Maybe this is a very silly question, but I don’t see how to print the pages of the PDF file double-sided, fold them, and then get the pages in the correct order in the booklet. For example, if page 2 is printed on the reverse side of page 1, then page 2a will be separated from page 2b by all the pages 3,…,11. From my point of view, the reverse side of page 1 should contain on the left the contents currently in page 2a, and on the right the contents currently in page 11b.

]]>This post is very useful for something (philosophical) I’m writing up now. Is there a way of citing it, or have versions of this appeared in print?

Best,

Neil

]]>A quick follow up on (2.): For sure the consistency strength will be way higher, but I’m wondering if such a j directly *implies* other class principles. So, for example, if I add such a j over NBG, can I get any class comprehension? There’s at least some places where they converge (e.g. in both $\Pi^1_1$-Comp and $NBG + \exists j: M \to V$ you can get a first-order truth definition.

I’m looking forward to speaking to Kameryn (we’ve had a couple of e-mail interchanges already).

]]>1. KM is equiconsistent with KM+, which has the class-choice principle, and so there is no increase in consistency strength to add class-choice. Indeed, it is conservative over KM for first-order assertions, since every KM model has a KM+ realization with the same sets.

2. The embeddings j:M to V, however, have a consistency strength on the order of “Ord is Ramsey” according to the Vickers and Welch result, so this consistency strength does go beyond KM. I think that this axiom is expressible in second-order set theory, since M and j are classes, and so it would fit into the hierarchy, but the strength is beyond KM. (Note that KM itself is weaker in consistency strength than ZFC + there is an inaccessible cardinal, which is fairly mild by large cardinal standards.

3. That is interesting. It seems to me that there is a huge diversity of models of the second-order theory, since most of the variations in first-order set theory carry over to this realm. So we can consider models of GBC with CH or GCH or not or with large cardinals and so on. A difference, however, is to look at how one can make changes in the second-order part of a model, while keeping the same first-order part. My student Kameryn Williams has some results in this perspective.

]]>1. I know that strong choice principles for classes are largely independent of KM, is there work examining the consistency strength of these principles? More generally, is there much of a space above KM?

2. I wonder where the Welch embedding $j: M \longrightarrow V$ might fit in here (say added over GBC or KM). While necessarily second-order, that doesn’t clearly fit into the template you’ve got there, and substantially goes beyond any of the theories outlined in terms of consistency strength (in virtue of the large cardinals it implies). [Years ago I had an idea to try and reverse over the existence of such an embedding, but couldn’t get anything other than the very trivial.]

3. More speculatively: Clearly this slots nicely into your Multiverse View. However, do you think there are substantial differences in argumentation for Multiversism between the first-order and second-order claims? It’s interesting because there seems to be less of a `zoo’ of worlds, and you don’t obviously get incompatibility (perhaps because of inattention due to metamathematical quibbles)? Is there a philosophical equivalent of CH and ~CH in the second-order case?

Best,

Neil

]]>Indeed, although the mathematical project here is inspired by the issue of actualism versus potentialism, it is not meant to be taken as a piece of potentialist mathematics. The situation here is not unlike a logician who might use classical logic in order to analyze the power of a particular intuitionistic logical system.

My view of the project is that we seem able to use this kind of nonstandard analysis to provide a way in part to make sense of some views, such as ultrafinitism, which otherwise have had such a difficult time to be coherently expressed. What we have here in our potentialist system is a mathematical model that appears to share many of the main features that the ultrafinitists assert about the nature of mathematical reality, and which has a kind of friendly simulation of those views, if one adopts a nonstandard perspective, but of which we can provide a full mathematical account using the actualist mathematics at our disposal.

The conclusion is that our analysis seems to provide reasons to expect the ultrafinitist/potentialist perspective to have S4 as its potentialist validities, while opening the door to S5 for sentences.

In addition, our analysis tends to highlight some issues with philosophical significance, which we haven’t otherwise seen discussed enough. For example, using the arbitrary-extension model of potentialism rather than the end-extension model corresponds to the philosophical idea for an ultrafinitist that one might gain access to some large numbers, without necessarily yet having access to all smaller numbers. For example, perhaps it makes sense for an ultrafinitist to be able to analyze the number googol-plex-bang, which is a truly enormous number yet having a comparatively small description, without yet being committed to the actual existence yet of all the smaller numbers.

In our potentialist system, we nevertheless prove that still S4 is the class of validities for this version of potentialism.

Ultimately, my view of the philosophical value of the mathematical project is that the potentialist systems we are studying seem to exhibit many of the features that the potentialists seem to want, while avoiding the problematic issues, such as the commonly mentioned awkwardness for an ultrafinitist to assert the existence of a largest number. So we feel that by analyzing these potentialist systems, we gain philosophical insight into the potentialist viewpoint.

]]>I may be off the mark, but historically isn’t the statement “Consider the collection of all the models…” precisely of the type that is rejected by those who only want to consider potential infinities ? I thought that they would not want to deal with as an actual object, but only talk about “ for as large as one may whish” (a never-ending process rather than a given object).

If so, then in what way is your work on “the distinction between actual and potential infinity”, rather than on “exploring the structure of all possible models” (possible, not potential) ?

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