mathematical-philosophymultiverse-of-setspeano-arithmeticset-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.
(It happens that I will be giving a talk this week on this topic at the Exploring the Frontiers of Incompleteness series at Harvard, which is focussing on the question of determinism in set theory. The materials for all the talks are posted on the EFI web page, which also includes videos and discussion forums.)
Set theory currently provides at least a robust informal treatment of the multiverse. What I mean by this is that it is part of standard set theoretic practice and understanding to point out when a set-theoretic assertion $\varphi$ can have different truth values in different set-theoretic universes. That is, set theorists are already paying attention to the multiverse, when they discuss such issues as independence, forcing absoluteness, indestructibility and so on, which all concern the issue of how set-theoretic assertions can vary in the mutliverse.
Another more formal treatment of the multiverse idea is provided by what I have described as the toy model formalization, where one considers a multiverse from the perspective of a much larger set-theoretic background. For example, in ZFC one may consider the multiverse of all countable models of set theory, or just of a portion of them. This is the approach that Gitman and I took in our consistency proof that you mentioned, A natural model of the multiverse axioms. One can also view the typical countable-transitive-model approach to forcing as an instance of the toy model formalization.
Perhaps the toy model formalization has some affinity with the idea of the axiom of universes in category theory, and such a formalization would lead to a tower of multiverse concepts, one existing at the level of each Grothendieck universe. So it seems clear that one may set up a formal theory of multiverses using ZFC as a background theory, supplemented with various universe axioms, and leading to a tower of multiverses.
But my opinion is that the toy model method of formalization is ultimately unsatisfying, since it leads one to adopt axioms and principles about the toy models, rather than about the background set theory, which is the intended goal.
Although many of the questions that we have about the full multiverse happen to be first-order expressible in the language of set theory, nevertheless many of the questions that we are interested in happen to be formalizable in first-order ZFC. For example, many of the concepts of set-theoretic geology are formalizable in first-order ZFC, such as the open question whether every two ground models of the universe have a deeper ground inside them. Similarly, the questions surrounding the modal logic of forcing are also first-order expressible. This is a happy phenomenon, when we are able to use our current formalization to express and understand the multiverse issues. But the troubling matter underlying your question is that some of our questions are indeed pushing up against our formalism, which may be inadequate to the task.
Thank you for your interest in my views on the set-theoretic multiverse.
Yes, indeed, the well-foundedness mirage axiom you mention is probably the most controversial of my multiverse axioms, and so allow me to explain a little about it.
The axiom expresses in a strong way the idea that we don't actually have a foundationally robust absolute concept of the finite in mathematics. Specifically, the axiom asserts that every universe of set theory is ill-founded even in its natural numbers from the perspective of another, better universe. Thus, every set-theoretic background in which we might seek to undertake our mathematical activity is nonstandard with respect to another universe.
My intention in posing the axiom so provocatively was to point out what I believe is the unsatisfactory nature of our philosophical account of the finite.
You might be interested in the brief essay I wrote on the topic, A question for the mathematics oracle, published in the proceedings of the Singapore workshop on Infinity and Truth. For an interesting and entertaining interlude, the workshop organizers had requested that everyone at the workshop pose a specific question that might be asked of an all-knowing mathematical oracle, who would truthfully answer. My question was whether in mathematics we really do have a absolute concept of the finite.
To explain a bit more, the naive view of the natural numbers in mathematics is that they are the numbers, $0$, $1$, $2$, and so on. The natural numbers, with all the usual arithmetic structure, are taken by many to have a definite absolute nature; arithmetic truth assertions are taken to have a definite absolute nature, in comparison for example with the comparatively less sure footing of set-theoretic truth assertions.
To be sure, many mathematicians and philosophers have proposed a demarcation between arithmetic and analysis, where the claims of number theory and arithmetic are said to have a definite absolute nature, while the assertions of higher levels of set theory, beginning with claims about the set of sets of natural numbers, are less definite. Nik Weaver, for example, has suggested that classical logic is appropriate for the arithmetic realm and intuitionistic logic for the latter realm, and a similar position is advocated by Solomon Feferman and others.
But what exactly does this phrase, "and so on" really mean in the naive account of the finite? It seems truly to be doing all the work, and I find it basically inadequate to the task. The situation is more subtle and problematic than seems to me to be typically acknowledged. Why do people find their conception of the finite to be so clear and absolute? It seems hopelessly vague to me.
Of course, within the axiomatic system of ZFC or other systems, we have a clear definition of what it means to be finite. The issue is not that, but rather the extent to which these internal accounts of finiteness agree with the naive pre-reflective accounts of the finite as used in the meta-theory.
