Getting rich quick with the Axiom of Choice

This paper proposes new get-rich-quick schemes that involve trading in a financial security with a non-degenerate price path. For simplicity the interest rate is assumed zero. If the price path is assumed continuous, the trader can become infinitely rich immediately after it becomes non-constant (if it ever does). If it is assumed positive, he can become infinitely rich immediately after reaching a point in time such that the variation of the log price is infinite in any right neighbourhood of that point (whereas reaching a point in time such that the variation of the log price is infinite in any left neighbourhood of that point is not sufficient). The practical value of these schemes is tempered by their use of the Axiom of Choice.


Introduction
This paper belongs to the area of game-theoretic probability (see, e.g., [11]). The advantage of game-theoretic probability for mathematical finance over the dominant approach of measure-theoretic probability is that it allows us to state and prove results free of any statistical assumptions even in situations where such assumptions are often regarded as essential (see, e.g., the probability-free Dubins-Schwarz theorem in [12] and the probability-free theory of stochastic integration in [9]). In this paper we consider the framework of an idealized financial market with one tradable security and assume, for simplicity, a zero interest rate.
The necessity of a non-trivial requirement of measurability (such as Borel, Lebesgue, or universal) is well known in measure theory (without measurability we have counter-intuitive results such as the Banach-Tarski paradox 1 [14]), and it is inherited by measure-theoretic probability. In game-theoretic probability measurability is usually not needed in discrete time (and is never assumed in, e.g., [11]); this paper, however, shows that, in continuous time, imposing some regularity conditions (such as Borel or universal measurability) is essential even in the foundations of game-theoretic probability. If such conditions are not imposed, the basic definitions of game-theoretic probability become uninteresting, or even degenerate: e.g., in the case of continuous price paths the upper probability of sets can take only two values: 1 (if the set contains a constant price path) or 0 (if not).
This paper constructs explicit trading strategies for enriching the trader given a well-order of the space of all possible price paths; however, such a wellorder exists only under the Axiom of Choice (which is, despite some anomalous corollaries, universally accepted). Since we cannot construct such well-orders, our strategies cannot be regarded as genuinely constructive. Therefore, they cannot be regarded as practical get-rich-quick schemes. Moreover, they do not affect the existing results of continuous-time game-theoretic probability, which always explicitly assume measurability (to the best of my knowledge).
Our trading strategies will be very simple and based on Hardin and Taylor's work on hat puzzles (going back to at least [5] and [4,Chapter 4]). These authors show in [6] (see also [7,Section 7.4]) that the Axiom of Choice provides us with an Ockham-type strategy able to predict short-term future (usually albeit not always), which makes it easy to get rich when allowed to trade in a security whose price changes in a non-trivial manner.
Section 2 is devoted to continuous price paths. The assumption of continuity allows us to use leverage and stop-loss strategies, and the trader can profit greatly and quickly whenever the price path is not constant. (We only consider trading strategies that never risk bankruptcy. It is clear that profiting from a constant price path is impossible.) In Section 3 we assume, instead of continuity, that the price path is càdlàg. To make trading possible we further assume that the price path is positive. It is impossible for the trader to become infinitely rich if the log price path has a finite variation. If the variation is infinite, there will be either points in time such that the variation of the log price is infinite in any of their left neighbourhoods or points in time such that the variation of the log price is infinite in any of their right neighbourhoods. Becoming infinitely rich is possible after points of the latter type. Standard stochastic models of financial markets postulate price paths that have such points almost surely.
In the short Section 4 we only assume that the price path is positive; the theory in this case is almost identical to the theory for positive càdlàg price paths.
The proofs of all our main results are collected in a separate section, Section 5; they are based on Hardin and Taylor's results. Appendix A provides a more general picture of predicting short-term future using the Axiom of Choice. It answers several very natural questions (and at the end asks 256 more questions answering just one of them).
Our definitions of the basic notions of continuous-time probability (such as stopping times) will be Galmarino-type (see, e.g., [2, Theorems 1.2 and 1.4]) and modelled on the ones in the technical report [12] (the journal version uses slightly different definitions), except that the requirements of measurability will be dropped. By "positive" I will mean "nonnegative", adding "strictly" when necessary. The restriction f | C of a function f : A → B to a set C is defined as f | A∩C ; this notation will be used even when C ⊆ A.

