Natural Density and Probability, Constructively

REPORTS ON MATHEMATICAL LOGIC 55 (2020), 41–59 doi:10.4467/20842589RM.20.002.12434

Samuele MASCHIO

NATURAL DENSITY AND PROBABILITY, CONSTRUCTIVELY

A b s t r a c t. We give a constructive account of the frequentist approach to probability, by means of natural density. Then we discuss some probabilistic variants of the Limited Principle of Omniscience.

.1 Introduction

The notion of probability had different interpretations in its historical development. The naïf classical interpretation, according to which the probability of an event is given by the ratio of the number of favorable cases to the number of all possible cases, was extended at least in two directions: the first one is the frequentist approach, which consists in extending the classical interpretation from a finite number of possible cases to a sequence of outcomes (“success” or “failure”) in a sequence of iterated trials; the second one is

Received 19 February 2019; resubmited 26 June 2019 Keywords and phrases: constructive probability, frequentist probability, natural density, limited principle of omniscience. AMS Subject Classiﬁcation: 03F65, 60A05. 42 SAMUELE MASCHIO NATURAL DENSITY AND PROBABILITY, CONSTRUCTIVELY 43

Kolmogorov’s axiomatic approach, which consists in defining a probability restricting it to these classes and/or by weakening the statement n (x = 0) ∀ n measure as a σ-additive function defined on a σ-algebra of events. by transforming it into P(x)=0and/or by strengthening the statement Nowadays, in classical mathematics, Kolmogorov’s axiomatic probability n (xn = 1) by transforming it into P(x) > 0 . We will study these variants ∃ is the standard notion of probability. The frequentist approach is instead in Section 3 and we will prove there that each one of them is either true, connected with the notion of natural density ([7]) which provides a notion false or equivalent to LPO itself, constructively. of size for subsets of natural numbers. Classically, the density of A N is Our treatment will be informal à la Bishop (see [1]), but it could be ⊆ defined as formalized in the extensional level of the Minimalist Foundation (see [6], [5]) A 0, ..., n 1 δ(A) = lim | ∩{ − }| with the addition of the axiom of unique choice and the axiom of countable n →∞ n choice. provided that this limit exists. Since classically the subsets of N are in bijective correspondence with binary sequences (like “success”/“failure” sequences), this notion of density makes sense for such sequences, too. .1.1 Some prerequisites Here we are interested in probability in an alternative mathematical framework, that is that of constructive mathematics in Bishop’s sense ([1]). Let us recall here the definition of Bishop real numbers from [1]. Bishop’s mathematics can be roughly defined as mathematics based on intu- + Definition 1.1. A Bishop real x is a sequence x(n) Q [n N ] of ∈ ∈ itionistic logic. In his development of analysis, Bishop used only direct and rational numbers such that constructive methods; for this reason its constructive analysis admits a com- putational interpretation. This means in particular that no principle like 1 1 x(n) x(m) + Law of Excluded Middle or Proof by Contraddiction can be used in construc- | − |≤n m tive mathematics. Moreover in his Constructive Manifesto ([1]), E.Bishop for every n N+ and m N+. was clear about constructivism: “The task of making analysis constructive ∈ ∈ Two Bishop reals x and y are equal if x(n) y(n) 2 for every n N+. is guided by three basic principles. First, to make every concept affirmative | − |≤ n ∈ The set of Bishop reals is denoted with R and the equality between Bishop (...) Second, to avoid definitions that are not relevant (...) Third, to avoid reals (which is an equivalence relation) is denoted with = . pseudogenerality (...)”. R In the context of constructive mathematics, the main reference for a treat- We also recall how some basic operations between Bishop reals and their ment of probability is Chan’s Notes on Constructive Probability ([3]). There, ordering are defined in [1]: a constructive axiomatic definition of probability space (in the spirit of Kol- x y morogov’s) is given. However this structure is not based on the notion of Definition 1.2. If and are Bishop reals, then event, but on the notion of random number following the style of Daniell 1. x + y is the Bishop real defined by integral. + Here we will explore the frequentist interpretation of probability from (x + y)(n) := x(2n)+y(2n) n N ; ∈ a constructive point of view. We will exploit the sequential nature of events in the frequentist approach to define their probability in a very direct way; 2. x y is the Bishop real defined by − then we will prove some properties of such a notion of probability in Section 2. + (x y)(n) := x(2n) y(2n) n N ; Among the principles which are refused in Bishop’s mathematics there − − ∈ is the so-called Limited Principle of Omniscience (LPO) for which every 3. x is non-positive if x(n) 1 for every n N+; binary sequence is constantly 0 or has a term equal to 1. Since our approach ≤ n ∈ to natural density will naturally lead to the definition of some particular 4. x R y if and only if x y is non-positive; classes of binary sequences, it is a natural question to ask what is the relation ≤ − between the several “probabilistic” variants of LPO which are obtained by 5. x is positive if there exists n N+ such that x(n) > 1 ; ∈ n 42 SAMUELE MASCHIO NATURAL DENSITY AND PROBABILITY, CONSTRUCTIVELY 43

