Hyperreal infinitesimal probabilities and definability

In order to assign non-zero probabilities to such things as a lottery ticket in an infinite fair lottery or hitting a specific point on a target with a uniformly distributed dart throw, some people have proposed using non-zero infinitesimal probabilities in a hyperreal field. Hajek and Easwaran criticized this on the grounds that we cannot mathematically specify a specific hyperreal field for the infinitesimal probability. If that were right, then if there are hyperreal infinitesimal probabilities for such a situation, nonetheless we would not be able to say what they are. But it’s not quite right: there is a hyperreal field that is "definable", or fully specifiable in the language of ZFC set theory.

However, for Hajek-Easwaran argument against hyperreal infinitesimal probabilities to work, we don’t need that the hyperreal field be non-definable. All we need is that the pair (^*R,α) be non-definable, where ^*R is a hyperreal field and α is the non-zero infinitesimal assigned to something specific (say, a single ticket or the center of the target).

But here is a fun fact, much of the proof of which comes from some remarks that Michael Nielsen sent me:

Theorem: Assume ZFC is consistent. Then ZFC is consistent with there not being any definable pair (^*R,α) where ^*R is a hyperreal field and α is a non-zero infinitesimal in that field.

[Proof: Solovay showed there is a model of ZFC where every definable set is measurable. But every free ultrafilter on the powerset of the naturals is nonmeasurable. However, an infinite integer in a hyperreal field defines a free ultrafilter on the naturals—given an infinite integer M, say that a subset A of the naturals is a member of the ultrafilter iff |M| ∈ ^*A. And a non-zero infinitesimal defines an infinite integer—say, as the floor of its reciprocal.]

Given the Theorem, without going beyond ZFC, we cannot count on being able to define a specific hyperreal non-zero infinitesimal probability for situations like a ticket infinite lottery or hitting the center of a target. Thus, if a friend of hyperreal infinitesimal probabilities wants to be able to define one, they must go beyond ZFC (ZFC plus constructibility will do).

When can we have exact symmetries of hyperreal probabilities?

In many interesting cases, there is no way to define a regular hyperreal-valued probability that is invariant under symmetries, where “regular” means that every non-empty set has non-zero probability. For instance, there is no such measure for all subsets of the circle with respect to rotations: the best we can do is approximate invariance, where P(A)−P(rA) is infinitesimal for every rotation. On the other hand, I have recently shown that there is such a measure for infinite sequences of fair coin tosses where the symmetries are reversals at a set of locations.

So, here’s an interesting question: Given a space Ω and a group G of symmetries acting on Ω, under what exact conditions is there a hyperreal finitely-additive probability measure P defined for all subsets of Ω that satisfies the regularity condition P(A)>0 for all non-empty A and yet is fully (and not merely approximately) invariant under G, so that P(gA)=P(A) for all g ∈ G and A ⊆ Ω?

Theorem: Such a measure exists if and only if the action of G on Ω is locally finite. (Assuming the Axiom of Choice.)

The action of G on Ω is locally finite iff for every x ∈ Ω and every finitely-generated subgroup H of G, the orbit Hx = {hx : h ∈ H} of x under H is finite. In other words, we have such a measure provided that applying the symmetries to any point of the space only generates finitely many points.

This mathematical fact leads to a philosophical question: Is there anything philosophically interesting about those symmetries whose action is locally finite? But I’ve spent so much of the day thinking about the mathematical question that I am too tired to think very hard about the philosophical question.

Sketch of Proof of Theorem: If some subset A of Ω is equidecomposable with a proper subset A′, then a G-invariant measure P will assign equal measure to both A and A′, and hence will assign zero measure to the non-empty set A − A′, violating the regularity condition. So, if the requisite measure exists, no subset is equidecomposable with a proper subset of itself, which by a theorem of Scarparo implies that the action of G is locally finite.

Now for the converse. If we could show the result for all finitely-generated groups G, by using ultraproduct along an ultrafilter on the partially ordered set of all finitely generated subgroups of G we could show this for a general G.

So, suppose that G is finitely generated and the orbit of x under G is finite for all x ∈ Ω. A subset A of G is said to be G-invariant provided that gA = A for all g ∈ G. The orbit of x under G is always G-invariant, and hence every finite subset of A is contained in a finite G-invariant subset, namely the union of the orbits of all the points in A.

