（注：[HT08b] は、https://xorshammer.com/2008/08/23/set-theory-and-weather-prediction/ SET THEORY AND WEATHER PREDICTION XOR’S HAMMER Some things in mathematical logic that I find interesting WRITTEN BY MKOCONNOR Blog at WordPress.com. AUGUST 23, 2008 で 引用されており、かれの”Here’s a puzzle”の元ネタと思われる。
１） さて、まず、[HT08b] より (抜粋) P91 1. INTRODUCTION. We often model systems that change over time as functions from the real numbers R (or a subinterval of R) into some set S of states, and it is often our goal to predict the behavior of these systems. Generally, this requires rules governing their behavior, such as a set of differential equations or the assumption that the system (as a function) is analytic. With no such assumptions, the system could be an arbitrary function, and the values of arbitrary functions are notoriously hard to predict. After all, if someone proposed a strategy for predicting the values of an arbitrary function based on its past values, a reasonable response might be, “That is impossible. Given any strategy for predicting the values of an arbitrary function, one could just define a function that diagonalizes against it: whatever the strategy predicts, define the function to be something else.” This argument, however, makes an appeal to induction: to diagonalize against the proposed strategy at a point t, we must have already defined our function for all s < t in order to determine what the strategy would predict at t.
In fact, the lack of well-orderedness in the reals can be exploited to produce a very counterintuitive result: there is a strategy for predicting the values of an arbitrary function, based on its previous values, that is almost always correct. Specifically, given the values of a function on an interval (?∞, t), the strategy produces a guess for the values of the function on [t,∞), and at all but countably many t, there is an ε > 0 such that the prediction is valid on [t, t + ε). Noting that any countable set of reals has measure 0, we can restate this informally: at almost every instant t, the strateg predicts some “ε-glimpse” of the future.
Nevertheless, we choose this presentation because we find it the most interesting, as well as pedagogically useful. For instance, “predicting the present” is a very natural way to think of the problem of guessing the value of f (t) based on f |(?∞, t).
2. THE μ-STRATEGY. (詳細は略すので、原文ご参照。ここに引用するには、数学記号が複雑過ぎるので。)
P92 3. PREDICTING THE PRESENT. (詳細は略すので、原文ご参照。ここに引用するには、数学記号が複雑過ぎるので。)
Corollary 3.4. If T = R and ∇ is <, then W0 is countable, has measure 0, and is nowhere dense.
What Corollary 3.4 tells us is that, if we model the universe as a function from the real numbers into some set of states, then the μ-strategy will correctly predict the present from the past on a set of full measure. (In the following section, we show that, on a set of full measure, it correctly predicts some of the future as well.) Note that these results concerning T = R are also valid when T is any interval of reals. One needs to be cautious about interpreting this as meaning that the μ-strategy is correct with probability 1. For a fixed true scenario, if one randomly selects an instant t in the interval [0,1] (or in R, under a suitable probability distribution), then Corollary 3.4 does tell us that the μ-strategy will be correct at t with probability 1. However, if one fixes the instant t, and randomly selects a true scenario, then the probability that the μ-strategy is correct at t under that scenario might be 0 or might not even exist, depending on how one defines the notion of a random scenario.
P95 W = {t ∈ R | the μ-strategy does not guess well at t }. Theorem 5.1. The set W is countable, has measure 0, and is nowhere dense.
（一部仮訳） 推論3.4が示していることは、実数からある状態の集合に対する関数とするuniverseをモデル化すると、μ戦略は過去からの現在を完全な尺度で正しく予測するということです。 （次のセクションでは、on a set of full measureで、将来の予測も正しく予測されることを示しています）。 T = Rに関するこれらの結果は、Tが実数の任意の区間である場合にも有効であることに留意されたい。
２） 次に[HT09] より (抜粋) P3126 The derivation of this from our main result uses the upward topology on α in which, as we mentioned, the scattered sets are the finite subsets of α. A known result that we extend here is Theorem 5.1 from [5] in which the present is predicted from an “infinitesimal” piece of the past, and the predictor is correct except on a countable set that is nowhere dense. In terms of our framework here, we have the topology on R in which the basic open sets are half-open intervals (w, x] (so f 〜x g if f and g agree on (w, x) for some w < x). It is known that the scattered sets here are countable and nowhere dense. The exact characterization of the error sets in this example (as scattered sets) was absent in [5].
[5] C. Hardin and A. Taylor, A peculiar connection between the axiom of choice and predicting the future, American Mathematical Monthly 115 (2008),
３） 最後に[（成書）The Mathematics of Coordinated Inference: A Study of Generalized Hat Problems]より (抜粋) P76 7.3 Corollaries
The second result we derive concerns the extent to which "the present can be predicted based on the past." Here, the exact characterization of the error sets occurs in Theorems 3.1 and 3.5 in [HT08b].
The derivation of this uses the topology on R in which the basic open sets are half-open intervals (w; x] (so f 〜x g if f and g agree on (w, x) for some w < x). It is known that the scattered sets here are countable and nowhere dense. The exact characterization of the error sets in this example (as scattered sets) was absent in [HT08b].
＜結論＞ １．以上より、[HT08b]（XOR’S HAMMERのパズル元ネタ）は、著者自身の手（[HT09]と[成書]と）で、否定されている。 ”The exact characterization of the error sets in this example (as scattered sets) was absent in [HT08b].” ２．元々、[HT08b]中で 「これをμ戦略が確率1で正しいと解釈することには注意が必要です。 固定されたfixed true シナリオの場合、区間[0,1]（またはRにおいて、適切な確率分布の下で）において瞬間tをランダムに選択すると、 推論3.4は、μ戦略がtで確率1で正しいことを教えてくれる。 しかし、瞬間tを固定してランダムにfixed true シナリオを選択すると、そのシナリオの下でμ戦略が正しい確率は0であるか、または存在しないかもしれません ランダムなシナリオの概念をどのように定義するかによって異なります。」と注意を入れていて、完全に嵌まっている訳では無かったが しかし、その”1. INTRODUCTION”には、思わせぶりなことが書いてあり、ミスリードだろう。
３．XOR’S HAMMERは、勿論、きちんと[HT08b]中の注意書きは読んでいて、意識してあくまで、”Here’s a puzzle”と断っていることを注意しておく。
