# Re: Paper and slides on indefiniteness of CH

Dear Pen and Sy,

I’m sorry about the delay. I’ll now focus the point at issue — the idea of “lengthenings” of V (and the reference to my work).

A. Preliminary Point: Actualism versus Potentialism

I would like to get a better sense of how you understand “V”. Let me begin with a distinction between two forms of actualism and potentialism; one concerning height, the other concerning width.

The height actualist maintains that in terms of height the universe of sets forms a “completed totality” while the height potentialist maintains that in terms of height the universe of sets is “open-ended” or “indefinitely extensible”. The notions of width actualism and width potentialism are defined similarly, except with “height” replaced by width. The former concerns the ordinals, the latter concerns the powerset operation.

This, to be sure, is a rough distinction. But it has a long history, going back to Aristotle. One way of formally regimenting this informal distinction is by employing intuitionistic logic for domains for which one is a potentialist and reserving classic logic for domains for which one is an actualist. This is the approach Sol takes in his work on semi-intuitionistic systems of set theory. (I won’t presuppose that regimentation in what I go on to say below.)

Some examples: Gödel was (probably) an actualist with regard to both width and height. Zermelo was (probably) an actualist with regard to width and a potentialist with regard to height. Sol is an actualist with regard to width up to $V_\omega$ and a potentialist with regard to width and height beyond that.

Question: Are you an actualist or a potentialist with regard to height? with regard to width?

One reason I ask is that in your account the symbol “V” appears to play a dual role, first for “a surrogate” — one of many countable transitive models of ZFC that provides “an accurate picture of V” — and second for “the real thing” (the thing of which the former models are supposed to provide pictures).

If you answer this question I think it will help me get a better grip on your view. But the main thing I want to address is the dual role of “V” your account.

B. The Idea of “Lengthening” V.

In your account you consider “lengthenings of V” and “broadenings of V”. The purported acceptability of the former is supposed to lend credence to the latter.

The idea of a “lengthening of V” is certainly one which, upon first appearance, is very strange. For it seems to fly in the face of the whole idea of V. Reinhardt once articulated this strangeness by saying something like this (I paraphrase): “It doesn’t seem to make a whole lot of sense to speak of “lengthening V” since if one takes, e.g., the powerset of V one would seem to have more sets and, being sets, these should be in V, by the very conception of V as the universe of all sets.”

If by “V” we mean “the real thing” then the idea of a “lengthening of V” doesn’t make any sense on an actualist conception. And it is unclear that it even makes sense on the potentialist conception. (The issue is whether on the potentialist conception it even makes sense to speak of “V” in this sense.) But, of course, if by “V” one means not “the real thing” but “a surrogate for the real thing” — like a countable transitive model of ZFC or a rank initial segment of the universe — then one can make sense (on either the actualist or potentialist conception) of a “lengthening of V”.

In the reflection principles paper I start of by presenting a philosophical dilemma, which I will here briefly summarize as follows: The actualist can make sense of V as “the real thing” (and so can motivate reflection) but cannot make sense of higher-order quantification over V. The potentialist, on the other hand, can make sense of higher-order quantification over “V” (by understanding this to be some surrogate, some rank initial segment) but has a hard time motivating reflection since now V (as “the real thing”) has evaporated.

It is in the attempt to have the best of both worlds that people have tried to come up with ideas that resolve this dilemma. I see Ackermann and Reinhardt (in a spirit different than that from the paraphrase above) as doing just this. Reinhardt considered “lengthenings” of V, in his work on (what I call) “extension principles” — the principles leading to extendible cardinals and beyond. In taking this step he introduces (what I call) the “theory of legitimate candidates”. On this conception there are many different legitimate candidates for “the true universe V”– call them $V$, $V'$, $V''$, etc — and the idea is that they should all resemble one another to various degrees that we can articulate. Your work is very much in this spirit, though you expand it to include “broadenings of V”, in addition to “lengthenings of V”.