Kolmogorov’s axiomatic approach, which consists in defining a probability restricting it to these classes and/or by weakening the statement n (x = 0) ∀ n measure as a σ-additive function defined on a σ-algebra of events. by transforming it into P(x)=0and/or by strengthening the statement Nowadays, in classical mathematics, Kolmogorov’s axiomatic probability n (xn = 1) by transforming it into P(x) > 0 . We will study these variants ∃ is the standard notion of probability. The frequentist approach is instead in Section 3 and we will prove there that each one of them is either true, connected with the notion of natural density ([7]) which provides a notion false or equivalent to LPO itself, constructively. of size for subsets of natural numbers. Classically, the density of A N is Our treatment will be informal à la Bishop (see [1]), but it could be ⊆ defined as formalized in the extensional level of the Minimalist Foundation (see [6], [5]) A 0, ..., n 1 δ(A) = lim | ∩{ − }| with the addition of the axiom of unique choice and the axiom of countable n →∞ n choice. provided that this limit exists. Since classically the subsets of N are in bijective correspondence with binary sequences (like “success”/“failure” sequences), this notion of density makes sense for such sequences, too. .1.1 Some prerequisites Here we are interested in probability in an alternative mathematical framework, that is that of constructive mathematics in Bishop’s sense ([1]). Let us recall here the definition of Bishop real numbers from [1]. Bishop’s mathematics can be roughly defined as mathematics based on intu- + Definition 1.1. A Bishop real x is a sequence x(n) Q [n N ] of ∈ ∈ itionistic logic. In his development of analysis, Bishop used only direct and rational numbers such that constructive methods; for this reason its constructive analysis admits a com- putational interpretation. This means in particular that no principle like 1 1 x(n) x(m) + Law of Excluded Middle or Proof by Contraddiction can be used in construc- | − |≤n m tive mathematics. Moreover in his Constructive Manifesto ([1]), E.Bishop for every n N+ and m N+. was clear about constructivism: “The task of making analysis constructive ∈ ∈ Two Bishop reals x and y are equal if x(n) y(n) 2 for every n N+. is guided by three basic principles. First, to make every concept affirmative | − |≤ n ∈ The set of Bishop reals is denoted with R and the equality between Bishop (...) Second, to avoid definitions that are not relevant (...) Third, to avoid reals (which is an equivalence relation) is denoted with = . pseudogenerality (...)”. R In the context of constructive mathematics, the main reference for a treat- We also recall how some basic operations between Bishop reals and their ment of probability is Chan’s Notes on Constructive Probability ([3]). There, ordering are defined in [1]: a constructive axiomatic definition of probability space (in the spirit of Kol- x y morogov’s) is given. However this structure is not based on the notion of Definition 1.2. If and are Bishop reals, then event, but on the notion of random number following the style of Daniell 1. x + y is the Bishop real defined by integral. + Here we will explore the frequentist interpretation of probability from (x + y)(n) := x(2n)+y(2n) n N ; ∈ a constructive point of view. We will exploit the sequential nature of events in the frequentist approach to define their probability in a very direct way; 2. x y is the Bishop real defined by − then we will prove some properties of such a notion of probability in Section 2. + (x y)(n) := x(2n) y(2n) n N ; Among the principles which are refused in Bishop’s mathematics there − − ∈ is the so-called Limited Principle of Omniscience (LPO) for which every 3. x is non-positive if x(n) 1 for every n N+; binary sequence is constantly 0 or has a term equal to 1. Since our approach ≤ n ∈ to natural density will naturally lead to the definition of some particular 4. x R y if and only if x y is non-positive; classes of binary sequences, it is a natural question to ask what is the relation ≤ − between the several “probabilistic” variants of LPO which are obtained by 5. x is positive if there exists n N+ such that x(n) > 1 ; ∈ n 44 SAMUELE MASCHIO NATURAL DENSITY AND PROBABILITY, CONSTRUCTIVELY 45

6. x< y if and only if y x is positive. .2.2 Actual events R − Moreover every rational number q is identified, as a Bishop real, with the Actual events are those potential events for which the sequence of rates of constant sequence x such that x(n)=q for every n N+. success can be shown constructively to converge to a Bishop real number. ∈ Definition 2.1. An actual event is a pair (e,γ) where e is a potential event and γ is a strictly increasing sequence of natural numbers such that .2 A constructive account of natural density 1 Φ(e)(γ(n)+i) Φ(e)(γ(n)+j) In this section we will introduce a constructive notion of frequentist prob- | − |≤n ability. We will draw a picture in which a structure of actual events (on + for every n N , i N and j N. ∈ ∈ ∈ which a notion of probability is defined) lies in between two nested Boolean We denote with the set of actual events and two actual events (e,γ) and algebras: that of potential events and that of regular events. A (e ,γ ) are equal, (e,γ)= (e ,γ ), if e = e . A P We now show that one can define a notion of probability for actual events. .2.1 Potential events We first show that there exists a subsequence of the sequence of frequencies of an actual event which is a Bishop real. A potential event e is a binary sequence Proposition 2.2. If (e,γ) is an actual event, then Φ(e) γ is a Bishop + ◦ e(n) 0, 1 [ n N ] real. ∈{ } ∈ We interpret potential events as sequences of outcomes (“success”=1or Proof. Suppose m n are in N+. Then γ(m) γ(n) and hence ≤ ≤ “failure”=0) in a sequence of iterated trials. 1 1 1 Potential events form a set with extensional equality, that is two poten- Φ(e)(γ(n)) Φ(e)(γ(m)) < + . | − |≤m n m tial events e and e are equal (we write e = e ) if e(n)=e (n) for every P n N+. This set, which we denote with , can be endowed with a structure This means that Φ(e) γ is a Bishop real. ∈ P ◦ of Boolean algebra as follows: Now we show that equal actual events give rise to equal Bishop reals. 1. the bottom is λn.0; ⊥ Proposition 2.3. If (e,γ) and (e ,γ ) are equal actual events, then 2. the top is λn.1; Φ(e) γ and Φ(e ) γ are equal Bishop reals. ◦ ◦ 3. the conjunction e e of e and e in is λn.(e(n)e (n)); Proof. Suppose (e,γ) and (e ,γ ) are equal actual events. Since, for ∧ P every positive natural number n, γ(n) γ (n) or γ (n) γ(n), and e = e , ≤ ≤ P 4. the disjunction of e e of e and e in is λn.(e(n)+e (n) e(n)e (n)); then for every n N+ ∨ P − ∈ 5. the negation e of e is λn.(1 e(n)); 1 2 ¬ ∈P − (Φ(e) γ)(n) (Φ(e ) γ )(n) = Φ(e)(γ(n)) Φ(e)(γ (n)) < . + ◦ − ◦ − ≤ n n 6. for every e, e , e e if and only if e(n) e (n) for every n N . ∈P ≤ ≤ ∈ This means that Φ(e) γ and Φ(e ) γ are equal Bishop reals. For every potential event e, one can define a sequence of rational numbers ◦ ◦ Thus we can give the following + Φ(e)(n) Q [ n N ] ∈ ∈ Definition 2.4. The function P : R is defined as follows: if (e,γ) n e(i) A→ by taking Φ(e)(n) to be i=1 . This sequence is called the sequence of is an actual event, then P(e,γ) := Φ(e) γ. The Bishop real number P(e,γ) n ◦ rates of success (or frequencies ) of the potential event e. is called the probability of the actual event (e,γ). 44 SAMUELE MASCHIO NATURAL DENSITY AND PROBABILITY, CONSTRUCTIVELY 45

