Notice that by Tarski's theorem on the undefinability of truth, the truth constraint is trivially satisfied by the standard conception of truth in set theory which takes the multiverse to contain a single element, namely, V. In other words, one would have to adopt a multiverse conception of the multiverse, a multiverse conception of the multiverse conception of the multiverse, and so on, off to infinity.
Theorem 4. In summary: There is evidence that the only way out is the fourth way out and this places the burden back on the pluralist—the pluralist must come up with a modified version of the generic multiverse. In , Chris Freiling presented an argument against CH by showing that the negation of CH is equivalent to Freiling's axiom of symmetry , a statement about probabilities.
This would complete the parallel with the first step. There appear to be four ways that the advocate of the generic multiverse might resist the above criticism.
How, then, is one to adjudicate between them?
So it passes the first test. One way to formalize this is by taking an external vantage point and start with a countable transitive model M. This motivates the shift to views that narrow the class of universes in the multiverse by employing a strong logic. For example, a strict finitist might be a non-pluralist about PA but a pluralist about set theory and one might be a non-pluralist about ZFC and a pluralist about large cardinal axioms and statements like CH.
One way of providing a foundational framework for such a view is in terms of the multiverse. Theorem 4. Putting everything together: It is very likely that this statement is in fact true ; so this line of response is not promising. So it passes the first test. This motivates the shift to views that narrow the class of universes in the multiverse by employing a strong logic.
Moreover, this latter fact is in conflict with the spirit of the Transcendence Principles discussed in Section 4. There is evidence coming from inner model theory which we shall discuss in the next section to the effect that the conjecture is in fact false. Theorem 5. How then shall one select from among these theories? Notice also that if one modifies the definability constraint by adding the requirement that the definition be uniform across the multiverse, then the constraint would automatically be met.
For ease of comparison we shall repeat these features here: The first step is based on the following result: Theorem 5. Is the generic multiverse conception of truth tenable? Let the generic multiverse conception of truth be the view that a statement is true simpliciter iff it is true in all universes of the generic multiverse.
More recently, Matthew Foreman has pointed out that ontological maximalism can actually be used to argue in favor of CH, because among models that have the same reals, models with "more" sets of reals have a better chance of satisfying CH Maddy , p. So the above response is not available to the advocate of the generic-multiverse conception of truth. Notice also that if one modifies the definability constraint by adding the requirement that the definition be uniform across the multiverse, then the constraint would automatically be met.
Skolem argued on the basis of what is now known as Skolem's paradox , and it was later supported by the independence of CH from the axioms of ZFC since these axioms are enough to establish the elementary properties of sets and cardinalities. There is also a related constraint concerning the definability of truth.
The multiverse conception of truth is the view that a statement of set theory can only be said to be true simpliciter if it is true in all universes of the multiverse. So it is reasonable to expect that this statement is resolved by large cardinal axioms.
Notice again that by Tarski's theorem on the undefinability of truth, the definability constraint is trivially satisfied by the degenerate multiverse conception that takes the multiverse to contain the single element V. In this section and the next we will switch sides and consider the pluralist arguments to the effect that CH does not have an answer in this section and to the effect that there is an equally good case for CH in the next section. Putting everything together: It is very likely that this statement is in fact true ; so this line of response is not promising. Foreman does not reject Woodin's argument outright but urges caution.