Intuitionistic nonstandard bounded modified realisability and functional interpretation

Annals of Pure and Applied Logic 169 (2018) 392–412

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Annals of Pure and Applied Logic www.elsevier.com/locate/apal

Intuitionistic nonstandard bounded modified realisability and functional interpretation ✩ Bruno Dinis a,1 , Jaime Gaspar b,c,∗,2 a

Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa, Campo Grande, Edifício C6, 1749-016 Lisboa, Portugal b School of Computing, University of Kent, Canterbury, Kent, CT2 7NF, United Kingdom c Centro de Matemática e Aplicações (CMA), FCT, UNL, Portugal

a r t i c l e

i n f o

Article history: Received 5 September 2016 Received in revised form 26 May 2017 Accepted 10 December 2017 Available online 12 December 2017 MSC: 03B20 03F25 03F35 11U10

a b s t r a c t We present a bounded modified realisability and a bounded functional interpretation of intuitionistic nonstandard arithmetic with nonstandard principles. The functional interpretation is the intuitionistic counterpart of Ferreira and Gaspar’s functional interpretation and has similarities with Van den Berg, Briseid and Safarik’s functional interpretation but replacing finiteness by majorisability. We give a threefold contribution: constructive content and proof-theoretical properties of nonstandard arithmetic; filling a gap in the literature; being in line with nonstandard methods to analyse compactness arguments. © 2017 Published by Elsevier B.V.

Keywords: Intuitionism Bounded functional interpretation Bounded modified realisability Majorisability Nonstandard arithmetic Transfer principle

✩ The authors are grateful to Fernando Ferreira for his availability for discussion and useful remarks and to an anonymous referee for his remarks and suggestions. The authors are however the only persons responsible for any mistakes. * Corresponding author. E-mail addresses: [email protected] (B. Dinis), [email protected], [email protected] (J. Gaspar). URLs: http://www.jaimegaspar.com, http://www.cs.kent.ac.uk/people/rpg/jg478 (J. Gaspar). 1 Funded by the grant SFRH/BPD/97436/2013 of the Fundação para a Ciência e a Tecnologia and by the grant UID/MAT/04561/2013 of the Centro de Matemática, Aplicações Fundamentais e Investigação Operacional / Fundação da Faculdade de Ciências da Universidade de Lisboa. 2 Funded by a Research Postgraduate Scholarship from the Engineering and Physical Sciences Research Council / School of Computing, University of Kent.

https://doi.org/10.1016/j.apal.2017.12.004 0168-0072/© 2017 Published by Elsevier B.V.

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1. Introduction In the past few years there has been a growing interest in the study of nonstandard arithmetic using realisabilities and functional interpretations. Particularly interesting for us are: 1. Ferreira and Gaspar’s functional interpretation [6] (which deals with majorisability); 2. Van den Berg, Briseid and Safarik’s realisability and functional interpretation [2] (which deals with finiteness). In their spirit we present: 1. a realisability (based on Ferreira and Nunes’s bounded modified realisability [7], and similar to Van den Berg, Briseid and Safarik’s realisability [2]); 2. a functional interpretation (based on Ferreira and Oliva’s functional interpretation [8], and Ferreira and Gaspar’s functional interpretation [6], and similar to Van den Berg, Briseid and Safarik’s functional interpretation [2]); of intuitionistic nonstandard arithmetic with nonstandard principles and we prove their soundness and characterisation theorems. Overall we give a threefold contribution: 1. we give as applications various results, for example about (a) the constructive content (such as bounded-program extraction and extraction of bounds on witnesses); (b) proof-theoretical properties (such as relative consistency results and independence results); of nonstandard arithmetic; 2. we fill an existing gap in the literature, for example (a) we show that if we restrict our realisability to the “purely external fragment” (where there are only quantifiers of the form ∃st and ∀st ), then we recover Ferreira and Nunes’s bounded modified realisability, and analogously if we restrict our functional interpretation to the “purely external fragment”, then we recover Ferreira and Oliva’s bounded functional interpretation; (b) we think that if we combine our intuitionistic functional interpretation with a suitable negative translation, then we obtain Ferreira and Gaspar’s classical functional interpretation (a result that we hope to publish soon); 3. we are in line with the argument that nonstandard methods can be used to analyse compactness arguments [8,7,6,5]. Our realisability and functional interpretation, like previous ones, make use of Nelson’s syntactic approach to nonstandard analysis called internal set theory [12,13] by extending the language with a new unary predicate st(t) (meaning “t is standard”). 2. Framework Let E-PAω and E-HAω be (respectively) Peano and Heyting arithmetics in all finite types with full extensionality and with primitive equality only at type 0. In the next definition, proposition and theorem, we recall well-known facts about the Howard–Bezem strong majorisability ≤∗σ [3,9].

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Definition 1. 1. The Howard–Bezem strong majorisability ≤∗σ is defined by recursion on the finite type σ by: (a) s ≤∗0 t :≡ s ≤0 t; (b) s ≤∗ρ→σ t :≡ ∀v ∀u ≤∗ρ v (su ≤∗σ tv ∧ tu ≤∗σ tv). 2. We say that xσ is monotone if and only if x ≤∗σ x. Proposition 2. We have: 1. E-HAω  x ≤∗σ y → y ≤∗σ y; 2. E-HAω  x ≤∗σ y ∧ y ≤∗σ z → x ≤∗σ z. Theorem 3 (Howard’s majorisability theorem [9]). For all closed terms tσ of E-HAω , there is a closed term t˜σ of E-HAω such that E-HAω  t ≤∗σ t˜. ω ω We introduce a nonstandard variant E-HAω st of E-HA analogously to the nonstandard variant E-PAst [6] ω of E-PA .

