Dear Pen and Peter,
I am sorry to have annoyed you with the issue of TR and Type 2 evidence, indeed you have made it clear many times that the TR does take such evidence into account. I got that! But in your examples, you vigorously hail the virtues of and other developments that have virtually no relevance for math outside of set theory, rather than Forcing Axioms, which provide real Type 2 evidence! As I said, I think you got it very right with your excellent “Defending”, but in your 2nd edition you might want to hail the virtues of Forcing Axioms well above , Ultimate L (should it be “ripe” by the time of your 2nd edition) or other math-irrelevant topics, giving FA’s their richly-earned praise for winning evidence of Types both 1 and 2.
I was really hoping for your reaction to the following, but I guess I ain’t gonna get it:
Hence my conclusion is that the only sensible move for us to make is to gather evidence from all three sources: Set theory as an exciting and rapidly-developing branch of math and as a useful foundation for math, together with evidence we can draw from the concept of set itself via the maximality of V in height and width. None of these three types (which I have labelled as Types 1, 2 and 3, respectively) should be ignored.
Let me make this more specific. Look at the following axioms:
- V = L
- V is not L, but is a canonical model of ZFC, generic over L
- Large Cardinal axioms like Supercompact
- Forcings Axioms like PFA
- AD in
- Cardinal Characteristics like
- (The famous) “Etcetera”
It seems that each of these has pretty good Type 1 evidence (useful for the development of set theory, with P’s and V’s).
But look! We can discriminate between these examples with evidence of Types 2 and 3! Type 2 comes down HARD for Forcing Axioms and V = L, as so far none of the others has done anything important for mathematics outside of set theory. And of course Type 3 kills V = L. So using all three Types of evidence, we have a clear winner, Forcing Axioms!
I expect that without a heavy use of Type 2 and Type 3 evidence, we aren’t going to get any consensus about set-theoretic truth using only Type 1 evidence.
Thanks again for your comments and the time you are putting in with the HP.
1. (Height Maximality, Transcending First-Order) #-generation provides a canonical principle that is compatible with V = L and yields all small large cardinals (those consistent with V = L). In the sense to which Hugh referred, its conjunction with V = L is a Shelavian “semi-complete” axiom encompassing all small large cardinals.
But of course #-generation is not first-order! That has been one of my predictions from the start of this thread: First-order axioms (FA’s, Ultimate L, , …) are inadequate for uncovering the maximal iterative conception. Height maximality demands lengthenings, width maximality demands “thickenings” in the width-actualist sense. We don’t need full second order logic (God forbid!) but only “Gödel” lengthenings of V (and except for the case of Height Maximality, very mild Gödel lengthenings indeed). We need the “external ladder” as you call it, as we can’t possibly capture all small large cardinals without it!
2. (Predictions and Verifications) The more I think about P’s and V’s (Predictions and Verifications), the less I understand it. Maybe you can explain to me why they really promote a better “consensus” than just the sloppy notion of “good set theory”, as I’m really not getting it. Here is an example:
When Ronald solved Suslin’s Hypothesis under V = L, one could have “predicted” that V = L would also provide a satisfying solution to the Generalised Suslin Hypothesis. There was little evidence at the time that this would be possible, as Ronald only had Diamond and not Square. In other words, this “prediction” could have ended in failure, as perhaps it would have been too hard to solve the problem under V = L or the answer from V = L would somehow be “unsatisfying”. Then in profound work, Ronald “verified” this “prediction” by inventing the “fine-structure theory” for L. In my view this is an example of evidence for V = L, based on P’s and V’s, perhaps even more impressive than the “prediction” that the properties of the Borel sets would extend all the way to via large cardinals (Ronald didn’t even need an appeal to anything “additional” like large cardinals, he did it all with V = L). Now one might ask: Did someone really “predict” that the Generalised Suslin Hypothesis would be satisfactorily solved under V = L? I think the correct answer to this question is: Who cares? Any “evidence” for V = L comes from the “good set theory”, not from the “prediction”.