Some mathematicians point to the various categoricity arguments as an explanation of why it is meaningful to speak of the natural numbers as a definite mathematical structure. Dedekind proved, after all, that there is up to isomorphism only one model $\langle\mathbb{N},S,0\rangle$ of the second-order Peano axioms, where $0$ is not a successor, the successor function $S$ is one-to-one, and $\mathbb{N}$ is the unique subset of $\mathbb{N}$ containing $0$ and closed under successor.
But to my way of thinking, this categoricity argument merely pushes off the problem from arithmetic to set theory, basing the absoluteness of arithmetic on the absoluteness of the concept of an arbitrary set of natural numbers. But how does that give one any confidence?
We already know very well, after all, about failures of absoluteness in set theory. Different models of set theory can disagree about whether the continuum hypothesis holds, whether the axiom of choice holds, and so with innumerable examples of non-absoluteness. Different models of set theory can disagree on their natural number structures, and even when they agree on their natural numbers, they can still disagree on their theories of arithmetic truth (see Satisfaction is not absolute). So we know all about how mathematical truth assertions can seem to be non-absolute in set theory.
Skolem pointed out that there are models of set theory $M_1$, $M_2$ and $M_3$ with a set $A$ in common, such that $M_1$ thinks $A$ is finite; $M_2$ thinks $A$ is countably infinite and $M_3$ thinks $A$ is uncountable. For example, let $M_3$ be any countable model of set theory, and let $M_1$ be an ultrapower by a ultrafilter on $\mathbb{N}$ in $M_3$, and let $A$ be a nonstandard natural number of $M_1$. So $M_1$ thinks $A$ is finite, but $M_3$ thinks $A$ has size continuum. If $M_2$ is a forcing extension of $M_3$, we can arrange that $A$ is countably infinite in $M_2$.
No amount of set-theoretic information in our set-theoretic background could ever establish that our current conception of the natural numbers, whatever it is, is the truly standard one, since whatever we assert to be true is also true in some nonstandard models, whose natural numbers are not standard.
The well-foundedness mirage axiom asserts that this phenomenon is universal: all universes are wrong about well-foundedness.
In defense of the mirage axiom, let me point out that whatever attitude toward it one might harbor, nevertheless the axiom cannot be seen as incoherent or inconsistent, because Victoria Gitman and I have proved that all of my multiverse axioms are true in the multiverse consisting of the countable computably saturated models of ZFC. So the axiom is neither contradictory nor incoherent. See A natural model of the multiverse axioms.
I have discussed my multiverse views in several papers.
Hamkins, Joel David, The set-theoretic multiverse, Rev. Symb. Log. 5, No. 3, 416-449 (2012). Doi:10.1017/S1755020311000359, ZBL1260.03103.
Hamkins, Joel David, A multiverse perspective on the axiom of constructibility, Chong, Chitat (ed.) et al., Infinity and truth. Based on talks given at the workshop, Singapore, July 25--29, 2011. Hackensack, NJ: World Scientific (ISBN 978-981-4571-03-6/hbk; 978-981-4571-05-0/ebook). Lecture Notes Series. Institute for Mathematical Sciences. National University of Singapore 25, 25-45 (2014). DOI:10.1142/9789814571043_0002, ZBL1321.03061.
Gitman, Victoria; Hamkins, Joel David, A natural model of the multiverse axioms, Notre Dame J. Formal Logic 51, No. 4, 475-484 (2010). DOI:10.1215/00294527-2010-030, ZBL1214.03035.
Hamkins, Joel David; Yang, Ruizhi, Satisfaction is not absolute, to appear in the Review of Symbolic Logic.
But finally, to address your specific question. Of course, there are specific finite numbers that will be finite with respect to any alternative set-theoretic background. As Michael Greinecker points out in the comments, the number 35253586543 has that value regardless of your meta-mathematical position. So of course, there are many proofs that are standard finite with respect to any of the alternative foundations.
Meanwhile, I find it very interesting to consider the situation where different foundational systems disagree on what is provable. In very recent work of mine, for example, we are looking at the theory of set-theoretic and arithmetic potentialism, where different foundational systems disagree on what is true or provable.
For example, recently with Hugh Woodin, I have proved that there is a universal finite set $\{x\mid\varphi(x)\}$, a set that ZFC proves is finite, and which is empty in any transitive model of set theory, but if the set is $y$ in some countable model of set theory $M$ and $z$ is any finite set in $M$ with $y\subset z$, then there is a top-extension of $M$ to a model $N$ inside of which the set is exactly $z$. The key to the proof is playing with the non-absolute nature of truth between $M$ and its various top-extensions.
Best Answer
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.