6. x< y if and only if y x is positive. .2.2 Actual events R − Moreover every rational number q is identified, as a Bishop real, with the Actual events are those potential events for which the sequence of rates of constant sequence x such that x(n)=q for every n N+. success can be shown constructively to converge to a Bishop real number. ∈ Definition 2.1. An actual event is a pair (e,γ) where e is a potential event and γ is a strictly increasing sequence of natural numbers such that .2 A constructive account of natural density 1 Φ(e)(γ(n)+i) Φ(e)(γ(n)+j) In this section we will introduce a constructive notion of frequentist prob- | − |≤n ability. We will draw a picture in which a structure of actual events (on + for every n N , i N and j N. ∈ ∈ ∈ which a notion of probability is defined) lies in between two nested Boolean We denote with the set of actual events and two actual events (e,γ) and algebras: that of potential events and that of regular events. A (e ,γ ) are equal, (e,γ)= (e ,γ ), if e = e . A P We now show that one can define a notion of probability for actual events. .2.1 Potential events We first show that there exists a subsequence of the sequence of frequencies of an actual event which is a Bishop real. A potential event e is a binary sequence Proposition 2.2. If (e,γ) is an actual event, then Φ(e) γ is a Bishop + ◦ e(n) 0, 1 [ n N ] real. ∈{ } ∈ We interpret potential events as sequences of outcomes (“success”=1or Proof. Suppose m n are in N+. Then γ(m) γ(n) and hence ≤ ≤ “failure”=0) in a sequence of iterated trials. 1 1 1 Potential events form a set with extensional equality, that is two poten- Φ(e)(γ(n)) Φ(e)(γ(m)) < + . | − |≤m n m tial events e and e are equal (we write e = e ) if e(n)=e (n) for every P n N+. This set, which we denote with , can be endowed with a structure This means that Φ(e) γ is a Bishop real. ∈ P ◦ of Boolean algebra as follows: Now we show that equal actual events give rise to equal Bishop reals. 1. the bottom is λn.0; ⊥ Proposition 2.3. If (e,γ) and (e ,γ ) are equal actual events, then 2. the top is λn.1; Φ(e) γ and Φ(e ) γ are equal Bishop reals. ◦ ◦ 3. the conjunction e e of e and e in is λn.(e(n)e (n)); Proof. Suppose (e,γ) and (e ,γ ) are equal actual events. Since, for ∧ P every positive natural number n, γ(n) γ (n) or γ (n) γ(n), and e = e , ≤ ≤ P 4. the disjunction of e e of e and e in is λn.(e(n)+e (n) e(n)e (n)); then for every n N+ ∨ P − ∈ 5. the negation e of e is λn.(1 e(n)); 1 2 ¬ ∈P − (Φ(e) γ)(n) (Φ(e ) γ )(n) = Φ(e)(γ(n)) Φ(e)(γ (n)) < . + ◦ − ◦ − ≤ n n 6. for every e, e , e e if and only if e(n) e (n) for every n N . ∈P ≤ ≤ ∈ This means that Φ(e) γ and Φ(e ) γ are equal Bishop reals. For every potential event e, one can define a sequence of rational numbers ◦ ◦ Thus we can give the following + Φ(e)(n) Q [ n N ] ∈ ∈ Definition 2.4. The function P : R is defined as follows: if (e,γ) n e(i) A→ by taking Φ(e)(n) to be i=1 . This sequence is called the sequence of is an actual event, then P(e,γ) := Φ(e) γ. The Bishop real number P(e,γ) n ◦ rates of success (or frequencies ) of the potential event e. is called the probability of the actual event (e,γ). 46 SAMUELE MASCHIO NATURAL DENSITY AND PROBABILITY, CONSTRUCTIVELY 47

.2.3 Properties of actual events Hence ( e,γ) is an actual event. Moreover for every n N+ ¬ ∈ We show here some basic properties of actual events and of the probabil- P( e,γ)(n) (1 P(e,γ))(n) = 1 P(e,γ)(n) (1 P(e,γ)(2n)) ity function P deﬁned on them. However, before proceeding, we need the | ¬ − − | | − − − | following 1 1 2 Lemma 2.5. If (e,γ) is an actual event and γ is a strictly increasing se- = P(e,γ)(2n) P(e,γ)(n) + < + | − |≤n 2n n quence of positive natural numbers such that γ(n) γ (n) for every n N , ≤ ∈ then (e,γ ) is an actual event. so P( e,γ)= 1 P(e,γ). ¬ R − Proof. Let n N+ and i, j N. Then Moreover actual events are closed under disjunction of incompatible events: ∈ ∈ Φ(e)(γ (n)+i) Φ(e)(γ (n)+j) − Proposition 2.9. If (e,γ) and (e ,γ ) are actual events with e e = ∧ P ⊥ 1 and η := λn.(γ(2n)+γ (2n)), then the pair (e e ,η) is an actual event. = Φ(e)(γ(n)+(γ (n ) γ(n)+i)) Φ(e)(γ(n)+(γ (n ) γ(n)+j)) ∨ − − − ≤ n e e = e e = e + e η since γ (n) γ(n) 0 and (e,γ) is an actual event. Proof. Since , we have that . Moreover is ∧ P ⊥ ∨ P + − ≥ stricly increasing, as γ and γ are so, and, for every n N , we have that First, we can prove that the probability of an actual event is always ∈ a value between 0 and 1: Φ(e + e )(n) = Φ(e)(n)+Φ(e )(n).

Proposition 2.6. If (e,γ) is an actual event, then 0 R P(e,γ) R 1. ≤ ≤ + + In particular, for every n N , i, j N, using subadditivity of the absolute Proof. Let (e,γ) be an actual event. For every m N , 0 Φ(e)(m) 1. ∈ ∈ 1 ∈ ≤ 1 ≤ value, we have that Thus, in particular, Φ(e)(γ(2n)) n and Φ(e)(γ(2n)) 1 n for every + − ≤ − ≤ n N . Hence 0 R P(e,γ) R 1. ∈ ≤ ≤ Φ(e + e )(η(n)+i) Φ(e + e )(η(n)+j) Moreover, P satisﬁes the property which is usually called strictness: − Φ(e)(γ(2n)+γ (2n)+i) Φ(e)(γ(2 n)+γ (2n)+j) ≤ − Proposition 2.7. ( , λn.n) is an actual event and P( , λn.n)= 0. + Φ(e )(γ (2n)+γ(2n)+i) Φ(e )(γ (2n)+γ(2n)+j) ⊥ ⊥ R − 1 1 1 Proof. This follows from the fact that Φ( )(n)=0for every n N+. + = . ⊥ ∈ ≤ 2n 2n n Then we show that actual events and P satisfy involution: Hence (e e ,η) is an actual event. ∨ Proposition 2.8. If (e,γ) is an actual event, then ( e,γ) is an actual ¬ We now show that the set of actual events with null probability is down- event and ward closed: P( e,γ)= 1 P(e,γ). ¬ R − Proof. Let (e,γ) be an actual event. Since Φ( e)(n)=1 Φ(e)(n) for Proposition 2.10. Let (e,γ) be an actual event with P(e,γ)=R 0 and + + ¬ − every n , it follows that for every n and m, m let e be a potential event such that e e, then (e , λn.γ(6n)) is an actual N N N ≤ ∈ ∈ ∈ event. Φ( e)(γ(n)+m) Φ( e)(γ(n)+m ) ¬ − ¬ 1 Proof. First notice that if e e, then Φ(e )(n) Φ(e)(n) for every + ≤ ≤ = Φ(e)(γ(n)+m ) Φ(e)(γ(n)+m) . n N . − ≤ n ∈ 46 SAMUELE MASCHIO NATURAL DENSITY AND PROBABILITY, CONSTRUCTIVELY 47