Consider the set F of all finite G-invariant subsets of Ω. It’s worth noting that every finite subset of G is contained in a finite G-closed subset: just take the union of the orbits under G. For A ∈ F, let P_A be uniform measure on A. Let F^* = {{B ∈ F : A ⊆ B}:A ∈ F}. This is a non-empty set with the finite intersection property. Let U be an ultrafilter extending F^*. Let ^*R be the ultraproduct of the reals over F with respect to U, and let P(C) be the equivalence class of the function A ↦ P_A(A ∩ C) on F. Note that C ↦ P_A(A ∩ C) is G-invariant for any G-invariant set A, so P is G-invariant. Moreover, P(C)>0 if C ≠ ∅. For let C′ be the orbit of some element of C. Then {B ∈ F : C′⊆B} is in F^*, and P_A(A ∩ C′) > 0 for all A such that C′⊆A, so the set of all A such that P_A(A ∩ C′) > 0 is in U. It follows that P(C′) > 0. But C′ is the orbit of some element x of C, so every singleton subset of C′ has the same P-measure as {x} by the G-invariance of P. So P({x}) = P(C′)/|C′| > 0, and hence P(C)≥P({x}) > 0.

Hyperreal modeling of infinitely many coin flips

A lot of my work in philosophy of probability theory has been devoted to showing that one cannot use technical means to get rid of certain paradoxes of infinite situations. As such, most of the work has been negative. But here is a positive result. (Though admittedly it was arrived at in the service of a negative result which I hope to give in a future post.)

Consider the case of a (finite or infinite, countable or not) sequence of independent fair coin flips. Here is an invariance feature we would like to have for our coin flips. Suppose that ahead of time, I designate a (finite or infinite) set of locations in the infinite sequence. You then generate the sequence of independent fair coin flips, and I go through my pre-designated set of locations, and turn over each of the coins corresponding to that location. (For instance, if you will make a sequence of four coin flips, and I predesignate the locations 1 and 3, and you get HTTH, then after my extra flipping set the sequence of coin flips becomes TTHH: I turned over the first and third coins.) The invariance feature we want is that no matter what set of locations I predesignate, it won’t affect the probabilistic facts about the sequence of independent fair coin flips.

This invariance feature is clearly present in finite cases. It is also present if “probabilistic facts” are understood according to classical countably-additive real-valued probability theory. But what if we have infinitely many coins, and we want to be able to do things like comparing the probability of all the coins being heads to all the even-numbered coins being heads, and say that the latter is more likely than the former, with both probabilities being infinitesimal? Can we still have our reversal-invariance property for all predesignated sets of locations?

There are analogous questions for other probabilistic situations. For instance, for a spinner, the analogous property is adding an extra predesignated rotation to the spinner once the spinner stops, and it is well-known that one cannot have such invariance in a context that gives us “enough” infinitesimal probabilities (e.g., see here for a strong and simple result).

But the answer is positive for the coin flip case: there is a hyperreal-valued probability defined for all subsets of the set of sequences (with fixed index set) of heads and tails that has the reversal-invariance property for every set of locations.

This follows from the following theorem.

Theorem: Assume the Axiom of Choice. Let G be a locally finite group (i.e., every finite subset generates a finite subgroup) and suppose that G acts on some set X. Then there is a hyperreal finitely additive probability measure P defined for all subsets of X such that P(gA)=P(A) for every A ⊆ X and g ∈ G and P(A)>0 for all non-empty A.

To apply this theorem to the coin-flip case, let G be the abelian group whose elements are sets of locations with the exclusive-or operation (i.e., A ⊕ B = (A − B)∪(B − A) is the set of all locations that are in exactly one of A and B). The identity is the empty set, and every element has order two (i.e., A ⊕ A = ∅). But for abelian groups, the condition that every finite subset generates a finite subgroup is equivalent to the condition that every element has finite order (i.e., some finite multiple of it is zero).

Mathematical notes: The subgroup condition on G in the Theorem entails that every element of G has finite order, but is stronger than that in the non-abelian case (due to the non-trivial fact that there are infinite finitely generated torsion groups). In the special case where X = G, the condition that every element of G have finite order is necessary for the theorem. For if g has infinite order, let A = {gⁿ : n ≥ 0}, and note that gA is a proper subset of A, so the condition that non-empty sets get non-zero measure and finite additivity would imply that P(gA)<P(A), which would violate invariance. It is an interesting question whether the condition that every finite subset generates a finite subgroup is also necessary for the Theorem if X = G.