Continuous price paths
Let Ω be the set C[0, 1] of all continuous functions ω : [0, 1] → R (intuitively, these are the potential price paths over the time interval [0,1]). An adapted process S is a family of functions S t : Ω → [−∞, ∞], t ∈ [0, 1], such that, for all ω, ω ′ ∈ Ω and all t ∈ [0, 1], The intuition is that τ (ω) is not affected if ω changes over (τ (ω), 1]. For any stopping time τ , a function X : Ω → R is said to be determined by time τ if, for all ω, ω ′ ∈ Ω, The intuition is that X(ω) depends on ω only via ω| [0,τ (ω)] . We will often simplify ω(τ (ω)) to ω(τ ) (occasionally, the argument ω will be omitted in other cases as well). The class of allowed trading strategies is defined in two steps. First, a simple trading strategy G consists of an increasing sequence of stopping times τ 1 ≤ τ 2 ≤ · · · and, for each k = 1, 2, . . ., a bounded function h k that is determined by time τ k . It is required that, for each ω ∈ Ω, τ k (ω) = 1 from some k on. To such G and an initial capital c ∈ R corresponds the simple capital process (with the zero terms in the sum ignored, which makes the sum finite); the value h k (ω) will be called the bet at time τ k (ω), and K G,c t (ω) will be referred to as the capital at time t.
Second, a positive capital process is any adapted process S that can be represented in the form where the simple capital processes K Gn,cn t (ω) are required to be positive, for all t and ω, and the positive series ∞ n=1 c n is required to converge in R. The sum (4) is always positive but allowed to take the value ∞. Since K Gn,cn 0 (ω) = c n does not depend on ω, S 0 (ω) also does not depend on ω and will sometimes be abbreviated to S 0 .
The upper probability of a set E ⊆ Ω is defined as where S ranges over the positive capital processes and 1 E stands for the indicator function of E. We say that a set E ⊆ Ω is null if P(E) = 0.
Remark 1. The intuition behind a simple trading strategy is that the trader is allowed to take positions h k , either long or short, in a security whose price at time t ∈ [0, 1] is denoted ω(t). The positions can change only at a discrete sequence of times τ 1 , τ 2 , . . ., which makes the definition (3) of the trader's capital at time t uncontroversial. To obtain more useful trading strategies, we allow the trader to split his initial capital into a countable number of accounts and to run a separate simple trading strategy for each account; none of the component simple trading strategies is allowed to go into debt. The resulting total capital at time t is given by (4). The upper probability P(E), defined by (5), is the smallest initial capital sufficient for superhedging the binary option on E.
The following theorem will be proved in Section 5.
Theorem 1. The set of all non-constant ω ∈ Ω is null. Moreover, there is a positive capital process S with S 0 = 1 that becomes infinite as soon as ω ceases to be constant: for all t ∈ [0, 1], Remark 2. A more popular version of our definition (5) was given by Perkowski and Prömel [9]. Perkowski and Prömel's definition is more permissive [9, Section 2.3], and so the first statement of Theorem 1 continues to hold for it as well if we allow non-measurable (but still non-anticipative) trading strategies. As all papers (that I am aware of) on continuous-time game-theoretic probability, the definitions given in [9] assume the measurability of all strategies.
Remark 3. The definitions of this section assume that the trader is permitted to short the security (which allows h k (ω) < 0) and borrow money (which allows leverage, H k (ω) > 1 in the notation of Remark 4 below). If shorting and borrowing are not permitted (in the notation of Remark 4, if H k are only permitted to take values in [0, 1]), Theorem 1 ceases to be true, but Theorem 3 is still applicable.

