Re: Paper and slides on indefiniteness of CH

Dear Peter,

Thanks for your message, especially the first half with helpful clarifications regarding the proper use of the terms “extrinsic” and “intrinsic”. I especially like your suggestion of considering “degrees of intrinsicness” (“relativised” versions) and this should fit very well with the HP. This is perhaps anticipated by Pen’s comment near the end of her 27.August message, where she reports that I might consider “returning to the proposal of a different conception of set. The challenge there is to do so without returning to the unappealing idea that ‘intrinsic justification’ and ‘set-theoretic truth’ are determined by a conception of the set-theoretic universe that’s special to a select group.” One could perhaps interpret “special to a select group” as “low degree of intrinsicness” in your sense.

But more to the point, Peter, the situation is as follows: Pen and I initiated a process of carefully examining the steps in the HP and got stuck. As Pen said in her 27.August message, one of the key steps is my claim to extract something new from the Maximal Iterative Conception (MIC). Her message includes a description of my claim, which includes the use of “lengthenings” and “thickenings” of mental pictures of V. Pen felt that there was “something off about a universe being ‘maximal in width’, but also having a ‘thickening'” and I replied that “lengthenings” were already implicit in reflection (Pen’s message includes my argument for that). At that point Pen decided it would be best to consult with you directly about the role of “lengthenings” in reflection and explicitly asked for your opinion. What do you think? I’d really appreciate your response because Pen and I have been waiting for it since August 27!

Regarding the second half of your message: I do not claim that the IMH is intrinsically justified! This is partly my fault, since my views have changed since Tatiana and I wrote our paper. In my 7.August message to Pen I say:

… as my views have evolved slightly since Tatiana and I wrote the BSL paper I’d like to take the liberty (see below) of fine-tuning and enhancing the picture you present above. My apologies for these modifications, but I understand that changes in one’s point of view are not prohibited in philosophy? ;)

I go on to say:

… what I came to realise is that the IMH deals only with “powerset maximality” and it is compelling to also introduce “ordinal maximality” into the picture. (I should have come to
that conclusion earlier, as indeed the existence of inaccessible cardinals is derivable from the intrinsic maximal iterative concept of set!)

And further on:

This was an important lesson for me and strongly confirms what you suggested: In the HP (Hyperuniverse Programme) we are not able to declare ultimate and unrevisable truths. Instead it is a dynamic process of exploration of the different ways of instantiating intrinsic features of universes, learning their consequences and synthesising criteria together with the long-term goal of converging towards a stable notion ….

I do understand that I said very different things in my original paper with Tatiana, but I tried to correct this in my 7.August e-mail to Pen. And I do realise that it is too much to ask that you sift through all of those e-mails I sent to Pen (there were many!) so I’m happy to repeat things now to sort out misunderstandings.

In abridged form: The HP starts with intrinisic features of V that follow from the Maximum Iterative Conception and then provides a method for turning these features into precise mathematical criteria which ultimately yield first-order statements. The process is dynamic, whereby the choice of criteria together with their first-order consequences can change over time, as indicated in the last quote above. So one does not arrive at “unrevisable intrinsically justified” statements, as sometimes criteria and their consequences are discarded as the programm progresses. This already happened to the IMH: it must be synthesised with ordinal-maximality, and if this is done using the Friedman-Honzik form of reflection (\#-generation) this removes its anti-large cardinal consequences.

The above description obviously leaves huge gaps, in particular it does not explain what the mathematical criteria are about and how they are chosen. But it gives the rough idea and I hope clarifies that the word “intrinsic” is intended to apply to features of V and only to criteria and their consequences after a lengthy (practice-independent) process of analysis and synthesis has occurred. I plan to examine each feature of the HP carefully in further discussions with Pen. But we are in need of your response to her message!


PS: The HP is not concerned with justifications of consistency. The consistency of large cardinals is taken as given in the programme.

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