I should say, before turning to your arguments, that I do not see how you are appealing to my work. My discussion of higher-order reflection principles is entirely critical. I present a philosophical critique of higher-order reflection, saying that I cannot make sense of them (on either the actualist or potentialist conception). But then not wanting to rest too much weight on a philosophical critique I consider a specific proposal, namely, that of Tait. I set aside the question of whether higher-order reflection principles make sense and are justified and I consider the technical question of how far Tait’s principles go, showing that they are either consistent and weak or inconsistent. But at no point do I endorse the idea that higher-order principles make sense and are intrinsically justified (or even justified at all). The discussion is entirely critical.

And regarding “the theory of legitimate candidates”: That is something I discussed briefly in the paper on reflection principles and at length in my dissertation. I tried to make philosophical sense of the idea as a way of reaching strong principles. I considered both height reflection and width reflection. I managed to get principles (which bear resemblance to your principles on sharp-generation) that imply $\text{AD}^{L(\mathbb R)}$ (by an easy core model induction argument). But my conclusion was that (a) I couldn’t make sense of the philosophical basis of this conception (“the theory of legitimate candidates”) and (b) I thought it would be a real stretch to say that such principles were intrinsically justified.

C. The Case for “Lengthenings of V”

The first argument:

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 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 Gödel and others had in mind when they spoke of reflection.

The question, of course, is whether this is legitimate, on either the actualist or the potentialist conception. As mentioned above, the challenge for the actualist is to make sense of full-second order set theory (over V). (Some have tried to do this by invoking a plural interpretation of the second-order quantifiers.) And the challenge for the potentialist is to make sense of the idea that there should be rank-initial segments of the universe that satisfy second-order reflection (something that looks like a posit).

The second argument:

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 Gödel was talking about.

Reinhardt pointed out (at one point, in an actualist spirit) that when one is reflecting one must be careful of what one reflects. For example, it obviously doesn’t make sense to reflect the property of having all and only the sets in V (`the real thing’). For this reason he prohibited reference “V” or anything involving “V” and instead observed that one can only reflect “structural properties” (roughly speaking, properties that are “internally characterizable without de re reference to V”). Gödel also followed this course (in his discussions with Wang.)

From this perspective the property you are reflecting is not a legitimate candidate for reflection. The point is that V simply has no lengthenings! Part of what we mean by V (on this point of view) is that it contains _all_ of the sets; there are no lengthenings. — that is part of what we _mean_ by V! To speak of sets outside of V involves essential reference to V itself and this feature, by Reinhardt’s criterion, prohibits it from being a candidate for reflection.

Of course, if one understand by “V” not “the real thing” but “a surrogate that resembles the real thing” then one can make sense of “lengthenings of V”. But if that is the understanding of “V” that you are working with then you don’t need to give an argument since you already have the conclusion from the start.

So, either the argument doesn’t work or you do not need to give it since you have presupposed an understanding of “V” in which it is immediate.

To summarize: There appears to be a dual use of “V” in your account. At times it refers to “the real thing”, at times it refers to “a surrogate that resembles the real thing” (like a countable transitive model of ZFC). It is on the latter understanding that you can speak of “lengthenings of V” and “broadenings of V”. And on this understanding you do not have to give an argument that such things exist since it is immediate. But then the challenge is to make a case for why the principles you endorse are intrinsically justified (or intrinsically plausible or even just plausible) on this understanding. I don’t see how such a case goes. When I reflect on the concept of countable transitive model of ZFC I get very little. You must be reflecting on a different concept, one that involves both the surrogates (among the countable transitive models of ZFC) and something else. What?

Best,
Peter

P.S. Thanks for the clarification on your retraction of your earlier claim that IMH is intrinsically justified. The shift — and your comments about the dynamical element of this investigation — make me think that you are really speaking of what is either intrinsically plausible or extrinsically justified.

The questions I raised at the end — (a) – (b) — apply to $\textsf{IMH}^\#$ just as much as they apply to IMH.

# 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.