Using the multiverse approach to decide the law of the exluded middle?

Question

Recently, in response to deciding the Continuum Hypothesis $CH$, Hamkins and Gitman have proposed consider a multiverse of set-theoretic universes, some in which $CH$ is true, some in which $\neg CH$ is true (and some in which $CH$ is not a relevant hypothesis?).

In formulating logical languages, there has been an outstanding problem: that of deciding the Law of the Excluded Middle $p\vee\neg p$. In classical logic $p\vee\neg p$ is true, but in intuitionistic logic this is not the case. With $CH$, the pragmatic mathematician tries to avoid invoking $CH$; if he assumes $CH$ or $\neg CH$, he will state it clearly. In everyday mathematics practice, the mathematician does use $p\vee\neg p$, but we do see an effort to give effective constructions, and hard estimates in analysis.

It seems that people are using classical logic basically because it's a core logic, in analogy to the constructible universe $L$ in which $CH$ is true. It is the logic first discovered (and often wrongly attributed as the Platonic choice) and the consistency and strength of other languages proven in terms of this core logic. Afterwards people come up with other logics, like Brouwer coming up with intuitionistic logic, in which a fundamental principle, the Law of the Excluded Middle, does not hold. It seems to me that this debate regarding the Law of Excluded Middle can be formalized using the multiverse approach.

So has anyone tried to use the multiverse approach towards considering these plurality of languages? Perhaps by considering a multiverse of topoi?

PS. The difficulty for arithemetizable syntax to describe continuous properties of geometric/measure-theoretic space is central (may I say?) to the difficulty of deciding $CH$. In a like fashion, the difference between classical and intuitionistic logic plays up in comparing the Dedekind-reals and the Cauchy-reals. In other words, continuous properties have different descriptions in these two logics. I'm hoping that work on the Law of the Excluded Middle will shed some light on trying to use discrete languages to model continuous properties. There's more where this comes from, but it is enough for now...

Answer 1

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:

1. There is a standard mathematical universe.
2. There are many mathematical universes.
3. There is a multiverse of mathematics.

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:

• Set theory studies the multiverse of models of ZF. The structure is studied via notions of forcing, and probably other things I am not aware of.
• Topos theory studies the multiverse of toposes. The structure of the multiverse is expressed as a 2-category of toposes (and geometric morphisms).

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:

• if the multiverse is too small, we will be puzzled by its ad hoc properties and we will look in vain for overall structure (imagine doing analysis with only rational numbers),
• if the multiverse is too big, its overall structure will be poor and it will include universes whose internal mathematics is too far removed from our own mathematical experience (imagine doing analysis on arbitrary rings--I am sure it's possible but it's unlike classical analysis).

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.

Answer 2

I offer a different perspective on the question, which I hope will be found useful.

One view of logic is strongly syntactic: develop a language, and develop rules for manipulating this language and classifying parts of this language (terms, well-formed formulae, sentences, derivation, proofs, theories, and so on). One of course would like to arrange things so that from a syntactic perspective one can easily define such subsets so as to say "for each x in this collection of objects, I can interpret x in this environment so that the interpretation is at the same time true, meaningful, and tells me something about the environment." .

Indeed, such a viewpoint helped in various areas of scientific and technological development. Compiler theory, software development, fuzzy logic are just three areas of "real-world" applications. In this "real-world", the focus is often on solving a relatively narrow class of problems with a small set of tools, as opposed to considering many alternative problems with a wide variety of tools. I call this an "engineering" perspective.

In my view, what you propose is a comparison of several logical systems in the context of some unifiable framework, loose enough to encompass all of the possibilities that you want to consider, but tight enough that you can resolve some of the potential questions you see. I call this a "scientific" perspective.

Speaking from ignorance, the closest I see in attempting such things are in fields like modal logic, algebraic logic, and certain forms of general proof theory. I don't know enough of modal logic to comment intelligently, but I see various logics being compared and contrasted in some papers on modal logic. I haven't worked in algebraic logic, but relational algebras, cylindric algebras, Heyting algebras, and more occurred in my studies in universal algebra: these structures were meant to put a strong algebraic component in studying and modeling logical systems. From my naive perspective, one of the intentions was to provide an algebraic way to "define, true, meaningful, and useful subsets of a language". I know (almost) less than nothing of proof theory, but I recall Kleene's "Introduction to Metamathematics" where he develops formalisms for both classical and intuitionistic logic in parallel, using notation similar to those found in proof theory, and I dimly recall Lindstroem's result on first-order logic being the strongest logic system in a sense (something about having both compactness and downward L-S theorems). So attempts have been made which are (as Gerry Myerson might say) "not a million miles away" from your proposal.

Of course, people who have spent time studying such things should weigh in and clarify the situation. Andrej Bauer has given a nice philosophical perspective, but I think he missed the aspects above. In particular, I think the "engineering" perspective can be brought more in line with the "scientific" perspective, in that while a "full" multiverse of logics could be studied, emphasis should be placed on the "favored" multiverse of logics that are currently or soon will be applied to give us some of the technological benefits mathematical logic can give society.

Gerhard "Ask Me About System Design" Paseman, 2015.10.08