Dear Pen, Sol and others,
It occurs to me that some of my disagreements with Pen and Sol could be resolved just by being clear about how the term “Set Theory” is used.
As I see it, Set Theory is three things:
- It is a branch of mathematics.
- It is a foundation for mathematics.
- It is the study of the concept of set.
Regarding 3: It is plain as pie that there is indeed a “concept of set”, familiar to schoolchildren who are victims of the “new math” (Venn diagrams, essentially). Even kids understand basic set-theoretic operations; probably once they are out of short pants they understand what we mean by powerset.
Now take a look at the “standard” axioms of ZFC. Why are they “standard”? It’s because we all seem to feel that they are “essential to Set Theory”. But there are two distinct sources for believing that:
As Boolos clarified in his paper on the iterative conception (IC), the axioms of Zermelo set theory are derivable from the concept of set as expressed by that conception. Replacement is not derivable from the IC, but it easily follows once we invoke Maximality, i.e. we strengthen the IC to the MIC (maximal iterative conception), also part of the concept of set.
As Pen has clearly expressed, the Axiom of Choice is a different matter: It does not follow from the MIC, but it does follow from the role of Set Theory as a foundation for mathematics. She can say this better than I, but the idea is that mathematics did much better once the old restrictive idea of set given by a rule was liberated through AC.
Now here we come to an important distinction that is ignored in discussions of Thin Realism: The Axiom of Choice didn’t get elected to the club because it is beneficial to the development of Set Theory! It got elected only because of its broader value for the development of mathematics outside of Set Theory, for the way it strengthens Set Theory as a foundation of mathematics. It is much more impressive for a statement of Set Theory to be valuable for the foundations of mathematics than it is for it to be valuable for the foundations of just Set Theory itself!
In other words when a Thin Realist talks about some statement being true as a result of its role for producing “good mathematics” she almost surely means just “good Set Theory” and nothing more than that. In the case of AC it was much more than that.
This has a corresponding effect on discussions of set-theoretic truth. Corresponding to the above 3 roles of Set Theory we have three notions of truth:
- True in the sense of Pen’s Thin Realist, i.e. a statement is true because of its importance for producing “good Set Theory”.
- True in the sense assigned to AC, i.e., a statement is true based on Set Theory’s role as a foundation of mathematics, i.e. because it is important for the development of areas of mathematics outside of Set Theory.
- True in the intrinsic sense, i.e., derivable from the maximal iterative conception of set.
- Pen’s model Thin Realist John Steel will go for Hugh’s Ultimate-L axiom, assuming certain hard math gets taken care of. Will he then regard it as “true” based on its importance for producing “good Set Theory”? I assume so. If not, then maybe Pen will have to look for a new Thin Realist.
- Examples here are much harder to find! What have axioms beyond ZFC done for areas of math outside of Set Theory? Surely forcing axioms have had some dramatic combinatorial consequences, but large cardinals haven’t yet had a similar impact. Descriptive Set Theory has had recent and major implications for functional analysis, but the DST being used is just part of good old ZFC. To understand this situation better I think it’s time for set-theorists to stop being so self-centered and to take a close look at independence outside of set theory, with the aim of seeing which axioms beyond ZFC are the most fruitful for resolving those cases of independence (I’m happy to lead the charge!).
- Small large cardinals come easily out of the MIC. Precisely what I am doing with the HP is to derive further consequences. Maybe the negation of CH! Work in progress.
Now I see absolutely no argument for rejecting any of these three notions of Truth in Set Theory. Nor do I see an argument that they should reach common conclusions! Maybe you’ll find this to be excessively diplomatic, taking the heat and excitement out of the Great Set Theory Truth Debate, but I’m sure that even if we agree to this proposed Grand Truce, we’ll still find interesting things to argue about.
As I understand it (I am happy to be corrected), Pen is no fan of Type 3 truth and Sol is no fan of Type 1 truth. OK, I have nothing against aesthetic preferences. But to say that an answer to the Continuum Problem based on one of these three takes on Truth is “illegitimate” is going too far. If someone is going to say that CH is true (or false) then she has to say what notion of Truth is being referenced. Indeed, maybe CH is Type 2 true but Type 3 false!
In any case, it is clearly very hard (but in my view possible) to come to conclusions about what is true in any of these senses. As I have emphasized in the HP (Type 3 truth), for me to make a verdict about CH I will have to first produce “optimal” maximality criteria and show that CH is decided in the same way by those criteria. That is very hard work. For Type 2 truth one would similarly have to show that the statements of Set Theory which are most fruitful for the further development of Set Theory as a foundation for mathematics converge on a theory which settles CH. We have barely begun an investigation of the class of such statements!
I am most pessimistic about Type 1 truth (Thin Realism). To get any useful conclusions here one would not only have to talk about “good Set Theory” but about “the Best Set Theory”, or at least show that all forms of “good Set Theory” reach the same conclusion about something like CH. Can we really expect to ever do that? To be specific: We’ve got an axiom proposed by Hugh which, if things work out nicely, implies CH. But then at the same time we have all of the “very good Set Theory” that comes out of forcing axioms, which have enormous combinatorial power, many applications and imply not CH. So it seems that if Type 1 truth will ever have a chance of resolving CH one would have to either shoot down Ultimate L, shoot down forcing axioms or argue that one of these is not “good Set Theory”. Pen, how do you propose to do that? Forcing axioms are here to stay as “good Set Theory”, they can’t be “shot down”. And even if Ultimate L dies, there will very likely be something to replace it. Why should we expect this replacement for Ultimate L to come to the same conclusion about CH that forcing axioms reach (i.e. that CH is false)?
Nevertheless, as a stubborn optimist I do still expect that at least one of these forms of truth will generate some useful conclusions. But I have given up on the idea that there is a unique, supreme notion of truth in Set Theory that overrides all others; there are at least three distinct and legitimate forms to be taken seriously (despite my pessimism about Thin Realism). And maybe there is even yet a another form of set-theoretic truth that I have overlooked.
Best to all,