# Is platonism regarding arithmetic consistent with the multiverse view in set theory?

A "truth" platonist for arithmetic believes, given a statement in the language of arithmetic, that the problem whether the statement is true has a definite answer.

Prof. Hamkins has argued for a multiverse view of set theory. Since different models of ZFC can have a different arithmetic (that is, model of the natural numbers), I wonder whether platonism regarding arithmetic is consistent with the multiverse view.

• I think it's a reasonable question, but this is not the right forum. Commented Jan 22, 2016 at 18:18
• Well, it's a philosophy question. This is a website for research mathematics. Maybe try the FOM discussion list? Commented Jan 22, 2016 at 18:28
• I think, questions on philosophy of mathematics may be interesting for many working mathematicians, thus I am against closing this question. Commented Jan 22, 2016 at 18:53
• Well, I hesitate to engage in a question that I think the community would disapprove of, but if it is going to stay open I would like to say that I defend exactly this point of view (truth platonism for arithmetic plus multiverse view of set theory) in the final chapter of my book on forcing. Commented Jan 22, 2016 at 22:08
• It seems to me that one straightforward answer is that, according to a theorem of Gitman and Hamkins, one of the axioms consistent with the "multiverse view" is that every model of set theory is ill-founded with respect to some other model. If this axiom holds, then the concept of a unique "standard" model of arithmetic becomes very doubtful. See "A natural model of the multiverse axioms", projecteuclid.org/euclid.ndjfl/1285765800 Commented Jan 22, 2016 at 23:52

## 3 Answers

The view you are suggesting is something close to what is held by Solomon Feferman, who holds that the objects and truths of arithmetic have a definite nature that is not shared when one moves up to higher-order objects, such as the collection of all sets of natural numbers. Feferman has long been known for the view that the continuum hypothesis is inherently vague, in a way that arithmetic is not, and this seems to be basically what you are talking about. See for example his article

There are several other papers for the EFI project exploring similar issues.

One interesting aspect of the view is the idea of using classical logic in the lower more-definite realm, and intuitionistic logic in the higher realm, where assertions such as the continuum hypothesis may have a less definite meaning. Nik Weaver has pointed out in the comments below that he had first proposed this dichotomizing idea in his 2005 article:

Finally, let me criticize your use of the term Platonism to imply a kind of singularist view of mathematical existence, whereas I have argued that it should instead imply only a kind of realism or definite existence. With this idea, the multiverse view itself is a kind of Platonism, where one gives up on the uniqueness of the existence of mathematical objects, but not on their objective existence. For example, on the multiverse view in set theory, there are many different concepts of set, each giving rise to its own set-theoretic universe, which are just as real as the set theory claimed by the universists.

• Joel, philosophers of mathematics sometimes distinguish between "objects platonism" (mathematical objects exist in some meaningful sense) and "truth platonism" (all the sentences of some well-defined theory have definite truth values). See here for instance. Commented Jan 22, 2016 at 22:05
• Yes, thanks, Nik, I am aware of those other usages, and my objection is also aimed at them. My view is that the term Platonism should have something to do with real or objective existence of some kind (in the Platonic realm), that is, with realism, rather than with the singularist versus pluralist issue. If for example we have two (or more) kinds of set-theoretic truth, they might both have an objective existence, and so it should be considered Platonist, but it wouldn't be using the terminology you cite. And I find that unfortunate. Commented Jan 22, 2016 at 22:37
• Also, I don't think it's quite fair to attribute the idea about dichotomizing between classical and intuitionistic logic in this way to Feferman ("he argues that"). I first proposed this in a paper from 2005, which, not to put too fine a point on it, I know that Feferman read in detail. He switched to this position several years later. Commented Jan 22, 2016 at 22:41
• Thank you for that! Yes, his arguments about CH go way back --- I think he was already suggesting this (maybe a little more obliquely) in the 1960s. Of course Weyl had said already something similar in 1918 in Das Kontinuum ... Commented Jan 22, 2016 at 23:11
• @cody No, I am not an arithmetic truth singularist, and I believe or at least worry that our concept of the finite might be less absolute and robust than is often proclaimed. You can see a brief essay I wrote on the topic at: jdh.hamkins.org/question-for-the-math-oracle. Also, I address the issue in my multiverse paper. Commented Mar 30, 2016 at 20:14

I'm still uncertain of its appropriateness here, but since Joel asked, here is a quote from my book that discusses this issue:

The two kinds of independence … in geometry and number theory offer us strikingly different paradigms. In both cases there is broad agreement about the correct interpretation of the independence results. For instance, no one nowadays would consider it meaningful to ask whether the parallel postulate is "really true" in some universal sense; it simply holds in some two-dimensional geometries and fails in others.