Definition 4. The nonstandard Heyting arithmetic in all finite types with full extensionality E-HAω st is obtained from E-HAω by enriching the language and the axioms of E-HAω as follows. 1. We add the standard predicates stσ (tσ ) (for each finite type σ). 2. We add the standardness axioms: (a) x =σ y ∧ stσ (x) → stσ (y); (b) stσ (y) ∧ x ≤∗σ y → stσ (x); (c) stσ (t) for each closed term t; (d) stσ→τ (x) ∧ stσ (y) → stτ (xy); (for each finite types σ and τ ). 0 0 (x)→(Φ(x)→Φ(x+1))) 3. We add the external induction rule Φ(0) ∀x∀x(st (or, equivalently, the corresponding 0 (st0 (x)→Φ(x)) axioms). 4. The logical axioms are extended to the formulas in the language of E-HAω st but the arithmetical axioms are not. The standardness axioms are (essentially) the same as used by Ferreira and Gaspar [6], and by Van den Berg, Briseid and Safarik [2], with the exception of the second one. The second standardness axiom has a clear meaning in type 0 (namely that the nonstandard natural numbers are an end-extension of the standard natural numbers [10, page 12]) but not in higher types. We will use the following convenient abbreviations. Definition 5. Let x be a tuple of variables x1 , . . . , xn of E-HAω st of length n, s and t be tuples of terms s1 , . . . , sn and t1 , . . . , tn (respectively) of E-HAω also of length n, and Φ(x) and Ψ(x) be formulas of E-HAω st st . We abbreviate: 1. 2. 3. 4. 5.

s ≤∗σ t by s ≤∗ t; s1 ≤∗ t1 ∧ · · · ∧ sn ≤∗ tn by s ≤∗ t; stσ (t) by st(t); st(t1 ) ∧ · · · ∧ st(tn ) by st(t); ∀x Φ(x) by ∀x Φ, and analogously for ∃x Φ(x);

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6. ∀x (Φ → Ψ) with Φ ≡ x ≤∗ x,

Φ ≡ st(x),

Φ ≡ x ≤∗ x ∧ x ≤∗ t,

Φ ≡ x ≤∗ t,

Φ ≡ st(x) ∧ x ≤∗ t,

Φ ≡ x ≤∗ x ∧ st(x), Φ ≡ x ≤∗ x ∧ st(x) ∧ x ≤∗ t

by (respectively) ˜st x Ψ, ∀x ˜ ≤∗ t Ψ, ∀st x ≤∗ t Ψ, ∀ ˜st x ≤∗ t Ψ, ˜ Ψ, ∀st x Ψ, ∀x ≤∗ t Ψ, ∀ ∀x and analogously for ∃x (Φ ∧ Ψ). Many times we have a formula ∀y ∃z Φ(z) and we take z as functions of y obtaining a new formula; then we denote those functions by Z (an uppercase italic letter), so the new formula is denoted by ∃Z ∀y Φ(Zy). Sometimes we have a formula ∀x ∃Z ∀y Φ(Zy) and we take Z as functions of x obtaining a new formula; then we denote those functions by Z (an uppercase roman letter), so the new formula is denoted by ∃Z ∀x, y Φ(Zxy). We introduce this convention in the next definition. Definition 6. Let x, y and z be tuples of variables. 1. If the variables z become functions of variables y, then we denote those functions by Z; 2. If the variables Z become functions of variables x, then we denote those functions by Z. We follow the following conventions by Nelson [12]. Definition 7. 1. We say that a formula of E-HAω st is internal if and only if the standard predicates do not occur in the formula and we denote the formula by lower case Greek letters φ, ψ, . . . . 2. We say that a formula of E-HAω st is external if and only if the standard predicates may occur in the formula and we denote the formula by upper case Greek letters Φ, Ψ, . . . . 3. Intuitionistic nonstandard bounded modified realisability Definition 8. The intuitionistic nonstandard bounded modified realisability b assigns to each formula Φ of ω b b ˜st E-HAω st the formulas Φ and Φb (a) of E-HAst such that Φ ≡ ∃ a Φb (a) accordingly to the following clauses where Φb (a) is the part inside square brackets below. For atomic formulas, we define: 1. Φb :≡ [Φ] for internal atomic formulas Φ (so the tuple a is empty); ˜st a [t ≤∗ a]. 2. st(t)b :≡ ∃ ˜st a Φb (a) and Ψb ≡ ∃ ˜st b Ψb (b), then we define: For the remaining formulas, if Φb ≡ ∃ 3. 4. 5. 6. 7.

˜st a, b [Φb (a) ∧ Ψb (b)]; (Φ ∧ Ψ)b :≡ ∃ b ˜st a, b [Φb (a) ∨ Ψb (b)]; (Φ ∨ Ψ) :≡ ∃ b ˜st B [∀ ˜st a (Φb (a) → Ψb (Ba))]; (Φ → Ψ) :≡ ∃ ˜st a [∀x Φb (a)]; (∀x Φ)b :≡ ∃ b ˜st a [∃x Φb (a)]. (∃x Φ) :≡ ∃

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We need the next lemmas to prove the soundness and characterisation theorems of b and to do the applications of b. Lemma 9. b 1. For all internal formulas φ of E-HAω st , we have φ ≡ φb (a) ≡ φ (so the tuple a is empty). ω 2. For all formulas Φ of E-HAst , we have (the equivalences being provable in E-HAω st ): ˜st A [∀ ˜st b ∀x ≤∗ b Φb (Ab)]; (a) (∀st x Φ)b ↔ ∃ ˜st x Φ)b ↔ ∃ ˜st A [∀ ˜st b ∀x ˜ ≤∗ b Φb (Ab)]; (b) (∀ ˜st b, a [∃x ≤∗ b Φb (a)]; (c) (∃st x Φ)b ≡ ∃ st b ˜ ˜st b, a [∃x ˜ ≤∗ b Φb (a)]; (d) (∃ x Φ) ≡ ∃ ∗ b st ˜ a [∃x ≤∗ t Φb (a)]; (e) (∃x ≤ t Φ) ≡ ∃ ˜ ≤∗ t Φ)b ≡ ∃ ˜st a [∃x ˜ ≤∗ t Φb (a)]. (f) (∃x