It’s hard for me to imagine a brand of “good set theory” doesn’t have its own P’s and V’s. Another example: I developed a study of models between L and 0# based on Ronald’s ground-breaking work in class-forcing, and that resulted in a rich theory in which a number of “predictions” were verifed, like the “prediction” that there are canonical models of set theory which lie strictly between L and (a pioneering question of Bob’s); but I don’t regard my work as “evidence” for , necessary for this theory, despite having “verified” this “prediction”. Forcing Axioms: Haven’t they done and won’t they continue to do just fine without the “prediction” that you mention for them? I don’t see what the “problem” is if that “prediction” is not fulfilled, it seems that there is still very good evidence for the truth of Forcing Axioms
I do acknowledge that Hugh feels strongly about P’s and V’s with regard to his Ultimate-L programme, and he likes to say that he is “sticking his neck out” by making “predictions” that might fail, leading to devastating consequences for his programme. I don’t actually believe this, though: I expect that there will be very good mathematics coming out of his efforts and that this “good set theory” will result in a programme of no less importance than what Hugh is currently hoping for.
So tell me: Don’t P’s and V’s exist for almost any “good set theory”? Is there really more agreement about how “decisive” they are than there is just for which forms of set theory are “good”?
You have asked me why I am more optimistic about a consensus concerning Type 3 evidence. The reason is simple: Whereas set theory as a branch of mathematics is an enormous field, with a huge range of different worthwhile developments, the HP confines itself to just one specific thing: Maximality of V in height and width (not even a broader sense of Maximality). Finding a consensus is therefore a much easier task than it is for Type 1. Moreover the process of “unification” of different criteria is a powerful way to gain consensus (look at the IMH, #-generation and their syntheses, variants of the ). Of course “unification” is available for Type 1 evidence as well, but I don’t see it happening. Instead we see Ultimate-L, Forcing Axioms, Cardinal Characteristics, …, developing on their own, going in valuable but distinct directions, as it should be. Indeed they conflict with each other even on the size of the continuum (omega_1, omega_2, large, respectively).
3. (So-Called “Changes” to the HP) OK, Peter here is where I take you to task: Please stop repeating your tiresome claim that the HP keeps changing, and as a result it is hard for you to evaluate it. As I explain below, you have simply confused the programme itself with other things, such as the specific criteria that it generates and my own assessment of its significance.
There have been exactly 2 changes to the HP-procedure, one on August 21 when after talking to Pen (and you) I decided to narrow it to the analysis of the maximality of V in height and width only (the MIC), leaving out other “features of V”, and on September 24 when after talking to Geoffrey (and Pen) I decided to make the HP-procedure compatible with width actualism. That’s it, the HP-procedure has remained the same since then. But you didn’t understand the second change and then claimed that I had switched from radical potentialism to height actualism! That was your fault, not mine. Since September 24 you have had a fixed programme to assess, and no excuse to say that you don’t know what the programme is.
Indeed there have been changes in my own assessment of the significance of the HP, and that is something else. I have been enormously influenced by Pen concerning this. I started off telling Sol that I thought that the CH could be “solved” negatively using the HP. My discussions with Pen gave me a much deeper understanding and respect for Type 1 evidence (recall that back then, which seems like ages ago, I accused Hugh of improperly letting set-theoretic practice enter a discussion of set-theoretic truth!). I also came to realise (all on my own) the great importance of Type 2 evidence, which I think has not gotten its due in this thread. I think that we need all 3 types of evidence to make progress on CH and I am not particularly optimistic, as current indications are that we have no reason to expect Types 1, 2 and 3 evidence to come to a common conclusion. I am much more optimistic about a common conclusion concerning other questions like PD and even large cardinals. Another change has been my openness to a gentler HP: I still expect the HP to come to a consensus, leading to “intrinsic consequences of the set-concept”. But failing a consensus, I can imagine a gentler HP, leading only to “intrinsically-based evidence”, analogous to evidence of Types 1 and 2.
Despite my best efforts, you still don’t understand how the HP handles maximality criteria. On 3.September, you attributed to me the absurd claim that both the IMH and inaccessible cardinals are intrinsically justified! I have been trying repeatedly to explain to you since then that the HP works by formulating, analysing, refining, comparing and synthesing a wide range of mathematical criteria with the aim of convergence. Yet in your last mail you say that “We are back to square one”, not because of any change in the HP-procedure or even in the width actualist version of the IMH#, but because of a better understanding of the way the translates into a property of countable models. I really don’t know what more I can say to get you to understand how the HP actually works, so I’ll just leave it there and await further questions. But please don’t blame so-called “changes” in the programme for the difficulties you have had with it. In any case, I am grateful that you are willing to take the time to discuss it with me.