Proposition 2.6. If (e,γ) is an actual event, then 0 R P(e,γ) R 1. ≤ ≤ + + In particular, for every n N , i, j N, using subadditivity of the absolute Proof. Let (e,γ) be an actual event. For every m N , 0 Φ(e)(m) 1. ∈ ∈ 1 ∈ ≤ 1 ≤ value, we have that Thus, in particular, Φ(e)(γ(2n)) n and Φ(e)(γ(2n)) 1 n for every + − ≤ − ≤ n N . Hence 0 R P(e,γ) R 1. ∈ ≤ ≤ Φ(e + e )(η(n)+i) Φ(e + e )(η(n)+j) Moreover, P satisﬁes the property which is usually called strictness: − Φ(e)(γ(2n)+γ (2n)+i) Φ(e)(γ(2 n)+γ (2n)+j) ≤ − Proposition 2.7. ( , λn.n) is an actual event and P( , λn.n)= 0. + Φ(e )(γ (2n)+γ(2n)+i) Φ(e )(γ (2n)+γ(2n)+j) ⊥ ⊥ R − 1 1 1 Proof. This follows from the fact that Φ( )(n)=0for every n N+. + = . ⊥ ∈ ≤ 2n 2n n Then we show that actual events and P satisfy involution: Hence (e e ,η) is an actual event. ∨ Proposition 2.8. If (e,γ) is an actual event, then ( e,γ) is an actual ¬ We now show that the set of actual events with null probability is down- event and ward closed: P( e,γ)= 1 P(e,γ). ¬ R − Proof. Let (e,γ) be an actual event. Since Φ( e)(n)=1 Φ(e)(n) for Proposition 2.10. Let (e,γ) be an actual event with P(e,γ)=R 0 and + + ¬ − every n , it follows that for every n and m, m let e be a potential event such that e e, then (e , λn.γ(6n)) is an actual N N N ≤ ∈ ∈ ∈ event. Φ( e)(γ(n)+m) Φ( e)(γ(n)+m ) ¬ − ¬ 1 Proof. First notice that if e e, then Φ(e )(n) Φ(e)(n) for every + ≤ ≤ = Φ(e)(γ(n)+m ) Φ(e)(γ(n)+m) . n N . − ≤ n ∈ 48 SAMUELE MASCHIO NATURAL DENSITY AND PROBABILITY, CONSTRUCTIVELY 49

+ Thus, for every n N and i, j N, using the subadditivity of the 2. P(e,γ1)+P(e ,γ2)= P(e,ε)+P(e ,ε) ∈ ∈ R absolute value function and an add-subtract trick, we have In order to conclude, it is sufficient to prove that Φ(e )(γ(6n)+i) Φ(e )(γ(6n)+j) − P(e,ε)+P(e ,ε)=R P(e e ,ε)+P(e e ,ε). Φ(e )(γ(6n)+i) + Φ(e )(γ(6 n)+j) ∨ ∧ ≤ Φ(e)(γ(6n)+i) + Φ( e)(γ(6n)+j) This follows from the fact that e(n)+e (n)=(e e )(n)+(e e )(n) for ≤| | | | + ∨ ∧ Φ(e)(γ(6n)+i) Φ(e)(γ(6n)) + Φ(e)(γ(6n)+j) Φ(e)(γ(6n)) every n N from which it follows that Φ(e)(n) + Φ(e )(n)=Φ(e e )(n)+ ≤| − | | − | ∈ + ∧ +2 Φ(e)(γ(6n)) Φ(e e )(n) for every n N . | | ∨ ∈ 1 1 2 1 + +2 = . ≤ 6n 6n 6n n The last inequality follows from the fact that (e,γ) is an actual event and .2.4 Regular events P(e,γ)=R 0. Hence (e , λn.γ(6n)) is an actual event. Among potential events, there are some events which can be considered sort Moreover P satisfies monotonicity: of “deterministic”, since their sequences of outcomes of trials have a periodic behaviour, up to a possible finite number of accidental errors in the recording Proposition 2.11. If (e,γ) and (e ,γ ) are actual events and e e , ≤ of the results. These events are here called regular. then P(e,γ) P(e ,γ ). Definition 2.13. Let π be a finite non-empty list of elements of 0, 1 ≤R { } with lenght (π). We define the potential event π∞ as the periodic sequence Proof. First of all if we take η to be the sequence λn.(γ(n)+γ (n)), defined by (e,η) (e ,η) + then, by Lemma 2.5, and are actual events (equal by definition π∞(n) := πmod(n 1,(π))+1 n N − ∈ to (e,γ) and (e ,γ ), respectively). In order to show that P(e,η) R P(e ,η), 1 ≤ + where we denote with πi the i-th component of the listπ. we must prove that (P(e,η) P(e ,η))(n) for every n N . But − ≤ n ∈ ( (e,η) (e ,η))(n) P P is equal to Definition 2.14. Let α and π be two finite lists of elements of 0, 1 − { } with π non-empty, and let (α) and (π) be their lenght, respectively. The P(e,η)(2n) P(e ,η)(2n)=Φ(e)(η(2n)) Φ(e )(η(2n)). − − potential event α.π∞ is a sequence having α as prefix and then having period 1 + π Hence we must prove that Φ(e)(η(2n)) Φ(e )(η(2n))+ for every n N . , that is the ultimately periodic sequence defined as follows: ≤ n ∈ But this is true, because from e e we deduce that Φ(e)(n) Φ(e )(n) for ≤ ≤ (α.π )(n) := α [n +,n (α)] every positive natural number n. ∞ n N ∈ ≤ + (α.π∞)(n) := π∞(n (α)) [n N ,n>(α)] Finally we prove that P satisfies a form of modularity: − ∈ where we denote with αi the i-th component of the list α. Proposition 2.12. If (e,γ1), (e ,γ2), (e e ,γ3), (e e ,γ4) are actual ∧ ∨ The potential events of this form are called regular. events, then