Positive càdlàg price paths
In this section we will prove an analogue of Theorem 1 for positive càdlàg price paths ω; the picture now becomes more complicated. Intuitively, ω : [0, 1] → [0, ∞) is a price path of a financial security whose price is known always to stay positive (such as stock, and from now on it will be referred to as stock). For simplicity in the bulk of this section we consider the price paths ω satisfying inf ω > 0; the case of general positive ω will be considered in Remark 7 at the end of the section. Therefore, we redefine Ω as the set of all ω : [0, 1] → [0, ∞) such that inf ω > 0. The definitions of adapted processes, stopping times, etc., stay literally as before (but with the new definition of Ω). Our goal will be to determine the sign of P(E) (i.e., to determine whether P(E) > 0) for a wide family of sets E ⊆ Ω.
Theorem 2. For any ω ∈ Ω, We can see that P(E) > 0 whenever E contains ω with var(log ω) < ∞. Therefore, in the rest of this section we will concentrate on ω ∈ Ω with var(log ω) = ∞. We start from a classification of such ω.
For any f : [0, 1] → R and t ∈ [0, 1], set var(f, Furthermore, for each ω ∈ Ω set The next lemma shows that the set of all ω ∈ Ω with infinite variation of their logarithm can be represented as Proof. The implication ⇐= is obvious, and so we only check =⇒. Suppose These neighbourhoods form a cover of [0, 1]. The existence of its finite subcover immediately implies that var(f ) < ∞.
The following theorem (proved in Section 5) tackles the second term of the union in (13).
. We will impose this requirement on the relative bets involved in a simple capital process (3); intuitively, H k (ω) is the fraction of the trader's capital invested in the stock at time τ k . In terms of the relative bets, the simple capital process (3) can be rewritten as Notice that this simple capital process is positive if and only if the relative bets H k are always in the range [0, 1] (since the stock price can shoot up or drop nearly to 0 at any time). A positive capital process (4) is predictable if the component simple capital processes K Gn,cn involve only predictable stopping times τ k and relative bets H k determined before τ k . Theorem 3 can be strengthened by requiring the positive capital process S to be, in addition, predictable. (Notice that predictability was automatic in the continuous case of Section 2.) This observation can be strengthened further. Let us say that a stopping time τ is strongly predictable if, for every ω ∈ Ω, there exists t < τ (ω) such that, for every A function X : Ω → R is said to be determined strictly before a stopping time τ if, for every ω ∈ Ω, there exists t < τ (ω) such that, for every ω ′ ∈ Ω, A positive capital process (4) is strongly predictable if the component simple capital processes K Gn,cn involve only strongly predictable stopping times τ k and relative bets H k determined strictly before τ k . Theorem 3 can be further strengthened by requiring the positive capital process S to be strongly predictable. A simple modification of the proof of Theorem 3 demonstrating this fact will be given in Remark 9.
Remark 5. In the context of the previous remark, imposing the requirement of being determined before τ k on the bets h k rather than relative bets H k would lead to a useless notion of a predictable positive capital process: all such processes would be constant. In the continuous case of Section 2 (where ω is not required to be positive), the notion of a predictable positive capital process is equivalent to that of a positive capital process, but the notion of a strongly predictable positive capital process, even as given in the previous remark, is useless: again, any such process is a constant.
Theorems 2 and 3 show that the only non-trivial part of Ω (as far as the sign of P is concerned) is Namely, Theorem 2 and the following result (also to be proved in Section 5) show that this part is really non-trivial: it has subsets of upper probability one and nonempty subsets (such as any singleton) of upper probability zero.
Theorem 4. The set Ω nt has upper probability one. Moreover, for each t ∈ (0, 1]. Remark 6. The following modification of variation (7) is often useful: var where u + := u ∨ 0; we will allow f : . Using this definition, (8) can be simplified (cf. [13, the end of Section 2]) to and in this form the equality becomes true for any positive càdlàg ω (with inf ω = 0 allowed). The modified versions of (9)-(12) are: var var Lemma 1 and Theorems 3 and 4 will continue to hold if we replace all entries of var by var + and all entries of I by J.
Remark 7. In this remark we allow the price path ω to take value zero. Redefine Ω as the set of all positive càdlàg functions ω : [0, 1] → [0, ∞), and consider the partition of Ω into the following three subsets: In other words, A is Ω as defined in the main part of this section, B is the set of all ω ∈ Ω that become zero at some point t in time and then never recover, and C is the set of all ω ∈ Ω such that ω(t 1 −) ∧ ω(t 1 ) = 0 and ω(t 2 ) > 0 for some t 1 < t 2 . Theorems 3 and 4, as stated originally or as modified in the previous remark, describe the sign of P for subsets of A. We can ignore the price paths in C: P(C) = 0 and, therefore, for any E ⊆ Ω, In the rest of this remark we will allow not only f : , and (19), but also any ω ∈ Ω (for the new definition of Ω) in (20)-(21). Theorem 3 will continue to hold for ω ∈ A ∪ B if we replace var by var + and I by J (as shown by the same argument, given in Section 5). Equation (15) in Theorem 4 can be rewritten as

Positive price paths
Let us now redefine Ω to be the set of all positive functions ω : satisfying inf ω > 0 (without any continuity requirements). The definitions of adapted processes, stopping times, etc., again stay as in Section 2. Theorems 3 and 4 will still hold, as shown by the same arguments in the next section. Remark 4 will still hold with the same definitions of predictable and strongly predictable positive capital processes. Remark 7 will still hold for Ω the set of all positive functions ω : [0, 1] → [0, ∞).