Proof. 1. The proof is by a simple induction on the length of φ (analogous to the proof of Lemma 12 below). 2. The proof is by simple calculations using that x ≤∗ t is an internal formula. For example, for point 2b: ˜st a Φb (a), Φb ≡ ∃

(assumption)

(x ≤∗ x)b ≡ x ≤∗ x,

(x ≤∗ x internal)

˜st b [x ≤∗ b], st(x)b ≡ ∃

(calculation)

˜st b [x ≤∗ x ∧ x ≤∗ b], (x ≤∗ x ∧ st(x))b ≡ ∃ ∗

˜st

˜st

∗

(calculation) ∗

(x ≤ x ∧ st(x) → Φ) ≡ ∃ A [∀ b (x ≤ x ∧ x ≤ b → Φb (Ab))], b  ˜st A [∀x ∀ ˜st b (x ≤∗ x ∧ x ≤∗ b → Φb (Ab))], ∀x (x ≤∗ x ∧ st(x) → Φ) ≡ ∃ b  ˜st A [∀ ˜st b ∀x ˜ ≤∗ b Φb (Ab)], ∀x (x ≤∗ x ∧ st(x) → Φ) ↔ ∃ b

˜st x Φ ≡ ∀x (x ≤∗ x ∧ st(x) → Φ). where ∀

(calculation) (calculation) (logic)

2

The formulas Φb (a) are monotone on a in the sense of the following lemma. ω ∗ Lemma 10 (monotonicity of b). For all formulas Φ of E-HAω ˜ → Φb (˜ a). st , we have E-HAst  Φb (a) ∧ a ≤ a

Proof. The proof is by a simple induction on the length of Φ. For example, in the case of →, we have ˜st a (Φb (a) → Ψb (Ba)), we also have E-HAω  Ψb (b) ∧ b ≤∗ ˜b → Ψb (˜b) by induction (Φ → Ψ)b (B) ≡ ∀ st ˜st a (Φb (a) → Ψb (Ba)) ˜st a (Φb (a) → Ψb (Ba)) ∧ B ≤∗ B ˜ → ∀ ˜ hypothesis, and we prove ∀ by noticing that ∗ ˜ ∗ ˜ ˜ B ≤ B implies Ba ≤ Ba (for monotone a) and so the induction hypothesis implies Ψb (Ba) → Ψb (Ba) (for monotone a). 2 ˜st -free formulas that has a role similar to the one of ∃-free ˜ Now we define the class of ∃ formulas [7], which st ˜ ˜ in turn reminds the well-known ∃-free formulas (with the difference that ∃ -free and ∃-free formulas allow disjunctions). ˜st Definition 11. We say that a formula of E-HAω st is ∃ -free if and only if it is built: 1. from atomic internal formulas s =0 t;

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2. 3. 4. 5. 6.

by by by by by

397

conjunctions ∧; disjunctions ∨; implications →; ˜ and ∃); ˜ quantifications ∀ and ∃ (so also ∀ ˜st (but not ∃ ˜st ), monotone standard universal quantifications ∀

˜st as a subscript as in Φ˜st , Ψ˜st , . . . . and we denote the formula by upper case Greek letters with    ˜st -free formulas Φ˜st of E-HAω , we have (Φ˜st )b ≡ (Φ˜st )b (a) (so the tuple a is empty) Lemma 12. For all ∃ st    and E-HAω  (Φ ) ↔ Φ . st st ˜ ˜ b st   ˜st , we Proof. The proof is by a simple induction on the length of Φ˜st . For example, in the case of ∀ st b st st ˜ ∗ ˜ ˜ ˜ have (∀ x Φ˜st ) ≡ ∃ A [∀ b ∀x ≤ b (Φ˜st )b (Ab)] (where by induction hypothesis A is empty), which is ˜st b ∀x ˜ ≤∗ b (Φ˜st )b (where (Φ˜st )b ↔ Φ˜st by induction hypothesis), which is equivalent to ∀ ˜st x Φ˜st . 2 ∀     ˜st Lemma 13. For all formulas Φ of E-HAω st , the formula Φb (a) is ∃ -free. Proof. The proof is by a simple induction on the length of Φ. For example, in the case of st(t), we have ˜st -free because it is built from formulas of the form r ≤0 s (which is an st(t)b (a) ≡ t ≤∗ a, which is ∃ abbreviation for an appropriate internal atomic formula) by means of ∀, → and ∧. 2 The following four principles, inspired by or even taken from Ferreira, Nunes and Gaspar’s articles [7,6], play an important role in the sequel as they are the characteristic principles of b. Definition 14. We define the following principles: 1. 2. 3. 4.

the the the the

˜st Y ∀ ˜st x ∃y ˜st x ∃ ˜st y Φ → ∃ ˜ ≤∗ Y x Φ; monotone choice mACω is all instances of ∀ ω st st ∗ ˜ z ∀x ∃y ≤ z Φ; realisation R is all instances of ∀x ∃ y Φ → ∃ ω ˜st x Ψ) → ∃ ˜st y (Φ˜st → ∃x ˜ ≤∗ y Ψ); independence of premises IP˜st is all instance of (Φ˜st → ∃  ω st st ∗ majorisability axioms MAJ are all instances of ∀ x ∃ y (x ≤ y).

The principles above generalise to tuples of variables, that is each principle provably implies in E-HAω st its variant where the single variables x, y, Y and z are replaced by (respectively) the tuples of variables x, y, Y and z (proof sketch: first we generalise from x to x by induction on the length of x and then we generalise from y, Y and z to y, Y and z by induction on the length of y, Y and z). ω ω The principle Rω implies the principle MAJω , that is E-HAω st + R proves all instances of MAJ (proof sketch: we take any standard x , we take Φ to be the formula y = x in Rω , then the premise of Rω is provable and the conclusion of Rω implies ∃st z (x ≤∗ z)). However, we keep explicitly mentioning MAJω (even in the presence of Rω ) because of its importance. The following two principles, the first one inspired by Ferreira and Oliva’s article [8], are used later on in applications of b. Definition 15. We define the following principles: ˜st x φ → ψ) → ∃ ˜st y (∀x ˜ ≤∗ y φ → ψ); 1. Markov’s principle Mω is all instances of (∀ 2. the law of excluded middle LEM is all instances of Φ ∨ ¬Φ. Now we present our main theorem about b. This theorem shows that b:

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ω ω ω ω ω 1. interprets E-HAω ˜ st + MAJ into E-HAst ; st + mAC + R + IP ω ω 2. extracts computational information t from proofs in E-HAst + mACω + Rω + IPω ˜ st + MAJ . 

