lo.logicmultiverse-of-setsset-theory
In J.D. Hamkins' multiverse view of set theory, every universe has an ill-founded $\mathbb{N}$ from the perspective of another universe. Does this mean that every proof in our universe can be seen as a nonstandard length proof from the perspective of some other universe, so that there is no real "truth" in the Multiverse?
Could this happen even for a simple concrete proof, like the fundamental theorem of arithmetic? Or are some some proofs standard-length in all universes?
I am not sure I understand all remarks that Colin made, and I disagree with some of them, but I can comment on the idea of "multiverse". Let us consider the following positions:
These are supposed to be informal statements. You may interpret "universe" as "model of set theory ZF(C)" or "topos" if you like, but I don't want to fix a particular interpretation.
I would call the first position "naive absolutism". While many mathematicians subscribe to it on some level, logicians have known for a long time that this is not a very useful thing to do. For example, someone who seriously believes that there is just one mathematical universe would reject model theory as fiction, or at least brand it as a technical device without real mathematical content. Such opinions can actually slow down mathematical progress, as is historically witnessed by the obstruction of "Euclidean absolutism" in the development of geometry.
The second position is what you get if you take model theory seriously. Set theorists have produced many different kinds of models of set theory. Why should we pretend that one of them is the best one? You might be tempted to say that "the best mathematical universe is the one I am in" but this leads to an unbearably subjective position and strange questions such as "how do you know which one you are in?" At any rate, it is boring to stay in one universe all the time, so I don't understand why some people want to stick to having just one cozy universe.
I need to explain the difference between the second and the third position. By "multiverse" I mean an ambient of universes which form a structure, rather than just a bare collection of separate universes. The difference between studying a "collection of universes" and studying a "multiverse" is roughly the same as the difference between Euclid's geometry and Erlangen program -- both study points and lines but conceptual understanding is at different levels. Likewise, a meta-mathematician might prove interesting theorems about models of set theory, or he might consider the overall structure of set-theoretic models.
It should be obvious at this point that there cannot be just one notion of "multiverse". I can think of at least two:
I do not mean to belittle set theory, but in a certain sense topos theory is more advanced than set theory because it uses algebraic methods to study the multiverse (I consider category theory to be an extension of algebra). In this sense the formulation of forcing in terms of complete Boolean algebras by Scott and Solovay was a step in the right direction because it brought set theory closer to algebra. Set theorists should learn from topos theorists that transformations between set-theoretic models are far more interesting than the models themselves.
In the present context the question "classical or intuitionistic logic" becomes "what kind of multiverse". If multiverse is "an ambient of universes, each of which supports the development of mathematics" then taking our multiverse to be either too small or to big will cause trouble:
Topos theory gains little by restricting to Boolean toposes. I have never heard a topos-theorist say "I wish all toposes were Boolean". Also, toposes occurring "in nature" (sheaves on a site) typically are not Boolean, which speaks in favor of intuitionistic mathematics.
An example of an ad hoc property in too small a multiverse occurs in set theory. We construct models of set theory by forming Boolean-valued sets which are then quotiented by an ultrafilter. What is the ultrafilter quotient for? The algebraic properties of Boolean-valued sets are hardly improved when we pass to the quotient, not to mention that it stands no chance of having an explicit description. A possible explanation is this: we are looking only at one part of the set-theoretic multiverse, namely the part encompassed by Tarskian model theory. Our limited view makes us think that the ultraquotient is a necessity, but the construction of Boolean-valued models exposes the ultraquotient as a combination of two standard operations (product followed by a quotient). We draw the natural conclusion: a model of classical first-order theory should be a structure that measures validity of sentences in a general complete Boolean algebra. The Boolean algebra $\lbrace 0,1 \rbrace$ must give up its primacy. What shall we gain? Presumably a more reasonable overall structure. At first sight I can tell that it will be easy to form products of models, and that these products will have the standard universal property (contrast this with ultraproducts which lack a reasonable universal property because they are a combination of a categorical limit and colimit). Of course, there must be much more.
How kind of you to take an interest in my paper. Please see also my blog post about the dream solution and the arxiv entry for the paper.
First, I shall make a quibble, and then I'll address your question at the end.
The quibble is that your quotation from the paper is not accurate. The full paragraph from the paper reads:
I have argued, then, that there will be no dream solution of the continuum hypothesis. Let me now go somewhat beyond this claim and issue a challenge to those who propose to solve the continuum problem by some other means. My challenge to anyone who proposes to give a particular, definite answer to CH is that they must not only argue for their preferred answer, mustering whatever philosophical or intuitive support for their answer as they can, but also they must explain away the illusion of our experience with the contrary hypothesis. Only by doing so will they overcome the response I have described, rejection of the argument from extensive experience of the contrary. Before we will be able to accept CH as true, we must come to know that our experience of the $\neg$CH worlds was somehow flawed; we must come to see our experience in those lands as illusory. It is insufficient to present a beautiful landscape, a shining city on a hill, for we are widely traveled and know that it is not the only one.
The difference is that it should say "extensive experience of the contrary" rather than "extensive evidence of the contrary", a difference that affects the meaning, since the point is that we have experience in both the CH and in the $\neg$CH worlds. In particular, there is a symmetry here, and I hope it was clear that implicitly include your variation as part of my intended meaning.
Now, let me consider your final question, which is very good.
I take the answer to be yes, these views are made coherent by what I have called the universe view in my paper The set-theoretic multiverse, from which the dream solution paper is adapted. The universe view is the view I am arguing against, and although I have attacked the universe view for being mistaken, I do not attack it as incoherent. The question is whether the alternative set-theoretic universes that we seem to have discovered via forcing and other methods exist as legitimate concepts of set or not. I have argued at length that they do. But the opposing universe view is that no, there is just one absolute background concept of set, and the purpose of set theory is to discover what is true there. This seems to be a perfectly coherent view. It is a view advanced explicitly by Daniel Isaacson, who I quote extensively in my dream solution paper, and also by Donald Martin, in his paper "Multiple universes of sets and indeterminism in set theory", Topoi 20, 5--16, 2001, among others.
Criticizing my argument, Peter Koellner has emphasized that one can view my account of the naturalist account of forcing, rather than providing evidence that forcing extensions are real, instead as the desired explanation of the illusion of forcing extensions of $V$. And perhaps this criticism is the detailed answer to your question. That is, Koellner argues that the details of the proof of the naturalist account of forcing is how one explains away the illusion of forcing. So that would seem to be a coherent view. My reply to that argument, in my multiverse paper, is that such an account of forcing seems fundamentally crippling to our mathematical intuition, if we must regard all talk of actual forcing extensions of $V$ as ever-more-fantastical simulations of the extensions inside $V$, something like the writings of the exotic-travelogue writer who never actually ventures west of sixth avenue, or the absurdity of the mathematician who insists that yes, the real numbers exist with a full Platonic existence, but the complex numbers do not; they must be simulated inside the reals, such as with ordered pairs. The multiverse perspective makes a philosophically simple position, taking the existence of the forcing extensions at face value, while nurturing a robust use of forcing that will ultimately aid our set-theoretical understanding.
Finally, let me say that I agree completely with Andrej's point about geometry, and I discuss this analogy in section 4 of my multiverse paper.
Best Answer
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.
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.