Notice that every periodic binary sequence π∞ is regular, since π∞ = [].π∞. P(e e ,γ4)+P(e e ,γ3)= P(e,γ1)+P(e ,γ2). ∨ ∧ R The goal of the following propositions is to show that regular events are Proof. Let ε := λn.(γ1(n)+γ2(n)+γ3(n)+γ4(n)). Since ε(n) γi(n) + ≥ actual. First we show that regular events without errors (that is periodic for every n N for i =1, 2, 3, 4, we can use Lemma 2.5 and obtain that: ∈ sequences) are actual events and their probability is equal to the frequency 1. P(e e ,γ4)+P(e e ,γ3)= P(e e ,ε)+P(e e ,ε) of their period. ∨ ∧ R ∨ ∧ 48 SAMUELE MASCHIO NATURAL DENSITY AND PROBABILITY, CONSTRUCTIVELY 49

Notice that every periodic binary sequence π∞ is regular, since π∞ = [].π∞. P(e e ,γ4)+P(e e ,γ3)= P(e,γ1)+P(e ,γ2). ∨ ∧ R The goal of the following propositions is to show that regular events are Proof. Let ε := λn.(γ1(n)+γ2(n)+γ3(n)+γ4(n)). Since ε(n) γi(n) + ≥ actual. First we show that regular events without errors (that is periodic for every n N for i =1, 2, 3, 4, we can use Lemma 2.5 and obtain that: ∈ sequences) are actual events and their probability is equal to the frequency 1. P(e e ,γ4)+P(e e ,γ3)= P(e e ,ε)+P(e e ,ε) of their period. ∨ ∧ R ∨ ∧ 50 SAMUELE MASCHIO NATURAL DENSITY AND PROBABILITY, CONSTRUCTIVELY 51

+ Proposition 2.15. For every ﬁnite non-empty list π =[π1, ..., πm], Proof. Suppose n N and i, j N with i j. Then, using the ∈ ∈ ≥ (π∞, λn.4nm) is an actual event and subadditivity of the absolute value function, m k=1 πk Φ(α.[0] )(2nm + i) Φ(α.[0] )(2nm + j) P(π∞, λn.4nm)= . ∞ ∞ R m | − | Φ(α.[0]∞)(2nm + i) + Φ(α.[0]∞)(2nm + j) + ≤| | | | Proof. For every n N and every i, j N, using the subadditivity of ∈ ∈ α1 + ... + αm α1 + ... + αm 2m 1 the absolute value function, we obtain + = . ≤ 2nm + i 2nm + j ≤ 2nm n

Φ(π∞)(4nm + i) Φ(π∞)(4nm + j) + | − | Hence (α.[0]∞, λn.2nm) is an actual event. Moreover for every n N m i mod(i,m) ∈ π (4n + )m πk = k=1 k m + k=1 m 4nm + i 4nm + i α1 + ... + αm m 1 2 P(α.[0]∞, λn.2nm)(n)= = < mod(j,m) 2nm ≤ 2nm 2n n m π (4n + j )m π k=1 k m k=1 k Thus (α.[0] , λn.2nm)= 0. − m 4nm + j − 4nm + j P ∞ R m i j Finally we prove that actual events are closed under shift to the right of k=1 πk (4n + m )m (4n + m )m terms and that probability is preserved by these shifts. ≤ m 4nm + i − 4nm + j mod(i,m) mod(j,m) Definition 2.17. If e is a potential event, then e+ is the potential event k=1 πk k=1 πk + + defined by 4nm + i 4nm + j e+(1) := 0, i j m (4n + )m (4n + )m 2 π + + m m k=1 k e (n + 1) := e(n)[n N ]. + ∈ ≤ 4nm + i − 4nm + j 4nm + Proposition 2.18. If (e,γ) is an actual event, then (e , λn.(γ(3n)+1)) 4n 1 1 1 1 + 1 + = + < is an actual event and P(e , λn.(γ(3n)+1))=R P(e,γ). ≤ − 4n +1 2n 4n +1 2n n + Proof. Using an add-subtract trick and the subadditivity of the absolute since for every n, m N and for every i N ∈ ∈ value function, we obtain that for n N+ and i, j N, the value i ∈ ∈ 4n (4n + m )m 1. Φ(e+)(γ(3n)+1+i) Φ(e+)(γ(3n)+1+j) 4n +1 ≤ 4nm + i ≤ − + Hence (π∞, λn.4nm) is an actual event. Moreover for every i N ∈ is less or equal than the sum of m k=1 πk + P(π∞, λn.4nm)(i)= . 1. Φ(e )(γ(3n)+1+i) Φ(e)(γ(3n)+i) , m | − | m 2. Φ(e)(γ(3n)+i) Φ(e)(γ(3n)+j) and k=1 πk | − | Hence P(π∞, λn.4nm)= . R m + 3. Φ(e)(γ(3n)+j) Φ(e )(γ(3n)+1+j) . The next step consists in proving that regular events definitely equal to | − | 0 0 Since for every m N are actual events and their probability is . ∈ Proposition 2.16. If α is a finite list of 0s and 1s with length m>0, m+1 e+(k) m e(k) m Φ(e)(m) Φ(e+)(m + 1) = k=1 = k=1 = · then (α.[0]∞, λn.2nm) is an actual event and P(α.[0]∞, λn.2nm)=R 0. m +1 m +1 m +1 50 SAMUELE MASCHIO NATURAL DENSITY AND PROBABILITY, CONSTRUCTIVELY 51