Theorem 16 (soundness theorem of b). For all formulas Φ of E-HAω st , if ω ω ω ω E-HAω ˜ st + MAJ  Φ, st + mAC + R + IP

then there are closed monotone terms t of appropriate types such that E-HAω st  Φb (t). Proof. The proof is by induction on the length of the derivation of Φ. The logical and arithmetical axioms and rules are dealt with in a way similar to the bounded modified realisability [7]. So let us focus on the standardness axioms and the characteristic principles (except MAJω , which follows from Rω ). We will sometimes use implicitly Lemmas 9 and 12. ˜st a (x = y ∧ x ≤∗ a → y ≤∗ Ba). It is ˜st B ∀ x = y ∧ st(x) → st(y). Its interpretation is equivalent to ∃ interpreted by B := λa . a. ˜st a (y ≤∗ a ∧ x ≤∗ y → x ≤∗ Ba). It is ˜st B ∀ st(y) ∧ x ≤∗ y → st(x). Its interpretation is equivalent to ∃ interpreted by B := λa . a. ˜st a (t ≤∗ a). It is interpreted by a closed term t˜ such that t ≤∗ t˜, which exists st(t). Its interpretation is ∃ by Howard’s majorisability theorem. ˜st a, b (x ≤∗ a ∧ y ≤∗ b → xy ≤∗ Cab). It is interpreted by ˜st C ∀ st(x) ∧ st(y) → st(xy). Its interpretation is ∃ C := λa, b . ab. st ˜ st ˜st Y ∀ ˜st x ∃y ˜ ≤∗ Y x Φ. Its interpretation is equivalent to ˜ ∀ x∃ yΦ → ∃ ˜st B, A (∀ ˜st c ∀x ˜ ≤∗ c ∃ ˜st y ≤∗ Bc Φb (Ac) → ˜st F, D ∀ ∃ ˜st e ∀x ˜ ≤∗ e ∃y ˜ ≤∗ Y x Φb (DBAe)). ˜st Y ≤∗ F BA ∀ ∃ ˜st c ∀x ˜ ≤∗ c ∃ ˜st y ≤∗ It is interpreted by F := λB, A . B and D := λB, A . A. Indeed, from the premise ∀ st st ∗ st ˜ e ∀x ˜ x∃ ˜ y ≤ Bx Φb (Ax), so (using monotonicity) we get the conclusion ∀ ˜ ≤∗ Bc Φb (Ac) we get ∀ ˜ ≤∗ Y x Φb (DBAe) with Y := B. e ∃y st ˜st z ∀x ∃y ≤∗ z Φ. Its interpretation is ∀x ∃ y Φ → ∃ ˜ ≤∗ Dba ∀x ∃y ≤∗ z Φb (Cba)). ˜st D, C ∀ ˜st b, a (∀x ∃y ≤∗ b Φb (a) → ∃z ∃ It is interpreted by D := λb, a . b and C := λb, a . a. ˜st x Ψ) → ∃ ˜st y (Φ → ∃x ˜ ≤∗ y Ψ). Its interpretation is equivalent to (Φ → ∃   ˜ ≤∗ Dba (Φ → ∃x ˜ ≤∗ y Ψb (Cba)) . ˜ ≤∗ b Ψb (a)) → ∃y ˜st D, C ∀ ˜st b, a (Φ → ∃x ∃ It is interpreted by D := λb, a . b and C := λb, a . a.

2

Theorem 17 (characterisation theorem of b). For all formulas Φ of E-HAω st , we have ω ω ω ω b E-HAω ˜ st + MAJ  Φ ↔ Φ . st + mAC + R + IP

Proof. The proof is by induction on the length of Φ. The cases of internal atomic formulas, ∧, ∨ and ∃ are easy, so let us focus on the remaining cases. We sometimes use implicitly Lemma 13.

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st(t). We have st(t) ↔

(MAJω , st(y) ∧ x ≤∗ y → st(x))

˜st a (t ≤∗ a) ≡ ∃

(definition)

st(t)b . →. We have (Φ → Ψ) ↔

(induction hypothesis)

˜st b Ψb (b)) ↔ ˜st a Φb (a) → ∃ (∃

(logic)

˜st a (Φb (a) → ∃ ˜st b Ψb (b)) ↔ ∀

(IPω ˜ st ) 

˜st b (Φb (a) → ∃ ˜˜b ≤∗ b Ψb (˜b)) ↔ ˜st a ∃ ∀

(monotonicity)

˜st b (Φb (a) → Ψb (b)) ↔ ˜st a ∃ ∀

(mACω )

˜st a ∃ ˜st b ≤∗ Ba (Φb (a) → Ψb (b)) ↔ ˜ ∀ ∃B

(monotonicity)

˜st a (Φb (a) → Ψb (Ba)) ≡ ˜ ∀ ∃B

(definition)

(Φ → Ψ)b . ∀. We have ∀x Φ ↔

(induction hypothesis)

˜ st a Φb (a) ↔ ∀x ∃

(Rω )

˜st a ∀x ∃˜ ˜a ≤∗ a Φb (˜ ∃ a) ↔

(monotonicity)

˜st a ∀x Φb (a) ≡ ∃

(definition)

(∀x Φ)b .

