A talk at the Philosophy and Model Theory conference held June 2-5, 2010 at the Université Paris Ouest Nanterre.

Set theorists commonly regard set theory as an ontological foundation for the rest of mathematics, in the sense that other abstract mathematical objects can be construed fundamentally as sets, enjoying a real mathematical existence as sets accumulate to form the universe of all sets. The Universe view—perhaps it is the orthodox view among set theorists—takes this universe of sets to be unique, and holds that a principal task of set theory is to discover its fundamental truths. For example, on this view, interesting set-theoretical questions, such as the Continuum Hypothesis, will have definitive final answers in this universe. Proponents of this view point to the increasingly stable body of regularity features flowing from the large cardinal hierarchy as indicating in broad strokes that we are on the right track towards these final answers.

A paradox for the orthodox view, however, is the fact that the most powerful tools in set theory are most naturally understood as methods for constructing alternative set-theoretic universes. With forcing and other methods, we seem to glimpse into alternative mathematical worlds, and are led to consider a model-theoretic, multiverse philosophical position. In this talk, I shall describe and defend the Multiverse view, which takes these other worlds at face value, holding that there are many set-theoretical universes. This is a realist position, granting these universes a full mathematical existence and exploring their interactions. The multiverse view remains Platonist, but it is second-order Platonism, that is, Platonism about universes. I shall argue that set theory is now mature enough to fruitfully adopt and analyze this view. I shall propose a number of multiverse axioms, provide a multiverse consistency proof, and describe some recent results in set theory that illustrate the multiverse perspective, while engaging pleasantly with various philosophical views on the nature of mathematical existence.

Slides | Article | see related Singapore talk

I have a question regarding your comments on Freiling’s Axiom of symmetry. on pg. 17 of the version you have on arXiv you write:

“Many mathematicians objected that that Freiling was implicitly assuming for a given function f that various sets were measurable, including importantly the set { (x,y)| y$\in$f(x)}”.

In Freiling’s Axiom of symmetry, f maps reals to countable sets of reals such that y$\in$f(x). Did you actually mean { (x,y)| y$notin$f(x)}? If you didn’t, can you show me an example of a countable non-measurable set? Thanks in advance,

that is, { (x,y)| y$\notin$f(x)}

A set is measurable just in case its complement is measurable, so the issue of $\in$ or $\notin$ doesn’t affect whether the set is measurable.

But in any case, even if each vertical slice is countable, this doesn’t meant that $\{(x,y)\mid y\in f(x)\}$ is measurable, and this is the whole point of this issue. This is a subset of the plane, and the conclusion that it has measure zero just because its vertical slices are all countable would only follow (by the Fubini theorem) provided that the set were measurable; and it is not generally measurable.

Thanks for the clarification. This ties in with Kai Hauser’s assertion in his paper “What Axioms Cannot Be” that $A_f$= {(x,y)| y$\notin$f(x)} and $A^{f}$= {(x,y)| x$\notin$f(y)} are not measurable. According to Hauser’s gloss on Freiling’s argument, “the formal statement of the conclusion of Freiling’s thought experiment is:

$\forall$f : [0,1]$\rightarrow$$[0,1]_\aleph_0$ ($\lambda^2$($A_f$)=1 $\land$ $\lambda^2$($A^{f}$)=1)

where $\lambda$ is the Lebesgue measure on $\mathbf B$, the Borel algebra in [0,1], and $\lambda^2$ is the product measure on $\mathbf B^2$, the product algebra in [0,1] $\times$ [0,1]. From this it follows that $\lambda^2$($A_f$ $\cap$ $A^{f}$)=1, hence this intersection is nonempty.”

My question is this: should one distinguish between the value of the proposition ‘$A_f$$\cap$$A^{f}$ is nonempty’ (a Boolean value) and the product measure $\lambda^2$($A_f$$\cap$$A^{f}$)=1? Certainly even if a subset A of [0,1] is nonmesurable, the proposition ‘A is nonempty’ should be assigned some boolean value. If all Freiling was trying to do with his ‘dart argument’ is to show that $A_f$$\cap$$A^{f}$$\neq$0 (and now the question becomes, did Freiling implicitly assume this in his argument), why should one bring measure theory into the mix (though Freiling is certainly guiltyof this too)?

I should have put

$\forall$f :[0,1]$\rightarrow$$[0,1]_\aleph_0$. Sorry

Let me try this again:

$[0,1]_\aleph_0$

Also, the actual title of the Hauser paper is “What New Axioms Could Not Be”– sorry. Refer to pg 3 regarding to what the right arrow refers to–I did not get the mathjax right and so what the right arrow refers to did not print at all.

Freiling argues essentially that the intersection is nonempty because it has measure one. But if I recall he does not refer explicitly to Lebesgue measure, but rather appeals to our pre-reflective ideas about how we should measure sets, and on these grounds argues that both those sets have full measure. The counterargument is that his pre-reflective ideas on measure do not account for the non-measurability phenomenon, which we know is widespread. In particular, his argument appears to presume that that subset of the plane (which Freiling does not speak about) is measurable.

Consider the following simplified thought experiment:

Assume $ZFC$. On the interval [0,1] form a Vitali set $\mathscr V$ and color the points associated with the members of $\mathscr V$ one color and the points associated with the members of its complement $\mathscr V^{/}$ on [0,1] a different color. Since $\mathscr V$ is non-measurable, $\mathscr V^{/}$ will be, too. This has the effect of partitioning [0,1] into two non-measurable sets, each having the cardinality of the continuum. Now imagine throwing darts at the interval [0,1] (assume also,for the sake of argument, that each dart thrown will always land in [0,1]). Obviously, each dart thrown will then land in either $\mathscr V$ or $\mathscr V^{/}$. Though each set is non- (Lebesgue)measurable, since the cardinality of each set is the cardinality of the continuum, is there any reason to assume, if, say, countably many darts are thrown that $\mathscr V$ and $\mathscr V{/}$ will not each contain countably many darts?

(I.e. the ‘probability’ of hitting either $\mathscr V$ or $\mathscr V^{/}$ is 1/2)?

Each arrow must hit either your Vitali set or its complement. That is the definition of a complement. Your question makes no sense otherwise, since you seem to try and assign a probability to a set which provably cannot be assigned a probability within the confines of the Lebesgue measure.

In addition to Asaf’s comment, I would add that one can find Vitali sets of very small outer measure (see this math.SE answer http://math.stackexchange.com/a/14623/413), that is, where it is covered by a measurable set of very small measure, and in this case, it would make sense to say that the odds are definitely not 1/2 – 1/2.

So you would say then, Prof. Hamkins, that the outer measures of $\mathscr V$ and $\mathscr V^{/}$ determine the ‘frequency’ (so to speak) of the darts hitting each set even though (because of each set’s non- measurability) one cannot assign a Lebesgue probability measure to either set?

No, I would say that because the sets are not measurable, there is no coherent way to assign such a `frequency’ to these sets. The sets that can support a coherent such assignment are precisely the measurable sets.

Yet on the other hand, you say that with $\mathscr V$ having very small outer measure and $\mathscr V^{/}$ having very large outer measure (relative to $\mathscr V$), the ‘odds’ of a dart hitting $\mathscr V^{/}$ rather than $\mathscr V$ are better than the ‘odds’ of a dart hitting $\mathscr V$. Since you also say that one cannot assign a coherent frequency to either $\mathscr V$ or $\mathscr V^{/}$ because they are both non-Lebesgue-measurable, were you using each set’s outer measure to infer the odds of a dart hitting each set as a “rule of thumb”?