Φ(π∞)(4nm + i) Φ(π∞)(4nm + j) + | − | Hence (α.[0]∞, λn.2nm) is an actual event. Moreover for every n N m i mod(i,m) ∈ π (4n + )m πk = k=1 k m + k=1 m 4nm + i 4nm + i α1 + ... + αm m 1 2 P(α.[0]∞, λn.2nm)(n)= = < mod(j,m) 2nm ≤ 2nm 2n n m π (4n + j )m π k=1 k m k=1 k Thus (α.[0] , λn.2nm)= 0. − m 4nm + j − 4nm + j P ∞ R m i j Finally we prove that actual events are closed under shift to the right of k=1 πk (4n + m )m (4n + m )m terms and that probability is preserved by these shifts. ≤ m 4nm + i − 4nm + j mod(i,m) mod(j,m) Definition 2.17. If e is a potential event, then e+ is the potential event k=1 πk k=1 πk + + defined by 4nm + i 4nm + j e+(1) := 0, i j m (4n + )m (4n + )m 2 π + + m m k=1 k e (n + 1) := e(n)[n N ]. + ∈ ≤ 4nm + i − 4nm + j 4nm + Proposition 2.18. If (e,γ) is an actual event, then (e , λn.(γ(3n)+1)) 4n 1 1 1 1 + 1 + = + < is an actual event and P(e , λn.(γ(3n)+1))=R P(e,γ). ≤ − 4n +1 2n 4n +1 2n n + Proof. Using an add-subtract trick and the subadditivity of the absolute since for every n, m N and for every i N ∈ ∈ value function, we obtain that for n N+ and i, j N, the value i ∈ ∈ 4n (4n + m )m 1. Φ(e+)(γ(3n)+1+i) Φ(e+)(γ(3n)+1+j) 4n +1 ≤ 4nm + i ≤ − + Hence (π∞, λn.4nm) is an actual event. Moreover for every i N ∈ is less or equal than the sum of m k=1 πk + P(π∞, λn.4nm)(i)= . 1. Φ(e )(γ(3n)+1+i) Φ(e)(γ(3n)+i) , m | − | m 2. Φ(e)(γ(3n)+i) Φ(e)(γ(3n)+j) and k=1 πk | − | Hence P(π∞, λn.4nm)= . R m + 3. Φ(e)(γ(3n)+j) Φ(e )(γ(3n)+1+j) . The next step consists in proving that regular events definitely equal to | − | 0 0 Since for every m N are actual events and their probability is . ∈ Proposition 2.16. If α is a finite list of 0s and 1s with length m>0, m+1 e+(k) m e(k) m Φ(e)(m) Φ(e+)(m + 1) = k=1 = k=1 = · then (α.[0]∞, λn.2nm) is an actual event and P(α.[0]∞, λn.2nm)=R 0. m +1 m +1 m +1 52 SAMUELE MASCHIO NATURAL DENSITY AND PROBABILITY, CONSTRUCTIVELY 53