Proof of Theorem: Let F be the partially ordered set whose elements are pairs (H, V) where H is a finite subgroup of G and V is a finite algebra of subsets of X closed under the action of H, with the partial ordering (H₁, V₁)≼(H₂, V₂) if and only if H₁ ⊆ H₂ and V₁ ⊆ V₂.

Given (H, V) in F, let B_V be the basis of V, i.e., a subset of pairwise disjoint non-empty elements of V such that every element of V is a union of (finitely many) elements of B_V. For A ∈ B_V and g ∈ H, note that gA is a member of V since V is closed under the action of H. Thus, gA = B₁ ∪ ... ∪ B_n for distinct elements B₁, ..., B_n in B_V. I claim that n = 1. For suppose n ≥ 2. Then g⁻¹B₁ ⊆ A and g⁻¹B₂ ⊆ A, and yet both g⁻¹B₁ and g⁻¹B₂ are members of V by H-closure. But since A is a basis element it follows that g⁻¹B₁ = A = g⁻¹B₂, and hence B₁ = B₂, a contradiction. Thus, n = 1 and hence gA ∈ B_V. Moreover, if gA = gB then A = B, so each member g of H induces a bijection of B_V onto itself.

Now let P_(H, V) be the probability measure on V that assigns equal probability to each member of B_V. Since each member of H induces a bijection of B_V onto itself, it’s easy to see that P_(H, V) is an H-invariant probability measure on V. And, for convenience, if A ∉ V, write P_(H, V)(A)=0.

Let F^* = {{B ∈ F : A ≼ B}:A ∈ F}. This is a nonempty set with the finite intersection property (it is here that we will use the fact that every finite subset of G generates a finite subgroup). Hence it can be extended to an ultrafilter U. This ultrafilter will be fine: {B ∈ F : A ≼ B}∈U for every A ∈ F. Let ^*R be the ultraproduct of the reals R over F with respect to U, i.e., the set of functions from F to R modulo U-equivalence. Given a subset A of X, let P(A) be the equivalence class of (H, V)↦P_(H, V)(A).

It is now easy to verify that P has all the requisite properties of a finitely-additive hyperreal probability that is invariant under G and assigns non-zero probability to every non-empty set.

Gunky ontology and virtual points

Gunk is subdivisible into smaller parts, and these are subdivisible into yet smaller parts, and this happens ad infinitum, with no smallest indivisible parts or atoms.

But here is an interesting fact: One can introduce ersatz atoms or virtual points into a gunky ontology, given some plausible mereological axioms. Suppose that O is a gunky object. Then the set M(O) of the parts of O has a partial order ≤ where x≤y if and only if x is a part of y. Now we can say that an ersatz atom of O is any ultrafilter on O with respect to the ordering.

Thus, ersatz atoms are subsets U of M(O) such that:

1. U is a non-empty proper subset of M(O)
2. if x is in U then everything that has x as a part is also in U
3. if x and y are in U, then there is a z in U such that z≤x and z≤y
4. U is maximal: any larger subset satisfying (1)-(3) is all of M(O).

We can then say that an ersatz atom U is an ersatz part of x∈M(O) provided that x∈U.

To get the existence of ersatz atoms, we need some axioms of mereology in addition to the Axiom of Choice. Fortunately, pretty weak mereological axioms suffice:

5. parthood is a partial ordering
6. O has two parts x and y that do not overlap

As usual, two things are said to overlap provided that there is something that is a part of both.

In general, given any two parts x and y that do not overlap, there will be an ersatz atom U that is an ersatz part of x but not of y. Let's further assume the strong supplementation axiom that if y is not a part of x, then there is a z that is a part of y such that z does not overlap with x. Then whenever x≠y, there will be an ersatz atom that's an ersatz part of one but not of the other. Hence, we can identify every part of O with a set of ersatz atoms. However, given gunkiness, not every set of ersatz atoms corresponds to a part. In particular, singleton sets of ersatz atoms do not correspond to parts.

So the gunk theorist can talk as if objects were made out of atoms. Now, if we have a gunky ontology, then I think we should take the parts to be non-fundamental, and grounded in the wholes rather than the other way around on pain of a grounding regress. But if we allow non-fundamental parts in our ontology, then one may worry that the gunkiness of the ontology is merely verbal and non-substantive, dependent on the verbal decision not to talk of the ersatz atoms as real parts.