2

In the next result we give some applications of b: 1. the first point is an equiconsistency result (“equiconsistency” meaning that two theories are both consistent or both inconsistent); 2. the second point is a bounded variant of the existence property (“bounded” meaning that instead of giving a term t such that Φ(t) said to be witnessing an existential quantification ∃x Φ(x), it gives a term t such that ∃x ≤∗ t Φ(x) said to be bounding the existential quantification ∃x Φ(x)); 3. the third, fourth and fifth points are term-and-rule variants of mACω , Rω and IPω ˜ st (“term” meaning that  some existential quantification ∃x Φ(x) is witnessed by some term t, and “rule” meaning that instead of Φ Φ → Ψ we have · · ·  Φ ⇒ · · ·  Ψ also denoted by Ψ ); 4. the sixth point is a conservation result (“conservation” meaning that if a stronger theory proves a formula of a certain form, then a weaker theory already proves it); 5. the seventh and eight points are independence results (“independence” meaning that a formula is neither provable nor refutable by a theory). Most of these results are inspired by similar folklore results.

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ω ω ω ω ω Application 18. Let E-HAω ˜ st + MAJ . st + P := E-HAst + mAC + R + IP ω 1. The theories E-HAω st + P and E-HAst are equiconsistent. ω st 2. If E-HAω st +P  ∃ x Φ, then there are closed monotone terms t of appropriate types such that E-HAst +P  ω st ∗ st ˜ ∃ x ≤ t Φ. Moreover, if Φ is ∃ -free, then in the conclusion we can replace E-HAst + P by E-HAω st . Analogously, if all quantifications of x are monotone. st st 3. If E-HAω st + P  ∀ x ∃ y Φ, then there are closed monotone terms t of appropriate types such that ω st ∗ ˜ z ∀x ≤ z ∃st y ≤∗ tz Φ. Also, in the conclusion we can replace ∀ ˜st z ∀x ≤∗ z ∃st y ≤∗ tz Φ E-HAst + P  ∀ ˜st x ∃st y ≤∗ tx Φ. Moreover, if Φ is ∃ ˜st -free, then in the conclusion we can replace E-HAω + P by by ∀ st E-HAω . Analogously, if all quantifications of x are monotone, or if all quantifications of y are monotone, st or if both. ω st 4. If E-HAω st +P  ∀x ∃ y Φ, then there are closed monotone terms t of appropriate types such that E-HAst + ω st ∗ st ˜ P  ∀x ∃ y ≤ t Φ. Moreover, if Φ is ∃ -free, then in the conclusion we can replace E-HAst +P by E-HAω st . Analogously, if all quantifications of x are monotone, or if all quantifications of y are monotone, or if both. st 5. If E-HAω ˜ st → ∃ x Ψ, then there are closed monotone terms t of appropriate types such that st + P  Φ ω ˜st -free, then in the conclusion we can replace E-HAst + P  Φ˜st → ∃st x ≤∗ t Ψ. Moreover, if Ψ is ∃ ω ω E-HAst + P by E-HAst . Analogously, if all quantifications of x are monotone. ˜st st ˜st , then E-HAω  ∀ ˜st x ∃st y Φ˜st . Analogously, if all quantifications of y are 6. If E-HAω st + P  ∀ x ∃ y Φ st  monotone. ω ω ω 7. If E-HAω st + P is consistent, then there is an instance Φ of M such that E-HAst + P  Φ and E-HAst + P  ¬Φ. ω ω 8. If E-HAω st + P is consistent, then there is an instance Φ of LEM such that E-HAst + P  Φ and E-HAst + P  ¬Φ.

Proof. The “Moreover, . . . ” parts are proved analogously to their preceding parts but using Lemma 12 ω instead of the characterisation theorem to remain in E-HAω st instead of going to E-HAst +P. The “Analogously, . . . ” parts are proved analogously to their proceedings parts and sometimes are also a corollary to their ω ˜st ˜ preceding parts (for example, a proof sketch for point 2: rewrite E-HAω st + P  ∃ x Φ as E-HAst + P  ω ˜ apply the first sentence of point 2 with Φ :≡ x ≤∗ x ∧ Φ ˜ getting E-HA + P  ∃st x ≤∗ ∃st x (x ≤∗ x ∧ Φ), st ω ∗ st ∗ ˜ ˜ ˜ t (x ≤ x ∧ Φ), and then rewrite this as E-HAst + P  ∃ x ≤ t Φ). We will sometimes use implicitly that closed terms are standard. 1. We have E-HAω st + P  0 =0 1 ⇒

(soundness)

 (0 =0 1)b ⇒

(definition)

E-HAω st

E-HAω st  0 =0 1. ω Trivially, E-HAω st  0 =0 1 ⇒ E-HAst + P  0 =0 1. 2. We have st E-HAω st + P  ∃ x Φ ⇒

(soundness)

 (∃x Φ)b (t, s) ⇒

(Lemma 9)

∗ E-HAω st  ∃x ≤ t Φb (s) ⇒

(t closed)

st ∗ E-HAω st  ∃ x ≤ t Φb (s) ⇒

(s closed monotone)

E-HAω st

B. Dinis, J. Gaspar / Annals of Pure and Applied Logic 169 (2018) 392–412

st ∗ b E-HAω ⇒ st  ∃ x ≤ t Φ

E-HAω st

401

(characterisation)

∗

+ P  ∃ x ≤ t Φ. st

3. We have st st E-HAω st + P  ∀ x ∃ y Φ ⇒

(soundness)

 (∀ x ∃ y Φ)b (t, s) ⇒

(Lemma 9)

˜st z ∀x ≤∗ z ∃y ≤∗ tz Φb (sz) ⇒ ∀

(tz standard)

E-HAω st E-HAω st E-HAω st

st

∗

˜st

st

∗

 ∀ z ∀x ≤ z ∃ y ≤ tz Φb (sz) ⇒

(sz monotone standard)

˜st z ∀x ≤∗ z ∃st y ≤∗ tz Φb ⇒ ∀

(characterisation)