(α) the sum above is equal to Since α.[0]∞ 0 .π = , using Proposition 2.9, we obtain that there ∧ P ⊥ exists γ such that (α.π∞,γ) is an actual event. Φ(e)(γ(3n)+i) Φ(e)(γ(3n)+j) (α) Finally, P(α.π∞,γ)= P(α.[0]∞,γ1)+P(0 .π∞,γ2) P( , λn.n) by + Φ(e)(γ(3n)+i) Φ(e)(γ(3n)+j) + R − ⊥ γ(3n)+i +1 | − | γ(3n)+j +1 Propositions 2.12 and 2.7 and, by Propositions 2.16, 2.18, 2.15 and 2.7, this (π) k=1 πk which is less or equal to is equal to (π) . 1 1 1 1 1 We conclude this section with the following result: + + 3 = γ(3n)+i +1 3n γ(3n)+j +1 ≤ 3n n Theorem 2.20. Regular events form a boolean algebra with the opera- since Φ(e)(m) 1 for every m N+, γ(3n) 3n (because γ is strictly in- tions inherited by the algebra of potential events. ≤ ∈ ≥ creasing) and (e,γ) is an actual event. Hence (e+, λn.(γ(3n)+1)) is an actual Proof. First of all = [0]∞ and = [1]∞. Suppose now that α.π∞ and event. Moreover, using the subadditivity of the absolute value function ⊥ β.ψ∞ are regular events. Then (α.π ) := ( α).( π) where α and π ¬ ∞ ¬ ¬ ∞ ¬ ¬ + are obtained by changing each term x of the finite lists to 1 x. Without Φ(e )(γ(3n)+1) Φ(e)(γ(n)) − − loss of generality, we can suppose that (β) (α). We have that: γ(3n) ≥ = Φ(e)(γ(3n)) Φ( e)(γ(n)) γ(3n)+1− α.π∞ β.ψ∞ = γ.ρ∞ Φ(e)(γ(3 n)) ∧ = Φ(e)(γ(3n)) Φ(e)(γ(n)) − − γ(3n)+1 α.π∞ β.ψ∞ = γ.ρ∞ ∨ 1 Φ(e)(γ(3n)) Φ(e)(γ(n)) + Φ(e)(γ(3n)) where (γ)=(γ)=(β), (ρ)=(ρ)=(π)(ψ), and ≤| − | γ(3n)+1 1 1 2 + < . γi := αi βi if i (α) ≤ n 3n +1 n ∧ ≤ γi := πmod(i (α) 1, (π))+1 βi if (α) subset of we define the following princi- + + E P LPO :( e )(( n N )(e(n)=0) ( n N )(e(n) = 1)) ples: ∀ ∈P ∀ ∈ ∨ ∃ ∈ 1. LPO[ ] def ( e ) [( n N+)(e(n)=0) ( n N+)(e(n)=1)]. Having introduced a notion of probability P on (some) binary sequences, we E ≡ ∀ ∈E ∀ ∈ ∨ ∃ ∈ can consider some variants of this principle by restricting it to some subsets 2. P-LPO[ ] def ( e )[P[e]= 0 ( n N+)(e(n) = 1)]. of potential events or by modifying the disjunction in it. E ≡ ∀ ∈E R ∨ ∃ ∈ First, we need some abbreviations: 3. PP-LPO[ ] def ( e )[P[e]= 0 P[e] > 0]. E ≡ ∀ ∈E R ∨ R + 1. List (A) is the set of finite lists of elements from a set A with positive Notice that LPO[ ] is equivalent to LPO, and that PP-LPO[ ] implies P P length; = . P A Let us now study the relation between these probabilistic versions of 2. Incr(N+, N+) is the set of strictly increasing sequences of positive nat- LPO when is , , or . As a direct consequence of Lemma 3.1 we ural numbers; E P A N R have: 3. := e ( γ Incr(N+, N+))((e,γ) ) ; A { ∈ P| ∃ ∈ ∈ A } Lemma 3.3. If is a subset of , then E P 4. ϕ(P[e]) means e ( γ Incr(N+, N+))((e,γ) ϕ(P(e,γ))) ∈A∧ ∀ ∈ ∈ A→ 1. PP-LPO[ ] P-LPO[ ]. whenever ϕ(x) is a proposition depending on a real number x; E → E + 2. LPO[ ] P-LPO[ ]. 5. := e ( α List( 0, 1 ))( π List ( 0, 1 ))(e = α.π∞) ; E → E R { ∈ P| ∃ ∈ { } ∃ ∈ { } P } Moreover, as a consequence of the definitions we have that 6. := e P[e]= 0 . N { ∈ P| R } Lemma 3.4. If are subsets of , then That is, , and are the subsets of of actual, regular and null events, E⊆E P A R N P respectively. 1. LPO[ ] LPO[ ]. Before proceeding, let us prove a simple, but very important fact. E → E 2. P-LPO[ ] P-LPO[ ]. Lemma 3.1. Let e ; E → E ∈P 3. PP-LPO[ ] PP-LPO[ ]. 1. if P[e] > 0, then ( n N+)(e(n) = 1); E → E R ∃ ∈ Some of these principles can be proven constructively, while one is con- 2. if ( n N+)(e(n)=0), then P[e]= 0. ∀ ∈ R structively false: Proof. 2. holds since ( n N+)(e(n) = 0) means exactly e = . ∀ ∈ P ⊥ Lemma 3.5. The following hold: Suppose now that (e,γ) and P(e,γ) > 0. As a consequence of the ∈ A R definition of > , there exists m N+ such that P(e,γ)(m) > 1 , i.e. 1. LPO[ ]. R ∈ m R γ(m) 2. P-LPO[ ]. R me(i) >γ(m) > 0. 3. PP-LPO[ ]. i=1 R Hence there exists n such that 1 n γ(m) and e(n)=1. 4. PP-LPO[ ]. ≤ ≤ N Thus ( n N+)(e(n)=1)and 1. is proved. ∃ ∈ 5. P-LPO[ ]. N 54 SAMUELE MASCHIO NATURAL DENSITY AND PROBABILITY, CONSTRUCTIVELY 55 plays an important role in constructive reverse mathematics (see [4]). It can We can now introduce a family of “probabilistic” versions of LPO. be formulated in our framework as follows Definition 3.2. For every subset of we define the following princi- + + E P LPO :( e )(( n N )(e(n)=0) ( n N )(e(n) = 1)) ples: ∀ ∈P ∀ ∈ ∨ ∃ ∈ 1. LPO[ ] def ( e ) [( n N+)(e(n)=0) ( n N+)(e(n)=1)]. Having introduced a notion of probability P on (some) binary sequences, we E ≡ ∀ ∈E ∀ ∈ ∨ ∃ ∈ can consider some variants of this principle by restricting it to some subsets 2. P-LPO[ ] def ( e )[P[e]= 0 ( n N+)(e(n) = 1)]. of potential events or by modifying the disjunction in it. E ≡ ∀ ∈E R ∨ ∃ ∈ First, we need some abbreviations: 3. PP-LPO[ ] def ( e )[P[e]= 0 P[e] > 0]. E ≡ ∀ ∈E R ∨ R + 1. List (A) is the set of finite lists of elements from a set A with positive Notice that LPO[ ] is equivalent to LPO, and that PP-LPO[ ] implies P P length; = . P A Let us now study the relation between these probabilistic versions of 2. Incr(N+, N+) is the set of strictly increasing sequences of positive nat- LPO when is , , or . As a direct consequence of Lemma 3.1 we ural numbers; E P A N R have: 3. := e ( γ Incr(N+, N+))((e,γ) ) ; A { ∈ P| ∃ ∈ ∈ A } Lemma 3.3. If is a subset of , then E P 4. ϕ(P[e]) means e ( γ Incr(N+, N+))((e,γ) ϕ(P(e,γ))) ∈A∧ ∀ ∈ ∈ A→ 1. PP-LPO[ ] P-LPO[ ]. whenever ϕ(x) is a proposition depending on a real number x; E → E + 2. LPO[ ] P-LPO[ ]. 5. := e ( α List( 0, 1 ))( π List ( 0, 1 ))(e = α.π∞) ; E → E R { ∈ P| ∃ ∈ { } ∃ ∈ { } P } Moreover, as a consequence of the definitions we have that 6. := e P[e]= 0 . N { ∈ P| R } Lemma 3.4. If are subsets of , then That is, , and are the subsets of of actual, regular and null events, E⊆E P A R N P respectively. 1. LPO[ ] LPO[ ]. Before proceeding, let us prove a simple, but very important fact. E → E 2. P-LPO[ ] P-LPO[ ]. Lemma 3.1. Let e ; E → E ∈P 3. PP-LPO[ ] PP-LPO[ ]. 1. if P[e] > 0, then ( n N+)(e(n) = 1); E → E R ∃ ∈ Some of these principles can be proven constructively, while one is con- 2. if ( n N+)(e(n)=0), then P[e]= 0. ∀ ∈ R structively false: Proof. 2. holds since ( n N+)(e(n) = 0) means exactly e = . ∀ ∈ P ⊥ Lemma 3.5. The following hold: Suppose now that (e,γ) and P(e,γ) > 0. As a consequence of the ∈ A R definition of > , there exists m N+ such that P(e,γ)(m) > 1 , i.e. 1. LPO[ ]. R ∈ m R γ(m) 2. P-LPO[ ]. R me(i) >γ(m) > 0. 3. PP-LPO[ ]. i=1 R Hence there exists n such that 1 n γ(m) and e(n)=1. 4. PP-LPO[ ]. ≤ ≤ N Thus ( n N+)(e(n)=1)and 1. is proved. ∃ ∈ 5. P-LPO[ ]. N 56 SAMUELE MASCHIO NATURAL DENSITY AND PROBABILITY, CONSTRUCTIVELY 57

