Tag Archives: Consistency

Re: Paper and slides on indefiniteness of CH

I see the importance you are attaching to your Ultimate L Conjecture – particularly getting a proof “by the current scenarios of course”. Care to make rough probabilistic predictions on when you will prove it “by the current scenarios of course”? Until then, your point of view seems to be that statements like Con(HUGE) are wide open, and you currently are not willing to declare any confidence in them.

Fascinating as this is, I think people here might be even more interested in the implications Con(EFA) arrows Con(PA) arrows Con(Z) arrows Con(ZFC) arrows Con(ZFC + measurable) arrows Con(ZFC + PD). Maybe you can comment on at least one of these arrows? — or maybe Peter Koellner?

Based on all my experience to date I have a conception of V in which PD holds. Based on that conception it is impossible for PD to be inconsistent. But that conception may be a false conception.

If it is a false conception then I do not have a conception of V to fall back on except for the naive conception I had when I first was exposed to set theory. This is why for me, if ZFC+PD is inconsistent I think that ZFC is suspect. This is not to say that I cannot or will not rebuild my conception to that of V which satisfies ZFC etc. But I would need to understand how all the intuitions etc. that led me to a conception V with PD went so wrong.

My question to those who feel V is just the integers (and maybe just a bit more) is: How do they assess that Con ZFC+PD is relevant to their conception? The conception of the set theoretic V with or without PD are very different conceptions with deep structural differences. I just do not see that happening yet to anywhere near the same degree in the case where the conception V  is just the integers. But as your work suggests this could well change.

But even so, somehow a structural divergence alone does not seem enough (to declare that Con ZFC+PD is an indispensable part of that conception).  Who knows, maybe there is an arithmetically based strongly motivated hierarchy of “large cardinals” and Con PD matches something there.

If one’s conception of V is the integers and one is never compelled to declare Con ZFC+PD as true based on whatever methodology one is using to refine that conception, then it seems to me that the only plausible conjecture one can make is that ZFC+PD is inconsistent. This was the cryptic point behind item (2) in my message which inspired your press releases.

To those still reading and to those who have participated, I would like to express my appreciation. But I really feel it is time to conclude my participation in this email thread. Classes are about to begin and I shall have to return to Cambridge shortly.

Regards,
Hugh

Re: Paper and slides on indefiniteness of CH

Dear Hugh,

I personally don’t feel the implication

If ZFC + PD is inconsistent then ZFC is inconsistent.

I have a couple of questions for you.

  1. I am under the impression that you are not committed to the consistency of ZFC + HUGE. Are you committed to the consistency of ZFC + LC roughly if and only if there is some good inner model theory for it? Are you also advocating a more general principle of this kind?
  2. Consider the statement: If ZFC is inconsistent then T is inconsistent. For how weak a T do you feel this? As an extreme, are you willing to take T down to EFA = exponential function arithmetic?
  3. As you can see from what I wrote about blurring pictures, my own view is one of relative clarity and therefore relative confidence. But you seem to have quite a different view, and I am wondering what you can say about your view (feelings, intuition)?

Harvey