed on page 23, and also at greater length in my paper Set-theoretic geology, G. Fuchs, J.D. Hamkins, J. Reitz.
Thanks for the question; I'm glad you're interested in it.
The multiverse view in set theory is a philosophical position offered in contrast to the Universe view, an orthodox position, which asserts that there is a unique background set-theoretic context or universe in which all our mathematical activity takes place. On the Universe view, there are definitive final answers to the question of whether a given mathematical statement, such as the Continuum Hypothesis, is true or not, and we seek to find these answers. On the Universe view, the fact that such a statement is independent of ZFC or another weak theory is regarded as a distraction from the question of determining whether or not it is ultimately true. For example, many set theorists regard the accumulating regularity consequences of large cardinals for properties of sets of reals as indicating that the large cardinal hierarchy is on the right track towards the final set-theoretic truth.
A paradox for the universe view, which I mention in the slides to which you link, is that the most powerful set-theoretic tools that have informed a half-century of research in set theory are most naturally understood as methods of constructing alternative set-theoretic universes. That is, from a given set-theoretic universe we can construct others, by means of forcing, ultrapowers, inner models, definability, large cardinal embeddings and so on. Indeed, we can often construct models of set theory to exhibit exact precise properties, and forcing especially has led to a staggering diversity of models.
The multiverse view takes these diverse models seriously, holding that there are diverse incompatible concepts of set, each giving rise to a set-theoretic universe in which they are instantiated. The set-theoretic tools provide a means of modifying any given concept of set to a closely related concept of set, whose resulting universes can be fruitfully compared in a single mathematical context. For example, we can understand the relationship between a ground model concept of set and that of its forcing extensions. Although the multiverse includes all the familiar models of set theory that we have built by forcing and other methods, it likely also includes universes arising from other set concepts that we have not yet imagined. There seems to be little reason that any two given concepts of set can be compared together in one set-theoretic context.
Now, the Universe view seems simulable inside the multiverse view by the idea of picking a single universe $V$, call it the actual universe if you like, and then restricting attention only to the universes that are somehow describable from the perspective of $V$.
But you ask about the other direction. There are a few ways to do it in a partial manner, but none of them seems fully satisfactory.
First, one can mathematise the concept of multiverse, by just considering a multiverse as a collection of (set) models of set theory. For example, with Victoria Gitman (p. 44 of slides), we showed that if ZFC is consistent, then the set of all countable computably-saturated models of ZFC forms a multiverse satisfying all the multiverse axioms that I mention there (and others). This is just a straight theorem of ZFC. Since this multiverse does not include the set-theoretic background universe $V$ in which the collection was formed, however, we can recognize that it is not the full multiverse in which we are interested, and this is the sense in which we wouldn't really want to limit ourselves to that multiverse.
Second, if one is interested only the generic multiverse---the part of the multiverse reachable by forcing, that is, by closing under forcing extensions and grounds in a zig-zag pattern---then one can formalize the whole set-up within ZFC. By describing exactly which forcing notions were used, one can index the models by their methods of construction. By this means, the concept of $\varphi$ is true throughout the generic multiverse of $V$'' is first-order expressible in $V$ in the language of set theory. This kind of analysis is a full answer to your question if you care only about the generic multiverse as opposed to the full multiverse.
But other researchers have cared about the full multiverse, and in general there is no satisfactory way to simulate it from the universe perspective. For example, the Inner Model Hypothesis of Sy Friedman (see also Friedman, Welch, Woodin), is described as the assertion that if the universe $V$ has an outer model with an inner model having a certain property, then there is already an inner model of $V$ with the property. Such a statement is explicitly appealing to the multiverse concept, but one cannot seem easily to formalize it within set theory. Instead, the official account of IMH backs off to the case where $V$ is a countable model of set theory, which takes it somewhat away from its initial multiverse sense.
So to finally come to an answer to your question, it doesn't seem possible to fully simulate the multiverse perspective from within the universe view, in a way that is fully satisfactory.
You ask, as a follow up, in this case why should we prefer one view to the other?
Well, these are philosophical positions on the fundamental nature of mathematical existence; this is a philosophical dispute rather than a mathematical one. Nevertheless, one's mathematical philosophy often suggests certain mathematical problems as interesting or solution methods as promising. The multiverse perspective naturally leads one to compare set-theoretic universes, and this led to the research on the modal logic of forcing and set-theoretic geology (mentioned in the slides). The Universe perpsective may lead elsewhere, perhaps towards an investigation of universes with highly structural features.
Surely the mathematicians' measure of a mathematical philosophy is the value of the mathematics to which it leads...
There are two ways that come to my mind right now of how the multiverse view can be "simulated" in the universe:
One is to assume that there is an actual countable transitive model $M$ of ZFC.
(Note that the existence of $M$ cannot be proved in ZFC alone.)
Now you can freely consider the set (!) of all forcing extensions and all inner models
of $M$.
Another way is not to talk about actual models of set theory but about forcing notions instead. Here forcing notions are partial orders or, which has certain technical advantages as long you are more interested in the big picture/general theory rather than specific consistency results, complete Boolean algebras.
Given a complete Boolean algebra $A$ you can
define a forcing language and assign truth values in $A$ to each sentence of your forcing
language. Without going into too much detail here, the forcing language allows it to get
some access to truth in a forcing extension of the universe by $A$.
In this sense you can talk about properties of a forcing extension by $A$ already in the "ground model", the single universe under consideration.