E-HAω st E-HAω st

st

∗

˜st

∗

+ P  ∀ z ∀x ≤ z ∃ y ≤ tz Φ. st

ω ∗ st ∗ ∗ ˜st ˜st st Also, E-HAω st + P  ∀ z ∀x ≤ z ∃ y ≤ tz Φ ⇒ E-HAst + P  ∀ x ∃ y ≤ tx Φ (by taking x := z). 4. We have st E-HAω st + P  ∀x ∃ y Φ ⇒

(soundness)

 (∀x ∃ y Φ)b (t, s) ⇒

(Lemma 9)

∗ E-HAω st  ∀x ∃y ≤ t Φb (s) ⇒

(t closed monotone)

E-HAω st

st

∗

 ∀x ∃ y ≤ t Φb (s) ⇒

(s closed monotone)

st ∗ b E-HAω ⇒ st  ∀x ∃ y ≤ t Φ

(characterisation)

E-HAω st E-HAω st

st

∗

+ P  ∀x ∃ y ≤ t Φ. st

5. We have st E-HAω ˜ st → ∃ x Ψ ⇒ st + P  Φ

(soundness)

st E-HAω ˜ st → ∃ x Ψ)b (t, s) ⇒ st  (Φ

(Lemma 9)

E-HAω st

(t closed)

 Φ˜st → ∃st x ≤∗ t Ψb (s) ⇒

(s closed monotone)

E-HAω st E-HAω st

∗

 Φ˜st → ∃x ≤ t Ψb (s) ⇒

E-HAω st

∗

 Φ˜st → ∃ x ≤ t Ψ st

⇒

b

(characterisation)

+ P  Φ˜st → ∃st x ≤∗ t Ψ.

6. Follows from point 4. 7. It is well known that there is an internal atomic formula φ(x, y, z) of E-HAω such that for all x, y, z ∈ N we have the following equivalence (where n ¯ denotes the numeral associated to n ∈ N): E-HAω  φ(¯ x, y¯, z¯) if and only if the Turing machine coded by x when given input coded by y halts with computation history coded by z. Let Φ :≡ (∀st z ¬φ → 0 =0 1) → ∃st z˜ (∀z ≤0 z˜ ¬φ → 0 =0 1). E-HAω st + P  Φ. We have E-HAω st + P  Φ ⇒

(logic)

E-HAω st + P  ∀x, y Φ ⇒

(soundness)

402

B. Dinis, J. Gaspar / Annals of Pure and Applied Logic 169 (2018) 392–412

st E-HAω st  (∀ x, y Φ)b (t) ⇒

(Lemma 9)

˜x, y˜ ∀x, y ≤0 x E-HAω ˜, y˜ (¬∀st z ¬φ → ∃˜ z ≤0 t˜ xy˜ ¬∀z ≤0 z˜ ¬φ) ⇒ st  ∀˜

(logic, arithmetic)

st st st E-HAω st  ∀ x, y (¬∀ z ¬φ → ¬∀ z ≤0 txy ¬φ).

If E-HAω st + P  Φ, then there is a term t as above and t induces a computable function that can be used to solve the halting problem, a contradiction. ω ω ω E-HAω st + P  ¬Φ. If E-HAst + P  ¬Φ, then E-HAst + P is inconsistent because E-HAst  ¬¬Φ. 8. Let φ be as above and Φ :≡ ∃st z φ(x, y, z) ∨ ¬∃st z φ(x, y, z), ˜ :≡ ∃st z φ(x, y, z) ∨ ∀st z ¬φ(x, y, z). Φ E-HAω st + P  Φ. We have E-HAω st + P  Φ ⇒

(logic)

˜ E-HAω st + P  ∀x, y Φ ⇒

(soundness)

st ˜ E-HAω st  (∀ x, y Φ)b (t) ⇒

(Lemma 9)

˜x, y˜ ∀x, y ≤0 x E-HAω ˜, y˜ (∃z ≤0 t˜ xy˜ φ(x, y, z) ∨ ∀z ¬φ(x, y, z)) ⇒ st  ∀˜

(logic, arithmetic)

st st st E-HAω st  ∀ x, y (∃ z ≤0 txy φ(x, y, z) ∨ ∀ z ¬φ(x, y, z)).

If E-HAω st + P  Φ, then there is a term t as above and t induces a computable function that can be used to solve the halting problem, a contradiction. ω ω ω E-HAω st + P  ¬Φ. If E-HAst + P  ¬Φ, then E-HAst + P is inconsistent because E-HAst  ¬¬Φ. 2 4. Intuitionistic nonstandard bounded functional interpretation Definition 19. The intuitionistic nonstandard bounded functional interpretation B assigns to each formula Φ ω B B ˜st a ∀ ˜st b ΦB (a; b) accordingly to the ≡ ∃ of E-HAω st the formulas Φ and ΦB (a; b) of E-HAst such that Φ following clauses where ΦB (a; b) is the part inside square brackets. For atomic formulas, we define: 1. ΦB :≡ [Φ] for internal atomic formulas Φ (so the tuples a and b are empty); ˜st a [t ≤∗ a] (so the tuple b is empty). 2. st(t)B :≡ ∃ ˜st a ∀ ˜st c ∀ ˜st b ΦB (a; b) and ΨB ≡ ∃ ˜st d ΨB (c; d) then we define: For the remaining formulas, if ΦB ≡ ∃ 3. 4. 5. 6. 7.

˜st a, c ∀ ˜st b, d [ΦB (a; b) ∧ ΨB (c; d)]; (Φ ∧ Ψ)B :≡ ∃ ˜st a, c ∀ ˜ ≤∗ e ΦB (a; b) ∨ ∀d ˜ ≤∗ f ΨB (c; d)]; ˜st e, f [∀b (Φ ∨ Ψ)B :≡ ∃ B st st ∗ ˜ ˜ ˜ (Φ → Ψ) :≡ ∃ C, B ∀ a, d [∀b ≤ Bad ΦB (a; b) → ΨB (Ca; d)]; ˜st a ∀ ˜st b [∀x ΦB (a; b)]; (∀x Φ)B :≡ ∃ B st ˜ st ˜ ˜ ≤∗ c ΦB (a; b)]. (∃x Φ) :≡ ∃ a ∀ c [∃x ∀b

We need the next lemma to prove the soundness and characterisation theorems of B.

