This note touches on three matters.
1. A correction to the SRM = strict reverse math system I presented.
The error and its correction are quite interesting. I’m writing a new paper on SRM which will have the right statement and break new SRM ground.
2. Formulations of the IMH = inner model hypothesis.
3. Concerning my need for a response to the Fields Medalist as to “why should I believe a subtle cardinal exists or is consistent?”
- Linearly ordered integral domain axioms.
- Finite interval. exists.
- Boolean difference. exists.
- Set addition. exists.
- Set multiplication. exists.
- Least element. Every nonempty set has a least element.
- . implies . is defined only if .
and said that the above is a conservative extension of EFA = exponential function arithmetic = .
8 needs to be changed to
This issue is not a problem in SRM formulations with finite sequences — we can get away with postulating that . In any case, one first proves that 1-7 above is conservative over PFA = polynomial function arithmetic = .
About IMH. Sy has just written to Pen Maddy,
“I am sorry that you take such a negative view of this programme’s philosophical merits. But at least you may have interest in the math that comes out of it!”
It appears that to some extent, Sy has followed a model of thought that I indicated in an earlier email. Namely, he felt some Foundational Traction with IMH, and then developed and fine tuned an associated philosophy. This is, for various reasons, evidently not received positively clearly by Pen and Peter, and to some extent, not by Hugh and John either.
I can see some Foundational Traction in the IMH. However, it needs to be reformulated in order to have more. I quote from “On the consistency strength of the Inner Model Hypothesis“:
The Inner Model Hypothesis (IMH): If a statement without parameters holds in an inner model of some outer model of V (i.e., in some model compatible with V), then it already holds in some inner model of V.
Equivalently: If a statement is internally consistent in some outer model of then it is already internally consistent in . This is formalised as follows. Regard as a model of Gödel-Bernays class theory, endowed with countably many sets and classes. Suppose that is another such model, with the same ordinals as . Then is an outer model of ( is an inner model of ) iff the sets of include the sets of and the classes of include the classes of . is compatible with iff and have a common outer model.
CAUTION: I AM NOT AN ACTIVE EXPERT IN THIS KIND OF HIGHER SET THEORY I AM A SEMI-CASUAL CONSUMER. SO WHAT I SAY BELOW MAY BE KNOWN, OR ONLY PARTIALLY TRUE, OR COMPLETELY FALSE.
They show that, using some PD (and hence a big enuf large cardinals by Martin/Steel), for all reals of sufficiently high Turing degree, the least transitive model of ZFC containing satisfies IMH. They also use IMH to get models of strong kinds of measurable cardinals using inner model theory for strong kinds of measurable cardinals (Mitchell). There seems to be good deep higher set theory here.
If I were writing that paper, I would have included some friendly warmup material to allow casual and semi-casual readers to get more out of it. In particular, a warmup proof that comes out of IMH, presumably using Jensen’s original covering theorem. And maybe a second warmup for . Also, there may be strategic warmups for the construction of models of IMH. But maybe there are some expositional notes on IMH that are more friendly??
The formulation is somewhat awkward for any kind of intrinsic treatment of set theoretic maximality. Clearly IMH has something to do with maximality, but the philosophy gets strained under the present formulation of IMH (at least the formulation from this IMH paper).
IMH as a statement in class theory paradoxically is formulated in terms of outer models of V. But according to maximality, V doesn’t have any (proper) outer models. So the outer models of V must be taken to be imaginary.
So one way to go is this:
INFORMAL HYPOTHESIS. If a sentence holds in some imaginary transitive model of ZFC containing all ordinals, then it holds in some actual transitive model of ZFC containing all ordinals.
An obvious candidate for “imaginary transitive model of ZFC” is given by class forcing, where of course one does not get involved with any actual generic objects – because V is in fact maximal and so generic objects don’t really exist. I.e., on this interpretation, a sentence holds in an “imaginary transitive model of ZFC” if and only if it is forced by all conditions in some appropriate class forcing that does exist. Appropriate means that it is ZFC friendly.
I guess that this is equivalent by Jensen coding to INFORMAL HYPOTHESIS. There is a real number such that the following holds. If a sentence holds in some imaginary transitive model of ZFC containing all ordinals, then it holds in some inner model of .
Now class forcing seems to be too technical to be a satisfactory interpretation of “imaginary transitive model of ZFC”. Instead, we can simply require that a sentence be consistent with a strong form of the axioms and rules of inference of infinitary logic applied to ZFC. This seems more general than class forcing. Using definable cone determinacy and Jensen coding, it does appear that the infinitary proof rule formulation can be shown to be consistent – by throwing it into countable models, where the infinitary axioms and rules of inference are complete and hence give actual models to do Jensen coding on.
Thus on this formulation, countable models only appear in a consistency proof, and not in the actual formulations.
With regard to my question about what the best answer is to that senior Fields Medalist who asks “Why is there a subtle cardinal, or why is it consistent with ZFC?” There is the following exchange so far on the FOM.
Rupert McCallum writes:
William Tait wrote an essay that appeared in “The Provenance of Pure Reason” called “Constructing Cardinals from Below” which discussed a set of reflection principles that justify SRP. Unfortunately Peter Koellner later observed that some of the reflection principles he considered were inconsistent. I wrote down my own thoughts in a recent Mathematical Logic Quarterly article about how one might find principled grounds for distinguishing the consistent ones from the inconsistent ones.
I sent in to FOM:
I’m sure that the FOM readers would be most interested if you could give a simple brief account of the ideas behind some of the reflection principles that work – at least in the sense that they can be obtained from standard large cardinal hypotheses. Of course, subtle cardinals themselves are based on a very simple idea – but that idea would not normally be characterized as reflection.
For just subtle, we have is essentially subtle if and only if is a cardinal such that for all binary relations on , there exists infinite such that the sections of at agree below .
Note that essentially subtle is closed upward, so it is not quite the same as being subtle. HOWEVER, the first subtle cardinal is exactly the first essentially subtle cardinal. ALSO “there exists a subtle cardinal” is equivalent to “there exists an essentially subtle cardinal”.
If FOM readers relate to your simple brief account, they can of course delve into publications. FOM readers can also get a chance to interact online starting from what you write.
PS: Maybe I see how to do this using some arguable reflection using multiple universes. Let’s consider two universes and , where is longer than . Let’s not worry about the most philosophically honest way to formalize this just yet.
Let be a binary relation on and let be a sentence that holds in . “Reflection” says that there exists kappa in V such that holds in . This seems to prove Con(ZFC + “there exists a subtle cardinal”). I think that if you use then you will get Con(SRP).