Re: Paper and slides on indefiniteness of CH

Dear Harvey,

I think it would be nice to revisit all of these topics. Let me say two things about the axiom “V = Ultimate L” and your request that it be presented in “generally understandable terms”.

(1) The development of inner model theory has involved a long march up the large cardinal hierarchy and has generally had the feature that when you build an inner model for one key level of the large cardinal hierarchy — say measurable, strong, or Woodin — you have to start over when you target the next level, building on the old inner model theory while adding a new layer of complexity (from measures to extenders, from linear iterations to non-linear iterations) — because the inner models for one level are not able to accommodate the large cardinals at the next (much as L cannot accommodate a measurable).

Moreover, the definitions of the inner models — especially in their fine-structural variety — are very involved. One essentially has to develop the theory in tandem with the definition. It looked like it would be a long march up the large cardinal hierarchy, with inner models and associated axioms of the form “V = M” of increasing complexity.

One of the main recent surprises is that things change at the level of a supercompact cardinal: If you can develop the inner model theory for a superpact cardinal then there is a kind of “overflow” — it “goes all the way” — and the model can accommodate much stronger large cardinals. Another surprise is that one can actually write down the axiom — “V = Ultimate L” — for the conjectured inner model in a very crisp and concise fashion.

(2) You will, however, find that the axiom “V = Ultimate L” may not meet your requirement of being explainable in “generally understandable terms”. It is certainly easy to write down. It is just three short lines. But it involves some notions from modern set theory — like the notion of a Universally Baire set of reals and the notion of \Theta. These notions are not very advanced but may not meet your demand or being “generally understandable”. Moreover, to appreciate the motivation for the axiom one must have some further background knowledge — for example, one has to have some knowledge of the presentation of HOD, in restricted contexts like L(\mathbb R), as a fine-structural inner model (a “strategic inner model”). Again, I think that one can give a high-level description of this background but to really appreciate the axiom and its motivation one has to have some knowledge of these parts of inner model theory.

I don’t see any of this as a shortcoming. I see it as the likely (and perhaps inevitable) outcome of what happens when a subject advances. For comparison: Newton could write down his gravitational equation in “generally understandable terms” but Einstein could not meet this demand for his equations. To understand the Einstein Field Equation one must understand the notions a curvature tensor, a metric tensor, and stress-energy tensor. There’s no way around that. And I don’t see it as a drawback. It is always good to revisit a subject, to clean it up, to make it more accessible, to strive to present it in as generally understandable terms as possible. But there are limits to how much that can be done, as I think the case of the Einstein Field Equations (now with us for almost 100 years) illustrates.

Best, Peter

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