Proofs of the theorems
The next result (Lemma 2) is applicable to all Ω considered in Sections 2-4. Fix a well-order of Ω, which exists by the Zermelo theorem (one of the alternative forms of the Axiom of Choice; see, e.g., [8,Theorem 5.1]). Let ω a , where ω ∈ Ω and a ∈ [0, 1], be the -smallest element of Ω such that ω a | [0,a] = ω| [0,a] . Intuitively, using ω a as the prediction at time a for ω is an instance of Ockham's razor: out of all hypotheses compatible with the available data ω| [0,a] we choose the simplest one, where simplicity is measured by the chosen well-order.
For any ω ∈ Ω set (in particular, 1 ∈ W ω ). The following lemma says, intuitively, that short-term prediction of the future is usually possible.
1. The set W ω is well-ordered by ≤. (Therefore, each of its points is isolated on the right, which implies that W ω is countable and nowhere dense.) Part 1 of Lemma 2 says, informally, that the set W ω is small. Part 2 says that at each time point t outside the small set W ω the Ockham prediction system that outputs ω t as its prediction is correct (over some non-trivial time interval). And part 3 says that even at time points t in W ω the Ockham prediction system becomes correct (in the same weak sense) immediately after time t.
Proof. Let us first check that W ω is well-ordered by ≤. Suppose there is an infinite strictly decreasing chain t 1 > t 2 > · · · of elements of W ω . Then we have ω t1 ≻ ω t2 ≻ · · · , which contradicts being a well-order.
Each point t ∈ W ω \ {1} is isolated on the right since W ω ∩ (t, t ′ ) = ∅, where t ′ is the successor of t. Therefore, W ω is nowhere dense. To check that W ω is countable, map each t ∈ W ω \ {1} to a rational number in the interval (t, t ′ ), where t ′ is the successor of t; this mapping is an injection.
Remark 8. It might be tempting to conjecture that, for any t ∈ W ω \ {1}, the function s → ω s does not depend on s ∈ (t, t ′ ), where t ′ is the successor of t.

Proof of Theorem 1
For each pair of rational numbers (a, b) such that 0 < a < b < 1 fix a strictly positive weight w a,b > 0 such that a,b w a,b = 1, the sum being over all such pairs. For each such pair (a, b) we will define a positive capital process K a,b such that K a,b will then achieve our goal (6).
It remains to construct such a positive capital process K a,b for fixed a and b. From now until the end of this proof, ω is a generic element of Ω. For each n ∈ {1, 2, . . .}, let D n := {k2 −n | k ∈ Z} and define a sequence of stopping times T n k , k = −1, 0, 1, 2, . . ., inductively by T n −1 := a, . . , where we set inf ∅ := b. For each n = 1, 2, . . ., define a simple capital process K n as the capital process of the simple trading strategy with the stopping times ω ∈ Ω → τ n k (ω) := T n k (ω) ∧ T n k (ω a ), k = 0, 1, . . . , the corresponding bets h n k that are defined as h n k (ω) := 2 2n ω a (τ n k+1 (ω a )) − ω(τ n k ) if ω τ n k (ω) = ω a and τ n k (ω) < b 0 otherwise, and an initial capital of 1. Since the increments of this simple capital process never exceed 1 in absolute value (and trading stops as soon as the prediction ω a is falsified), its initial capital of 1 ensures that it always stays positive. The final value K n b (ω) is Ω(2 n ) (to use Knuth's asymptotic notation) unless b] or ω| [a,b] is constant. This completes the proof of Theorem 1.

