As I understand it, there is a program in set theory to produce an ultimate, canonical model of set theory which, among other things, positively answers the Continuum Hypothesis and various questions about large cardinals. My question is about the motivation of such a program. I'll present some of the thoughts that motivate my question and then state my question more precisely at the end.
Naively, one would think that a good first order theory for some subject ought to be able to decide any first-order expressible question about that subject. Given certain reasonable restrictions on what "good" must entail, this is impossible, due to Godel. Still, there is a sense that even though a good theory cannot answer every first order question, it should answer a reasonable subset of them. Further still, there is a sense that for any given subject, there are certain first order propositions that a good theory should not only decide, but decide to be true.
What I just said may sound vague, so in the next two paragraphs I'll add examples of things I personally ought to be true or decidable in a good theory. The point of these examples is simply to provide background and motivation to the question, and clarify any vagueness, the actual content of those paragraphs is just my opinion and not the real point of the question I'm asking here.
Things that ought to be true
Robinson Arithmetic doesn't decide induction, but induction ought to hold in any good theory of numbers. Replacement and Foundation aren't decided by Zermelo set theory, but they ought to hold in any decent theory of sets, even if it took decades for these axioms to join the rest of Zermelo's axioms. Now there may be interesting theories extending Robinson Arithmetic in which induction fails, and interesting theories extending Zermelo set theory in which Replacement or Foundation fail, but these aren't good theories of numbers or sets, respectively. Now I cannot prove induction or Replacement, and I may not be able to convince the extreme skeptic that my beliefs are nothing more than the result of cultural bias and upbringing. Nonetheless, I can confidently assert that induction is true amd Replacement holds. And even if I were tempted to prove these claims based on some more fundamental assumptions, the skeptic could just as well question those assumptions, and this leads to infinite regress.
Things that ought to be decidable
A good theory of numbers ought to decide Goldbach's Conjecture. A good theory of sets probably ought to decide CH and the existence of various large cardinals. Again, I can't prove these claims, I'm simply making normative claims about what ought to be true of a theory if it is to be regarded as good. On the other hand, a good theory need not decide its own consistency. Excessively contrived formulas need not be decidable by a good theory either (e.g. formulas constructed simply to prove a certain theory is incomplete).
Main Question
I feel that CH and the existence or non-existence of large cardinals should be decidable in a good set theory, or in the same vein, if there are to be some canonical models of set theory, they should all decide these questions the same way.
But various prominent set theorists${}^{\dagger}$ believe further that CH should be true and large cardinals should exist in the "true $V$." What are some of the motivations for these beliefs?
[EDIT]
Although I feel this question is different from ones that have already been asked here, let me further distinguish my question by adding that I've heard the usual responses such as:
- There are models with very nice structural properties in which large cardinals exist
- Large cardinal axioms form a surprisingly linear hierarchy
- They decide many natural questions
- They fit together in a way that gives a nice, coherent picture of the universe
- Why not add them?
These responses implicitly appear to justify large cardinal axioms on some non-classical, non-platonist notion of truth – some combination of an aesthetic/pragmatic/coherentist theory of truth. So perhaps I should refine my question to be:
What motivates many set theorists to evaluate new axioms by these non-classical standards (or am I way off base)?
I should emphasize that I'm not only interested in the justification of these new axioms, but the motivation behind justifying these axioms the way that set theorists appear to justify them, as these axioms seem to be justified in a categorically different way from how Peano's axioms or Zermelo's axioms are justified.
[/EDIT]
${}^{\dagger}$This Wikipedia article on Large Cardinals mentions the Cabal for instance.
Secondary Question
I've made some specific claims about specific statements in specific theories that I feel ought to be true or ought to at least be decidable. Admittedly I haven't given any general explanations for what sort of things ought to be true, false, decidable, or neither, I've just stated my opinion on a few specific sentences. I doubt one could give a totally general account distinguishing the class of problems that ought to be decidable from the class of problems that needn't be (e.g. make a categorical distinction between statements "like" CH versus statements "like" Godel's self-referential sentence). I don't think it's the type of question amenable to total generalization or formalization. Nonetheless:
Can anyone shed some light on the apparent distinction between questions that a good theory ought to decide and those a good theory needn't decide?
Best Answer
Two weeks ago a conference was held on precisely the topic of your question, the Workshop on Set Theory and the Philosophy of Mathematics at the University of Pennsylvania in Philadelphia. The conference description was:
Speakers included Hugh Woodin, Justin Moore, John Burgess, Aki Kanamori, Tony Martin, Juliette Kennedy, Harvey Friedman, Andreas Blass, Peter Koellner, John Steel, James Cummings, Kai Hauser and myself. Bob Solovay also attended.
Several speakers have made their slides available on the conference page, and I believe that they are organizing a conference proceedings volume.
Without going into any details, let me say merely that in my own talk (slides here) I argued against the position that there should be a unique theory as in your question, by outlining the case for a multiverse view in set theory, the view that we have multiple distinct concepts of set, each giving rise to its own set-theoretic universe. Thus, the concept of set has shattered into myriad distinct set concepts, much as the ancient concepts of geometry shattered with the discovery of non-euclidean geometry and the rise of a modern geometrical perspective. On the multiverse view, the CH question is a settled question---we understand in a very deep way that the CH and $\neg$CH are both dense in the multiverse, in the sense that we can easily obtain either one in a forcing extension of any given universe, while also controlling other set-theoretic phenomenon. I also gave an argument for why the traditionally proposed template for settling CH---where one finds a new natural axiom that implies CH or that implies $\neg$CH---is impossible.
Meanwhile, other speakers gave arguments closer to the position that you seem to favor in your question. In particular, Woodin described his vision for the Ultimate L, and you can see his slides.