B. Dinis, J. Gaspar / Annals of Pure and Applied Logic 169 (2018) 392–412

403

Lemma 20. B 1. For all internal formulas φ of E-HAω st , we have φ ≡ φB (a) ≡ φ (so the tuple a is empty). ω 2. For all formulas Φ of E-HAω st , we have (the equivalences being provable in E-HAst ): B st ˜ st ∗ ˜ ˜ ˜ (a) (∀x Φ) ≡ ∃ A ∀ c, b [∀x ≤ c ΦB (Ac; b)]; ˜st A ∀ ˜st c, b [∀x ≤∗ c ΦB (Ac; b)]; (b) (∀st x Φ)B ≡ ∃ st B st ˜ st ˜ ˜ ˜ ≤∗ c ΦB (Ac; b)]; (c) (∀ x Φ) ≡ ∃ A ∀ c, b [∀x ∗ B st ˜ st ˜ (d) (∀x ≤ t Φ) ≡ ∃ a ∀ b [∀x ≤∗ t ΦB (a; b)]; ˜ ≤∗ t Φ)B ≡ ∃ ˜st a ∀ ˜ ≤∗ t ΦB (a; b)]; ˜st b [∀x (e) (∀x st B st st ˜ ˜ ˜ ≤∗ d ΦB (a; b)]; (f) (∃ x Φ) ↔ ∃ c, a ∀ d [∃x ≤∗ c ∀b st B st st ∗ ˜ x Φ) ↔ ∃ ˜ c, a ∀ ˜ ≤ c ∀b ˜ d [∃x ˜ ≤∗ d ΦB (a; b)]; (g) (∃ ˜st a ∀ ˜st c [∃x ≤∗ t ∀b ˜ ≤∗ c ΦB (a; b)]; (h) (∃x ≤∗ t Φ)B ↔ ∃ ∗ B st ˜ st ˜ ∗ ˜ ˜ ˜ (i) (∃x ≤ t Φ) ↔ ∃ a ∀ c [∃x ≤ t ∀b ≤∗ c ΦB (a; b)].

Proof. 1. The proof is by a simple induction on the length of φ (see the proof of Lemma 22 below). 2. The proof is by simple calculations using that x ≤∗ t is an internal formula. For example, for point 2g: ˜st a ∀ ˜st b Φb (a; b), ΦB ≡ ∃ ∗

(assumption)

∗

(x ≤∗ x internal)

(x ≤ x) ≡ x ≤ x, B

˜st c [x ≤∗ c], st(x)B ≡ ∃ ∗

˜st

∗

(calculation) ∗

(x ≤ x ∧ st(x)) ≡ ∃ c [x ≤ x ∧ x ≤ c], b

(calculation)

˜st c, a ∀ ˜st b [x ≤∗ x ∧ x ≤∗ c ∧ ΦB (a; b)], (x ≤∗ x ∧ st(x) ∧ Φ)B ≡ ∃ ∗

˜st

∗

˜st

∗

∗

˜ ≤ d (x ≤ x ∧ x ≤ c ∧ ΦB (a; b))], (∃x (x ≤ x ∧ st(x) ∧ Φ)) ≡ ∃ c, a ∀ d [∃x ∀b B

˜st c, a ∀ ˜ ≤∗ c ∀ ˜st d [∃x ˜st b ≤∗ d ΦB (a; b)], (∃x (x ≤∗ x ∧ st(x) ∧ Φ))B ↔ ∃ ˜st x Φ ≡ ∃x (x ≤∗ x ∧ st(x) ∧ Φ). where ∃

(calculation) (calculation) (logic)

2

The formulas ΦB (a; b) are monotone on a in the sense of the following lemma. ω ∗ Lemma 21 (monotonicity of B). For all formulas Φ of E-HAω ˜ → st , we have E-HAst  ΦB (a; b) ∧ a ≤ a ΦB (˜ a; b).

Proof. The proof is by a simple induction on the length of Φ. For example, in the case of →, we have ˜ ≤∗ Bad ΦB (a; b) → ΨB (Ca; d), we have E-HAω  ΨB (c; d) ∧ c ≤∗ c˜ → ΨB (˜ (Φ → Ψ)B (C, B; a, d) ≡ ∀b c; d) st ˜ ≤∗ Bad ΦB (a; b) → ΨB (Ca; d)) ∧ C, B ≤∗ C, ˜ ≤∗ ˜ → (∀b ˜ D by induction hypothesis, and we prove (∀b ˜ ΦB (a; b) → ΨB (Ca; ˜ d)) by noticing that C, D ≤∗ C, ˜ D ˜ implies Ca, Bad ≤∗ Ca, ˜ Bad ˜ (for monotone a, d) Bad ˜ ≤∗ ˜ ˜ ˜ ΦB (a; b) → ∀b and so the induction hypothesis implies ΨB (Ca; d) → ΨB (Ca; d) and trivially ∀b ≤∗ Bad Bad ΦB (a; b) (for monotone a, d). 2 B Lemma 22. For all internal formulas φ of E-HAω st , we have φ ≡ φB (a; b) (so the tuples a and b are empty) and φB ≡ φ.