Proof of Theorem 2
We will follow the proof of Proposition 2 in [13] (that proposition considers measurable strategies, but the assumption of measurability is not essential there). Let us check the equivalent statement (17). If c < var + (log ω), we can find a partition 0 = t 0 < t 1 < · · · < t n = 1 of [0, 1] such that (16)). By investing all the available capital into ω at time t i−1 whenever (log ω(t i ) − log ω(t i−1 )) + > 0 (i.e., whenever ω(t i ) > ω(t i−1 )), the trader can turn 1 into at least e c . This proves the inequality ≤ in (17). And it is clear that this is the best the trader can do without risking bankruptcy.
For further (obvious) details, see the proof of Proposition 2 in [13].

Proof of Theorem 3
The proof will use the fact that inf I + ω ∈ I + ω when I + ω = ∅. Notice that I + ω = J + ω , where J + ω is defined in Remark 6. In this subsection we construct a positive capital process S such that S 0 < ∞ and S 1 = ∞ whenever I + ω = ∅; moreover, it will satisfy (14). Namely, we define S via its representation (4) with the components K Gn,cn (ω), where ω is a generic element of Ω, defined as follows: • c n = 1/n 2 (which ensures that the total initial capital n 1/n 2 is finite); G n will consist of stopping times denoted as τ n 1 , τ n 2 , . . . and bets denoted as h n 1 , h n 2 , . . .; • if I + ω = ∅, set τ n 1 (ω) = τ n 2 (ω) = · · · = 1 and h n 1 (ω) = h n 2 (ω) = · · · = 0 (intuitively, G n never bets, which makes this part of the definition nonanticipatory); in the rest of this definition we will assume that I + ω = ∅ and, therefore, inf I + ω < 1; • set a := inf I + ω ; we know that a ∈ I + ω and a < 1; • in view of Lemma 2, set ω a+ := ω t for t ∈ (a, a + ǫ) for a sufficiently small ǫ (such that t → ω t does not depend on t ∈ (a, a + ǫ)); • set d := (a + c)/2 and define and h n k (ω a+ ), k = 1, 2, . . ., in such a way that (which is required implicitly by the definition of the positivity of S and is equivalent to the relative bets being in the range [0, 1]) and Let us check (14). Suppose the antecedent of (14) holds for given t ∈ [0, 1] and ω ∈ Ω. Using the notation introduced in the previous paragraph (and suppressing the dependence on ω and n, as before), we can see that t > a. From some n on we will have d + 2 −n < t and ω s = ω a+ for all s ∈ (a, d + 2 −n ), and so the divergence of the series n e n /n 2 implies that S t (ω) = ∞.
After changing all G n in this way, we will obtain a positive capital process that is strongly predictable and still satisfies (14).

Proof of Theorem 4
We will be proving (15) We will be particularly interested in ω ξ such that lim i→∞ ω ξ (t i ) exists in (0, ∞); we can then extend ω ξ to an element ω ξ→ of Ω that is constant over [t, 1]. We will call such ω ξ extendable; for them ω ξ→ exists and is an element of the set {ω ∈ Ω | I − ω = {t}, I + ω = ∅} in (15). Let us check that no positive capital process S grows by a factor of at least 1 + ǫ, where ǫ > 0 is a given constant, on each extendable ω ξ . Suppose, on the contrary, that a given S satisfies S 0 = 1 and for all extendable ω ξ . Consider any representation of S in the form (4). This proof uses methods of measure-theoretic probability; our probability space is Ξ equipped with the canonical filtration (F i ) and the power of the uniform probability measure on {−1, 1}: F i consists of all subsets of Ξ that are unions of cylinders {(ξ 1 , ξ 2 , . . .) ∈ Ξ | ξ 1 = c 1 , . . . , ξ i = c i }, and the measure of each such cylinder is 2 −i . This is a discrete probability space without any measurability issues (the simple idea of using such a "poor" probability space was used earlier in, e.g., [11,Section 4.3]).
To simplify formulas we use the notation ω i := ω ξ (t i ) and K i := S ti (ω) for any ω ∈ Ω that agrees with ω ξ over the interval [0, t i ] (there is no dependence on such ω, and the dependence on ξ is suppressed, as usual in measure-theoretic probability). According to (28), ω i is a martingale. Let us check that where b i is the total bet of all G n immediately before time t i after observing ω ξ | [0,ti) (the only issue in this check is convergence). Formally, where k(n, i) := max{k | τ n k < t i } and (τ n k ) and (h n k ) are the stopping times and bets of G n . The series (31) converges in [0, ∞] as its terms are positive (to ensure the positivity of each K Gn,cn ). Since the capital process S is positive and ω can drop almost to 0 at any time, we have b i ∈ [0, K i−1 /ω i−1 ] (this follows from the analogous inclusions for the component simple capital processes). We can see that (30) is indeed true. Let us define α i ∈ [0, 1] by the condition Being a martingale transform of ω i , K i is also a martingale. Combining (28), (30), and (32), we can see that the recurrences for the two martingales are Let us check that log ω i converge in R for almost all ξ as i → ∞. This follows from Taylor's formula (where θ i ∈ [0, 1]), the almost sure convergence of i ξ i /(i + 1) (which follows from Kolmogorov's two series theorem), and the convergence of i (i+1) −2 . We can see that ω ξ is almost surely extendable. In the same way we can demonstrate the convergence of log K i in R, but we will not need it.
By Fatou's lemma, we have, for any ξ ∈ Ξ with extendable ω ξ , Another application of Fatou's lemma and the fact that almost all ω ξ are extendable show that the chain is well defined and correct. This contradicts our assumption (29).

