Source
Wenmackers (2011), LIT-163, the PhD thesis Philosophy of Probability: Foundations, Epistemology, and Computation, chapter 2, read in full in NOTE-139 with chapters 1 and 3–6.
What was actually shown
The problem. A fair lottery on ℕ = {1, 2, 3, …} should satisfy four intuitions (§2.2.1.2): FAIR (every ticket equally likely), ALL (every set of tickets has a probability), SUM (set probabilities supervene on ticket probabilities) and LABEL (invariance under relabelling). Real-valued probability cannot have FAIR with countable additivity, since each ticket would get 0 and the tickets together 1. The thesis keeps FAIR as “non-negotiable” and gives up LABEL.
The construction (eqs. 2.20–2.21). For a set A, take the partial counts #(A ∩ {1..n}), take their class num(A) modulo a free ultrafilter U (the Benci–Di Nasso numerosity, with num(ℕ) = α), and normalise: P_num(A) = num(A)/α = [⟨#(A ∩ {1..n})/n⟩]_U. Every ticket gets 1/α, ℕ less one ticket gets 1 − 1/α, and each residue class mod m gets a value whose standard part is 1/m (§2.6.4.2).
Additivity (eq. 2.27, §2.5.2). For disjoint {A_n}, num of the union is a sum over the hyperextended family indexed by *ℕ, and past some K in *ℕ all terms vanish, so the additivity is in effect hyperfinite. No function into a non-standard set can be countably additive, because countable sums are not defined there (§2.5.2.2).
The standard part (eq. 2.30). st ∘ P_num equals the Hahn–Banach generalised asymptotic density, for every choice of ultrafilter.
Non-uniqueness (§2.6.2). There is one solution per free ultrafilter. For example P_num(Odd) is ½ + 1/(2α) or exactly ½, depending on whether the odd numbers are in U.
What could have come out otherwise. Normalised numerosity could have failed to be well defined on all of 𝒫(ℕ), or its standard part could have disagreed with density. NOTE-139 rates the construction, the additivity theorem and the standard-part result as strong, by proof.
What this does not say
- That the infinite fair lottery is solved uniquely. Which ultrafilter, and so which probabilities for sets like the odd numbers, is left open. The author sees “no convincing reasons” to constrain it (§2.6.2).
- That the probability is countably additive. The thesis’s “infinitely additive” means a sum over *ℕ of the star-extended family, which collapses to a hyperfinite sum (NOTE-139, Corrections).
- That it is preferable to de Finetti’s finitely additive solution. The author concedes the two are empirically indistinguishable (§2.6.4), so the preference is intuitive (C5, weak).
- That it is constructive or label-free. It needs the axiom of choice, and probabilities change under relabelling.
- That it is an endorsement of Regularity in general. The thesis says it is not (§2.2.1.2, §2.7). Its later critics (Pruss, “Infinitesimals are too small for countably infinite fair lotteries”) are not held in the record.
- That it gives any computational method. Its standard part is ordinary asymptotic density, so nothing here bears on zero-probability events in continuous models.
Connections
- The same “rounding error” diagnosis, carried to belief, gives Stratified Belief (THEORY-647). The thesis’s opinion-dynamics result is THEORY-649.
- The ultrafilter machinery is the one Turner presents as propositional theories (LIT-145, THEORY-595).