I just sent this posting to the FOM email list:
I am participating in a small email group concerning higher set theory, focusing originally on whether the continuum hypothesis is a “genuine” problem. The discussion has partly gone into issues surrounding the foundations of higher set theory. The ideas in this posting were inspired by the interchange there.
CAUTION: I am not an active expert in this kind of higher set theory, and so what I say may be either known, partly true, or even false.
As a modification of existing ideas concerning “maximality” I offer the following axiom over MK class theory with the global axiom of choice.
DEFINITION 1. Let be a sentence of set theory. We say that is set theoretically consistent if and only if ZFC + is consistent with the usual axioms and rules of infinitary logic. At the minimum, the standard axioms and rules of , with quantifiers ranging over , with epsilon,=, but one may consider the much stronger language where set length blocks of quantifiers are used. I think that for what I am going to do, it makes no difference, so that there is stability here.
Then we can formulate in class theory,
POSTULATE A. If a sentence is set theoretically consistent then it holds in some transitive model of ZFC containing all ordinals. It follows that it holds in some , a real.
Now what happens if we relativize this in an obvious way?
DEFINITION 2. Let be an infinite ordinal. Let be a sentence of set theory. We say that is set theoretically consistent relative to if and only if ZFC + is consistent with the usual axioms and rules of infinitary logic together with “all subsets of alpha are among the actual subsets of alpha”, where the latter is formulated in the obvious way using infinitary logic.
POSTULATE . If a sentence is set theoretically consistent relative to then it holds in some transitive model of ZFC containing all ordinals.
POSTULATE C. Postulate holds for all infinite .
To prove consistency, or consistency of this for some alpha, we seem to need
- An extension of Jensen’s coding the universe where we add a subset of + that codes the universe without adding any subsets of . This has probably been done.
- Cone determinacy for subsets of .E.g., using the equivalence relation if and only if are interdefinable over . I don’t know if this has been done.
Turning now to SRM once again from this link, let’s look at:
- 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 .
I originally said that 1-8 is a conservative extension of EFA, which is wrong. I corrected this by saying that 1-7,9 is a conservative extension of EFA. There are some subtle points, and it is possible that I might need some standard laws of exponentiation, maybe just , or even less, BUT: much easier is to use the full 1-9. 1-9 is a conservative extension of EFA. I really like the Foundational Traction here, as I unravel the subtleties of the situation properly.
Now to justifications of SRP.
On Sep 13, 2014, at 2:18 AM, Rupert McCallum wrote:
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.
Bill Tait wrote:
Thanks for the announcement, Rupert; I look forward to reading the paper.
In the interests of immodesty, let me mention that here is a bit of unclarity: I considered reflection principles for and used the to derive the existence of -ineffable cardinals. I also proved the consistent relative to a measurable. Peter showed that they are consistent relative to .
What was unfortunate was certainly not that Peter found the inconsistent for , rather it was my proposing them.
This illustrates a systematic problem with attempts at philosophical foundations of higher set theory. One formulates a reasonable looking idea, makes some associated philosophy, and obtains some partial information. Then perhaps somebody shows there is an inconsistency. Then one adjusts the philosophy to explain why the stuff that seems fine was a good idea, and the stuff that turned sour was, in retrospect, a bad idea.
Of course, so far, we have seen that proposals that survive a lot of attention and work, especially detailed structure (not so clear what this means for proposals compatible with V = L), invariably have not, so far, led to inconsistencies. Kunen with Reinhardt’s wasn’t around too long, and didn’t have any structure theory. However, I1,I2,I3 have been around a long time, but not too much structure theory, still a fairly substantial amount of research. What confidence should we have that they are consistent?
I don’t have any problem with this process as research – there is little or no alternative. BUT, in order to respond to inquiries like “Why should I believe subtle cardinals exist, or that they are consistent with ZFC?” from top core mathematicians, it is not going to be well received in anything like its present form. Probably also my approach with is still not innovative enough.
Having said this, it still appears that there is a kind of comfort zone with ZFC. But I explain this as follows. ZF is a kind of unique transfer from the finite world, and I have reported on this some time ago on the FOM.
I have just seen an article by Rupert McCallum which FOM readers may find of interest. It also contains a good list of references. A quick glance reminds me of the SRP characterization in terms of countable transitive models with outside elementary embeddings.
My feeling at the moment is that this area of justifying cardinals relatively low down (SRP hierarchy = subtle cardinal hierarchy = ineffable cardinal hierarchy) needs a really striking simple new idea.