Conclusion
This paper shows that some assumptions of regularity (apart from being nonanticipative, such as universal measurability) should be imposed on continuoustime trading strategies even in game-theoretic probability. This is not a serious problem in applications since only computable trading strategies can be of practical interest, and computable trading strategies will be measurable under any reasonable computational model. There are many interesting directions of further research, such as: • Is it possible to extend Theorems 1, 3, and 4 to the case where only the most recent past is known to the trader, as in [7, p. vii and Section 7.3] and [6, Section 5]?
• Is it possible to extend Theorems 1, 3, and 4 to the case of the trader without a synchronized watch (see [7,Section 7.7] or [1])?
Remember that is a well-order on Ω, which in this appendix can be any of the Ω considered in the main part of the paper (unless Ω is explicitly pointed out). The Ockham prediction system is: • given ω| [0,t) , the prediction ω t− for the rest of ω is defined as the -smallest • given ω| [0,t] , the prediction for the rest of ω is ω t , as defined in Section 5: • the revised prediction ω t+ at the time t ∈ [0, 1) is ω t ′ for any t ′ ∈ (t, 1] such that the function s → ω s is constant over (t, t ′ ]. The existence of ω t+ was shown in part 3 of Lemma 2 (and was already used in the proof of Theorem 3 in Subsection 5.3). By definition, ω 1+ is undefined, but notice that ω t and ω t− are defined for all t ∈ [0, 1] (in particular, ω 0− is the -smallest element of Ω).
We have the following three dichotomies for time points t ∈ [0, 1] for the purpose of short-term prediction of a given ω ∈ Ω: • t is future-successful if ω t = ω t+ ; in particular, 1 is not future-successful.
This gives us a partition of all t ∈ [0, 1] into 2 3 = 8 classes. We say that t is (−, 0, +)-successful (for the short-term prediction of ω using the Ockham prediction system) if t is simultaneously past-successful, present-successful, and future-successful; this is the highest degree of success. More generally, we will include − (respectively, 0, +) in the designation of the class of t if and only if t is past-(respectively, present-, future-) successful. The t that are ()-successful are not successful at all: they are not past-successful, not present-successful, and not future-successful. We will use notation such as C (−,0,+) ω and C () ω for denoting the set of t of the class indicated as the superscript. For example, C (−,0) ω is the class of t ∈ [0, 1] that are past-and present-successful but not future-successful for ω.
The definition (22) can be expressed in our new terminology by saying that W ω are the t ∈ [0, 1] that are not future-successful (which agrees with 1 ∈ W ω ); in our new notation, Let us modify the definition (22) by setting in particular, {0, 1} ⊆ F ω . This is the set of times t ∈ [0, 1] when the Ockham prediction system fails in the weakest possible sense that can be expressed via our three dichotomies; namely, F ω is the set [0, 1] \ C (−,0,+) of t ∈ [0, 1] that are not (−, 0, +)-successful. If t ∈ [0, 1]\F ω , the Ockham prediction system correctly predicts ω| [t1,t2] already at time t 1 , where (t 1 , t 2 ) ∋ t is a neighbourhood of t.
It is always true that W ω ⊆ F ω , and even the set F ω is still small: The set F ω is well-ordered by ≤. (Therefore, each of its points is isolated on the right, which implies that F ω is countable and nowhere dense.) Proof. We can modify the argument in the proof of Lemma 2, part 1. Suppose there is an infinite strictly decreasing chain t 1 > t 2 > · · · of elements of F ω .
Part 1 of Lemma 2 is a special case of Lemma 3, since any subset of a wellordered set is well-ordered. We can also see that each of the eight classes apart from C (−,0,+) ω is well-ordered. The following lemma shows that F ω splits [0, 1] into intervals of constancy of t → ω t .
Lemma 4. For any t ∈ F ω \{1}, the function s → ω s is constant on the interval (t, t ′ ), where t ′ is the successor of t in F ω .
Proof. Take any t ∈ F ω \ {1}, its successor t ′ ∈ F ω , and any t ′′ ∈ (t, t ′ ). Set It suffices to show that t 1 = t and t 2 = t ′ . Suppose, e.g., that t 1 > t. This implies t 1 / ∈ F ω . By the definition of F ω , there is a neighbourhood of t 1 in which s → ω s is constant and, therefore, ω s = ω t ′′ ; this, however, contradicts the definition (34) of t 1 .
Let us check that the analogue of Lemma 4 still holds for W ω in place of F ω if Ω = C[0, 1] but fails in general.

