Re: Paper and slides on indefiniteness of CH

Dear Sy,

I write with your permission to summarize for the group a brief exchange we had in private. Before that exchange began, you had agreed to these three points:

  1. The relevant concept is the familiar iterative conception, which includes a rough idea of maximality in ‘height’ and ‘width’.
  2. To give an intrinsic justification or intrinsic evidence for a set-theoretic principle is to show that it is implicit in the concept in (1).
  3. The HP is a method for extracting more of the implicit content of the concept in (1) than has heretofore been possible.

We then set about exploring how the process in (3) is supposed to work, beginning with more careful attention to the iterative conception in (1). You summarize it this way:

“Maximal” means “as large as possible”, whether one is talking about

a. Vertical or ordinal-maximality: the ordinal sequence is “as long as possible”, or about

b. Horizontal or powerset-maximality: the powerset of any set is “as large as possible”.

In other words there is implicitly a “comparative” (and “modal”) aspect to “maximality”, as to be “as large as possible” can only mean “as large as possible within the realm of ‘possibilities'”.

Thus to explain ordinal- and powerset-maximality we need to compare different possible mental pictures of the set-theoretic universe. In the case of ordinal-maximality we need to consider the possibility of two mental pictures P and P* where P* “lengthens” P, i.e. the universe described by P is a rank initial segment of the universe described by P*. We can now begin to explain ordinal-maximality. If a picture P of the universe is ordinal-maximal then any “property” of the universe described by P also holds of a rank initial segment of that universe. This is also called “reflection”.

In the case of powerset maximality we need to consider the possibility of two mental pictures P and P* of the universe where P* “thickens” P, i.e. the universe described by P is a proper inner model of the universe described by P*.

There seemed to me to be something off about a universe being ‘maximal in width’, but also having a ‘thickening’. Citing Peter Koellner’s work, you replied that reflection actually involves ‘lengthenings’ (to which the ‘thickenings’ would be analogous), because it appeals to higher-order logics:

Reflection has the appearance of being “internal” to V, referring only to V and its rank initial segments. But this is a false impression, as “reflection” is normally taken to mean more than 1st-order reflection. Consider 2nd-order reflection (for simplicity without parameters):

({*}) If a 2nd-order sentence holds of V then it holds of some V_\alpha.

This is equivalent to:

({*}{*}) If a 1st-order sentence holds of V_{\text{Ord} + 1} then it holds of some V_{\alpha + 1},

where \text{Ord} denotes the class of ordinals and V_{\text{Ord} + 1} denotes the (3rd-order) collection of classes. In other words, 2nd-order reflection is just 1st-order reflection from V_{\text{Ord} + 1} to some V_{\alpha + 1}. Note that V_{\text{Ord} + 1} is a “lengthening” of V = V_\text{Ord}. Analogously, 3rd order reflection is 1st-order reflection from the lengthening V_{\text{Ord} + 2} to some V_{\alpha + 2}. Stronger forms of reflection refer to longer lengthenings of V.

1st-order forms of reflection do not require lengthenings of V but are very weak, below one inaccessible cardinal. But higher-order forms yield Mahlo cardinals and much more, and this is what Goedel and others had in mind when they spoke of reflection.

Another way of seeing that lengthenings are implicit in reflection is as follows. In its most general form, reflection says:

({*}{*}{*}) If a “property” holds of V then it holds of some V_\alpha.

This is equivalent to:

({*}{*}{*}{*}) If a “property” holds of each V_\alpha then it holds of V.

[({*}{*}{*}) for a "property" is logically equivalent to ({*}{*}{*}{*}) for the negation of that "property".]

OK, now apply ({*}{*}{*}{*}) to the property of having a lengthening that models ZFC. Clearly each V_\alpha has such a lengthening, namely V. So by ({*}{*}{*}{*}), V itself has lengthenings that model ZFC! One can then use this to infer huge amounts of reflection, far past what Goedel was talking about.

I am not assuming that everybody is a “potentialist” about V. Even the Platonist can have mental images of the lengthenings demanded for reflection. And without such lengthenings, reflection has been reduced to a principle weaker than one inaccessible cardinal.

Now given that lengthenings are essential to ordinal-maximality isn’t it clear that thickenings are essential to powerset-maximality? We can then begin to explain powerset-maximality as follows: A picture P of the universe is powerset-maximal if any “property” of the universe described by a thickening of P also holds of the universe described by some thinning of P. What I called the weak-IMH is the “follow your nose” mathematical formulation of this notion of powerset-maximality for first-order properties.

I proposed that we consult with Peter Koellner directly about this business of ‘lengthenings’ in reflection, and this is why we again address the group.

(So, what do you think, Peter?)

Finally, you suggested that you might consider retracting (2) above and 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.

All best,
Pen

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