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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?

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?

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This reads more like an editorial than a mathematics question. The discussion of slavery seems especially off-topic. –  S. Carnahan Nov 6 '10 at 7:27
It's not clear to me that CH "should be" solvable. CH is of technical interest for fields like combinatorics and topology, and historically was of great importance, but that just makes it interesting, not solvable. It's not obvious that anything in the nature of "pure hereditary sets" causes either CH or its negation to hold - and the current evidence is strongly to the contrary. –  Carl Mummert Nov 6 '10 at 11:39
See these two articles by Shelah, specially the first one: Logical Dreams, The Future of Set Theory. –  Kaveh Nov 9 '10 at 7:05

2 Answers 2

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:

Hilbert, in his celebrated address to the Second International Congress of Mathematicians held at Paris in 1900, expressed the view that all mathematical problems are solvable by the application of pure reason. At that time, he could not have anticipated the fate that awaited the first two problems on his list of twenty-three, namely, Cantor's Continuum Hypothesis and the problem of the consistency of an axiom system adequate to develop real analysis. The Gödel Incompleteness Theorems and the Gödel-Cohen demonstration of the independence of CH from ZFC make clear that continued confidence in the unrestricted scope of pure reason in application to mathematics cannot be founded on trust in its power to squeeze the utmost from settled axiomatic theories which are constitutive of their respective domains. The goal of our Workshop is to consider the extent to which it may be possible to frame new axioms for set theory that both settle the Continuum Hypothesis and satisfy reasonable standards of justification. The recent success of set theorists in establishing deep connections between large cardinal hypotheses and hypotheses of definable determinacy suggests that it is possible to find rational justification for new axioms that far outstrip the evident truths about the cumulative hierarchy of sets, first codified by Zermelo and later supplemented and refined by others, in their power to settle questions about real analysis. The Workshop will focus on both the exploration of promising mathematical developments and on philosophical analysis of the nature of rational justification in the context of set theory.

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.

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One philosophical issue that occurs to me is that is isn't clear whether the multiverse view is about multiple concepts of set, or about a single concept of set that is not fully specified. Of course one can say every model of set theory represents its own concept of "set", but that seems reversed somehow. For example there is no standard model of a group but we don't think of different groups as giving different concepts. In geometry the Erlangen program provided a unified concept that could cover various models of geometry. –  Carl Mummert Nov 6 '10 at 12:03
@Carl: Either way, it's a multiverse. –  Andrej Bauer Nov 6 '10 at 19:33
Thanks Joel, interesting slides. A couple follow-up questions: 1. You claim that the multiverse view allows one to remain a platonist, we now become platonists about the existence of separate universes. This doesn't seem satisfying for someone seeking a firm ontology, since it leads to a regress - why not talk about multimultiverses? (I'm not sure I believe any longer that mathematics should have a firm ontology). –  Amit Kumar Gupta Nov 6 '10 at 19:44
2. Naively, there is just one $\mathbb{N}$. Your multiverse view requires there to be many $\mathbb{N}$'s, akin to category theory having a general notion of a natural numbers object. Have you always been okay with the idea that there are many different $\mathbb{N}$'s? Otherwise, how did this picture emerge for you? –  Amit Kumar Gupta Nov 6 '10 at 19:45

As was pointed out in some answers to this question, since the large cardinal axiom are linearly ordered by consistency strength, there is a natural direction in which we can strengthen set theory.
Since we want to work in a strong theory, it makes sense to assume the existence of large cardinals. If sufficient large cardinals exist, a lot of questions about projective sets of real and so on are decided, which is good.

On the other hand, the existence of large cardinals alone does not decide CH.
However, Woodin is working on finding new axioms that are "natural" and do decide things like CH. The ideas involved in this are things like "the universe should be canonical in some sense, but still sufficiently rich" and "forcing should not have to much of an effect on the set-theoretic universe".

Until some time ago, these considerations indicated that the size of the set of real numbers should be $\aleph_2$, recent results of Woodin seem to hint at CH.

Shelah takes a completely different view on this: ZFC is the right foundation of mathematics, and much more is provable in ZFC than was initially thought after the invention of forcing. For example, a version of the generalized continuum hypothesis actually holds after removing some "initial noise" from the picture.
Unfortunately, most of these results are not really concerned with objects of everyday mathematics.

I don't know whether anyone has compiled a list of statements that should be decidable. CH should certainly be on the list, but it seems that mainstream mathematics is happy with (and, to a large extent, not really interested in) the current status of the foundations of mathematics.

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Hi Stefan, thanks for the response. I would say Replacement is a natural axiom basically because it's intuitive. It seems to reflect an intuition about how sets should behave. The large cardinal axioms may be linearly ordered, but I don't necessarily see why that fact makes them "natural." The fact that they settle many questions also doesn't seem to be a good argument for their truth, just their pragmatic utility. "$V=L$ settles many questions too. –  Amit Kumar Gupta Nov 6 '10 at 16:24
Dear Amit, the striking point is that whenever someone came up with a plausible set-theoretic statement whose consistency strength was above that of ZFC, the consistency strength fit in somewhere in the large cardinal hierarchy (in some cases the true strength of a statement is not known. But there are no indications that these statements don't fit into the hierarchy). Usually you would exspect that there are some statements that cannot be compared in strength, but this has not happened yet. –  Stefan Geschke Nov 6 '10 at 17:51
Comment continued: This adds a lot of credibility to large cardinals: Whenever someone tries to strengthen ZFC in consistency strength (i.e., not just by adding axioms equiconsistent with ZFC such as CH or MA), the strengthening goes in the same direction: towards large cardinals. So, if you want the strongest natural theory, add large cardinal axioms. –  Stefan Geschke Nov 6 '10 at 17:57
By the way, Randall Holmes has an argument that says that full Replacement is actually not as intuitive as it may seem. The point is, if you build the von Neumann hierarchy of sets (iterating the power set operation) you wind up with a picture of the universe that looks like it should satisfy Replacement. However, this is only obvious for instances of Replacement with formulas that are absolute. Hence Randall proposes a weaker form of Replacement that only talks about formulas of low complexity, Replacement for $\Sigma_2$-formulas. – –  Stefan Geschke Nov 6 '10 at 18:09
Hi Stefan. I agree that these features of the large cardinal hierarchy are striking. And I agree that adding them to ZFC gives us more provability strength. But I'm not fully convinced that this adds credibility to them. Peano's axioms are justified on a classical account of truth; we accept them because they reflect the way we think numbers truly behave. (continued in the next comment) –  Amit Kumar Gupta Nov 6 '10 at 19:52

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