Lemma 5.
If Ω = C[0, 1], for any t ∈ W ω \{1}, the function s → ω s is constant on the interval (t, t ′ ), where t ′ is the successor of t in W ω . Otherwise, there are a well-order on Ω, ω ∈ Ω, and t ∈ W ω ∩ (0, 1) such that the successor t ′ of t in W ω is in (0, 1) and the function s → ω s is not constant over the interval (t, t ′ ),

Proof.
Suppose Ω = C[0, 1], t ∈ W ω \ {1}, and t ′ is the successor of t in W ω . Let t * ≤ t ′ be the successor of t in F ω . By Lemma 4, ω t+ (s) = ω(s) for each s ∈ (t, t * ), and so our continuity assumption implies that ω t+ (s) = ω(s) for each s ∈ (t, t * ]. This shows that t ′ = t * , and so s → ω s is constant on (t, t ′ ) = (t, t * ). Now suppose Ω = C[0, 1]. In the remaining proofs we will freely use the fact that the sum of two well-orders is again a well-order [10, Lemma 3.5(2)], where the sum of orders 1 and 2 on two disjoint sets X 1 and X 2 , respectively, is defined by for any x, x ′ ∈ X 1 ∪ X 2 [10, Definition 1.29]. This implies that any well-order on part of Ω can be extended to a well-order on the whole of Ω. Consider a well-order on Ω that starts from the càdlàg functions defined by if t ∈ [1/2, 3/4] t + 1/4 otherwise (in this order). For ω := ω 4 , we have W ω = {1/4, 3/4}, 3/4 is the successor of 1/4 in W ω , and the function s → ω s is not constant over the interval (1/4, 3/4) (it changes its value at s = 1/2 from ω 2 to ω 3 ).
The following result (easy but tiresome) shows that each of the eight classes may be non-empty, apart from the case Ω = C[0, 1], where there are six potentially non-empty classes (two of them being subsets of {0}). In particular, it implies that W ω = F ω is possible and, moreover, F ω \ W ω may contain any c ∈ (0, 1) (which is also clear from the proof of Lemma 5). It remains to consider, for this Ω, the classes containing 0 in the superscript. Let the well-order on Ω start from i.e., ω 1 ≺ ω 2 ≺ · · · . We have C  Lemma 6 shows that the number of different unions (such as (33)) that can be formed from the classes C (··· ) ω is very large (namely, 2 8 = 256), and many of these are potentially interesting. For each of the unions we can ask what sets in [0, 1] can be represented as such a union for different ω. We will answer this question only for the union (33), which plays the most important role in this paper.
Proof. It suffices to consider Ω = C[0, 1], which will imply the analogous statement for any other Ω considered in this paper. Let α be the ordinal that is isomorphic to W [8, Theorem 2.12], and let β ∈ α → w β ∈ W be the unique isomorphism [8, Corollary 2.6] between α and W . Let be a well-order that starts from (ω β ) β∈α , in the usual order of β ∈ α, such that ω β (t) := t if t < w β w β otherwise.
An analogue of Lemma 7 (however, with ⊇ in place of =) is contained in Theorem 3.5 of [6].