6. PP-LPO[ ]. Lemma 3.6. LPO[ ] implies LPO[ ]. Hence LPO[ ], LPO[ ] and ¬ P N P P A LPO[ ] are all equivalent. Proof. Let us prove one by one of the statements above. N Proof. Let e be a potential event. We can define a non-decreasing 1. LPO[ ] holds: if e = α.π∞ is in , then sequence e in as follows: R P R P + + ( n N )(e(n) = 0) ( n N )(e(n)=1) e(1) := e(1), ∀ ∈ ∨ ∃ ∈ + (α) (π) (α) (π) e(n + 1) := e(n) e(n + 1) [n N ]. is equivalent to α + π =0 α + π > 0. ∨ ∈ i=1 i j=1 j ∨ i=1 i j=1 j e Since equality of natural numbers is decidable, we can conclude. As a consequence of the definition of , 1. ( n N+)(e(n)=1)if and only if ( n N+)(e(n) = 1), and 2. This is a consequence of 1. and Lemma 3.3. ∃ ∈ ∃ ∈ + + (π) 2. ( n N )(e(n)=0)if and only if ( n N )(e(n)=0). j=1 πj ∀ ∈ ∀ ∈ 3. If e = α.π∞ is in , then P[e]=R Q and we can thus P R (π) ∈ Using e we define another potential event α(e) as follows: decide whether P[e]=R 0 or P[e] >R 0. α(e)(1) := e(1) 4. This is obvious, since e means, by definition, that P[e]= 0. + ∈N R α(e)(n + 1) := e(n + 1) e(n)[n N ] − ∈ 5. This is a consequence of 4. and Lemma 3.3. Since e is non-decreasing, α(e) takes value 1 for at most one input. In 6. There exist potential events e for which there is no γ such that (e,γ) particular α(e) is in , ∈ N . Consider for example the sequence e which starts with a 1 and a 0 + + A ( n N )(e(n) = 1) if and only if ( n N )(α(e)(n)=1) and then alternates a group of 2 3n ones and a group of 2 3n zeros ∃ ∈ ∃ ∈ · · increasing n at each step and + + 101100111111000000... ( n N )(e(n)=0)if and only if ( n N )(α(e)(n)=0). ∀ ∈ ∀ ∈ 1 3 If LPO[ ] holds, then n n + N In this case Φ(2 3 )= and Φ(4 3 )= for every n N . · 2 · 4 ∈ + + ( n N )(α(e)(n)=1) ( n N )(α(e)(n)=0), After these first three lemmas, the situation about the remaining variants ∃ ∈ ∨ ∀ ∈ can be summarize as follows which is equivalent to ( n N+)(e(n) = 1) ( n N+)(e(n) = 0), holds. ∃ ∈ ∨ ∀ ∈ Thus from LPO[ ] follows LPO[ ]. LPO[ ] /P/-LPO[ ] N P P P After the proof of this lemma the situation is the following:

LPO[ ] LPO[ ] LPO[ ] PP-LPO[ ] LPO[ ] /P/-LPO[ ] oo PP-LPO[ ] P ≡ A ≡ N A A A A P-LPO[ ] /P/-LPO[ ] LPO[ ] P A N Lemma 3.7. P-LPO[ ] implies LPO[ ], thus P-LPO[ ], P-LPO[ ] A P A P However, the left side of the previous diagram collapses: and LPO[ ] are equivalent. P 56 SAMUELE MASCHIO NATURAL DENSITY AND PROBABILITY, CONSTRUCTIVELY 57

LPO[ ] LPO[ ] LPO[ ] PP-LPO[ ] LPO[ ] /P/-LPO[ ] oo PP-LPO[ ] P ≡ A ≡ N A A A A P-LPO[ ] /P/-LPO[ ] LPO[ ] P A N Lemma 3.7. P-LPO[ ] implies LPO[ ], thus P-LPO[ ], P-LPO[ ] A P A P However, the left side of the previous diagram collapses: and LPO[ ] are equivalent. P 58 SAMUELE MASCHIO NATURAL DENSITY AND PROBABILITY, CONSTRUCTIVELY 59

Proof. Let e be a potential event and let ϕ : N+ N be the surjective .References → primitive recursive function sending each positive natural number n to the exponent of 2 in its prime factorization. [1] E. Bishop, D. S. Bridges, Constructive analysis, Springer 1985. Let e be the potential event defined by [2] D. Bridges, F. Richman, Varieties of Constructive Mathematics, Cambridge Univer- sity Press 1987. + e(n) := e(ϕ(n)+1)[n N ]. [3] Y.K. Chan, Notes on constructive probability theory, Ann. Probability 2 (1974), 51-75 ∈ We have that ( n N+)( e(n) = 1) if and only if ( n N+)(e(n) = 1). [4] H. Ishihara, Reverse Mathematics in Bishop’s Constructive Mathematics, Philosophia ∃ ∈ ∃ ∈ Moreover e , because (e, λn.2n+2) . Since for every n>3 Scientae (2006), 43–59. ∈A ∈ A [5] M.E. Maietti, A minimalist two-level foundation for constructive mathematics, An- n 1 n+2 e(n) − e(k) nals of Pure and Applied Logic 160:3 (2009), 319–354. (P(e, λn.2 ))(n 3) = + − 2n 1 2k [6] M.E. Maietti, G. Sambin, Toward a minimalist foundation for constructive mathe- − k=1 matics, In: From Sets and Types to Topology and Analysis: Practicable Foundations + e(n) for Constructive Mathematics (Eds. L. Crosilla, P. Schuster), Oxford University Press it follows that P[e]=R n=1∞ 2n . + 2005, pp. 91–114. In particular, P[e]=R 0 if and only if ( n N )(e(n)=0). ∀ ∈ + Assuming P-LPO [ ], we obtain P(e)=0 ( n N )(e(n)=1)which [7] I. Niven, The asymptotic density of sequences, Bull. Amer. Math. Soc. 6:11 (1951), A ∨ ∃ ∈ 420–434. is equivalent to ( n N+)(e(n)=1) ( n N+)(e(n)=0). ∃ ∈ ∨ ∀ ∈ We have hence shown that P-LPO[ ] implies LPO[ ]. A P After the proof of this lemma the situation is the following: Dipartimento di Matematica “Tullio Levi-Civita” PP-LPO[ ] LPO[ ]( LPO[ ] LPO[ ] P-LPO[ ] P-LPO[ ]) Università di Padova A → P ≡ A ≡ N ≡ P ≡ A However we have the following 35131 Padova (PD), Italy [email protected] Lemma 3.8. LPO[ ] implies PP-LPO[ ]. P A Proof. Assume LPO[ ]. Since we have assumed countable choice, P trichotomy for Bishop reals follows (see [2]) and thus PP-LPO[ ] holds. A As consequence of all the lemmas above we have the following Theorem 3.9. LPO[ ] LPO[ ] LPO[ ] P-LPO[ ] P-LPO[ ] PP-LPO[ ]. P ≡ A ≡ N ≡ P ≡ A ≡ A Thus we have proved that no one of the variations on LPO which we have introduced is different from a true statement, a false statement or LPO itself. Thus, from one side the hope of having an authentic “probabilistic” variant of LPO has failed; however the results above provide new statements equivalent to LPO, namely LPO[ ], LPO[ ], P-LPO[ ], P-LPO[ ] and A N A P PP-LPO[ ]. A Acknowledgements The author would like to thank Francesco Ciraulo, Hannes Diener, Milly Maietti, Fabio Pasquali and Giovanni Sambin for fruitful discussions about the subject of this paper. The author thanks EU-MSCA-RISE project 731143 “Computing with Infinite Data” (CID) for supporting the research. 58 SAMUELE MASCHIO NATURAL DENSITY AND PROBABILITY, CONSTRUCTIVELY 59