In contrast, although the arithmetical expression of the consistency of $PA$ is independent of $PA$, it is still widely regarded as true. To take a more extreme example, consider the formal system $MC$ = $ZFC$ + "measurable cardinals exist". Few would suggest that the sentence ${\rm Con}(MC)$ which arithmetically expresses that $MC$ is a consistent system might lack a well-defined truth value. Yet ${\rm Con}(MC)$ is presumably independent of $PA$, indeed, presumably even independent of $ZFC$.

… Should we suppose that the continuum hypothesis, for example, has a definite truth value in a well-defined canonical model? Or is there a range of models in which the truth value of the continuum hypothesis varies, none of which has any special ontological priority?

Forcing tends to push us in the latter direction. It creates the impression that there is a range of equally valid models of $ZFC$, and that one can always pass to a larger model in which the value of $2^{\aleph_0}$ changes … In an influential series of recent papers, Hamkins has vigorously argued for the position that there is no canonical model of $ZFC$, a position that he calls "the multiverse view".

… A picture emerges [from discussion omitted here] of a mathematical universe which is composed of countable structures that have absolute properties and which includes a range of countable models of $ZFC$ in which the truth values of questions like the continuum hypothesis can vary. Thus, with regard to independence phenomena, if we take "set theory" to be the theory of surveyable collections then it has an absolute meaning and behaves like number theory, but questions like the continuum hypothesis cannot even be posed; if we take it to be the theory of individuals in some model of $ZFC$ then it has a variable meaning and behaves like geometry.

(Now that the question has been bumped up to the front page:) Re-reading the answer by Hamkins, I notice that he did not respond to the question directly. I think that both views are possible (and I personally lean toward the negative answer but I am not a professional logician).

To give an affirmative answer, one would have to accept the distinction elaborated in the other answers, following (in alphabetical order) Feferman and Weaver: there is a difference between integers and higher-order sets; the former can be seen as somehow fixed, and the latter as varying from one universe to another. One still has to account for the countable models of the Gitman-Hamkins "toy model" of the multiverse, where the integers themselves vary from univese to universe. To this, one would have to reply that all these models are non-standard. Furthermore, Hamkins mentioned that the toy model may have spurious properties that are not obligatory for a "generic" (in a generic sense) multiverse.

Giving a negative answer is more straightforward IMHO. Here one can accept the toy model at face value, and point out that the integers vary from universe to universe, and therefore are not "fixed" (defeating any Platonism/realism about the integers). Furthermore, the toy model (built via ultraproducts) has a natural predicate of standardness that turns it into a model of an axiomatic theory of nonstandard analysis such as BST (a variant of IST). More precisely, for a pair of adjacent universes, the smaller one is the standard core (in the sense of nonstandard analysis) of the larger one.

• We are talking about the consistency of a philosophical theory. How are you going to bring into contradiction the view that there are two immaterially separate realms, one of which is inhabited by natural numbers and the other only by sets, if you have consistent descriptions of one and the other? Why can't they be peacefully next to each other? Commented Aug 30 at 0:54
• I'm a platonist about TA. (I believe there is the set of all natural numbers and all definable sets and relations in it, as well as the set of all true propositions about that realm.) But I am very skeptical about the existence of uncountable sets. Therefore, I am not a proponent of the view that there is both the realm of natural numbers and a full-blooded set multiverse also inhabited by uncountable sets, but this is not because I find such a view to be inconsistent. Commented Aug 30 at 22:59
• The OP explicitly spoke of ZFC. In the context of ZFC, the (von Neumann) natural numbers are just a type of set. The burden is on those who argue for a distinction between numbers and sets (of the type discussed in this question as well as the answers) to explain why some types of sets should be "fixed" and others "variable" as one passes from universe to universe within Hamkins' multiverse. I am not saying htat an explanation is impossible; only that one needs to be given. @Juan Commented Sep 1 at 13:36
• The Original Post. @Juan P.S. I should mention that I didn't downvote your answer. Commented Sep 1 at 15:56
• The topic of the post is the compatibility of two philosophical views regarding mathematics. Mr. Hamkins has some philosophical views, and so do I. They are determined by our philosophical intuitions (mine are reinforced by the intuitive notion of a finite series of interconnected infinitely accelerating Turing machines, with the output tape of one machine serving as the input tape for the next machine, with an arithmetic formula given on the start tape). Commented Sep 2 at 12:22