Proof. The proof is by a simple induction on the length of φ. For example, in the case of →, we have ˜st C, B ∀ ˜ ≤∗ Bad φB (a; b) → ψB (Ca; d)] (where C, B and a, d are empty by induction ˜st a, d [∀b (φ → ψ)B ≡ ∃ hypothesis), which is φB → ψB (where φB ≡ φ and ψB ≡ ψ by induction hypothesis), which is φ → ψ. 2

404

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Lemma 23. For all formulas Φ of E-HAω st , the formula ΦB (a; b) is internal. Proof. The proof is by a simple induction on the length of Φ. For example, in the case of st(t), we have st(t)B (a) ≡ t ≤∗ a, which is internal because it is built from formulas of the form r ≤0 s (which is an abbreviation for an appropriate internal atomic formula) by means of ∀, → and ∧. 2 The following seven principles, inspired by or even taken from Ferreira, Oliva and Gaspar’s articles [6,8], play an important role in the sequel as they are the characteristic principles of B. Definition 24. We define the following principles: ˜st Y ∀ ˜st x ∃y ˜st x ∃ ˜st y Φ → ∃ ˜ ≤∗ Y x Φ; the monotone choice mACω is all instances of ∀ ˜st z ∀x ∃y ≤∗ z Φ; the realisation Rω is all instances of ∀x ∃st y Φ → ∃ ω st ∗ ˜ the idealisation I is all instances of ∀ z ∃x ∀y ≤ z φ → ∃x ∀st y φ; ∗ ˜st ˜st ˜st ˜st ˜ the independence of premises IPω ˜ st is all instances of (∀ x φ → ∃ y Ψ) → ∃ z (∀ x φ → ∃y ≤ z Ψ); ∀ ˜st x φ → ψ) → ∃ ˜st y (∀x ˜ ≤∗ y φ → ψ); Markov’s principle Mω is all instances of (∀ ω ˜st u, v (∀x ≤∗ u φ ∨ ∀y ≤∗ v ψ) → the bounded universal disjunction principle BUD are all instances of ∀ st st ∀ x φ ∨ ∀ y ψ; 7. the majorisability axioms MAJω are all instances of ∀st x ∃st y (x ≤∗ y).

1. 2. 3. 4. 5. 6.

The principles above, where there are single variables x, y, Y and z instead of tuples of variables x, y, Y and z, generalise to tuples of variables, that is each principle provably implies in E-HAω st its variant where the single variables x, y, Y and z are replaced by (respectively) the tuples of variables x, y, Y and z (proof sketch: first we generalise from x to x by induction on the length of x and then we generalise from y, Y and z to y, Y and z by induction on the length of y, Y and z). The principles above, where there are tuples of variables u, v, x, y and z instead of single variables u, v, x, y and z, seemingly do not generalise to tuples of variables (by induction on the length of the tuples). ω ω The principle Iω implies the principle BUDω , that is E-HAω st + I proves all instances of BUD (proof ω sketch: rewrite the disjunction in the premises of BUD as an existential conjunction of implications getting ˜st u, v ∃x ((x =0 0 → ∀x ≤∗ u φ) ∧ (x =0 0 → ∀y ≤∗ v ψ)), that is ∀ ˜st u, v ∃x ∀x, y ≤∗ u, v ((x =0 0 → φ) ∧ ∀ ω st (x =0 0 → ψ)), apply I getting ∃x ∀ x, y ((x =0 0 → φ) ∧ (x =0 0 → ψ)), that is ∃x ((x =0 0 → ∀st x φ) ∧ (x =0 0 → ∀st y ψ)), and rewrite the existential conjunction of implications back as a disjunction getting the conclusion of BUDω ). The principle Rω implies the principle MAJω (see page 397). However, we keep explicitly mentioning BUDω and MAJω (even in the presence of Iω and Rω ) because of their importance. Now we present our main theorem about B. This theorem shows that B: ω ω ω ω ω ω ω ω 1. interprets E-HAω ˜ st + M + BUD + MAJ into E-HAst ; st + mAC + R + I + IP∀ ω ω ω ω ω 2. extracts computational information t from proofs in E-HAst +mAC +Rω +Iω +IPω ˜ st +M +BUD +MAJ . ∀

Theorem 25 (soundness theorem of B). For all formulas Φ of E-HAω st , if ω ω ω ω ω ω ω E-HAω ˜ st + M + BUD + MAJ  Φ, st + mAC + R + I + IP∀

then there are closed monotone terms t of appropriate types such that ˜st E-HAω st  ∀ b ΦB (t; b). Proof. The proof is by induction on the length of the derivation of Φ. The logical and arithmetical axioms and rules are dealt with in a way similar to the bounded functional interpretation [8]. So let us focus on the

B. Dinis, J. Gaspar / Annals of Pure and Applied Logic 169 (2018) 392–412

405

standardness axioms and the characteristic principles (except BUDω and MAJω , which follow respectively from Iω and Rω ). We will sometimes use implicitly Lemmas 20 and 22. ˜st a (x = y ∧ x ≤∗ a → y ≤∗ Ba). It is interpreted by ˜st B ∀ x = y ∧ st(x) → st(y). Its interpretation is ∃ B := λa . a. ˜st a (y ≤∗ a ∧ x ≤∗ y → x ≤∗ Ba). It is interpreted by ˜st B ∀ st(y) ∧ x ≤∗ y → st(x). Its interpretation is ∃ B := λa . a. ˜st a (t ≤∗ a). It is interpreted by a closed term t˜ such that t ≤∗ t˜, which exists st(t). Its interpretation is ∃ by Howard’s majorisability theorem. ˜st a, b (x ≤∗ a ∧ y ≤∗ b → xy ≤∗ Cab). It is interpreted by ˜st C ∀ st(x) ∧ st(y) → st(xy). Its interpretation is ∃ C := λa, b . ab. st ˜ st ˜st Y ∀ ˜st x ∃y ˜ ≤∗ Y x Φ. Its interpretation is equivalent to ˜ ∀ x∃ yΦ → ∃ ˜st J, C, G, F ∀ ˜st E, A, k, l ∃ ˜ f ≤∗ GEAkl, F EAkl ∀x ˜ ≤∗ Eg ∀b ˜ ≤∗ f ΦB (Ag; b) → ˜ ≤∗ g ∃y (∀g, ˜ ≤∗ Y x ∀d ˜ ≤∗ h ΦB (CEAi; d)). ˜ h ≤∗ k, l ∀x ˜ ≤∗ i ∃y ∃Y ≤∗ JEA ∀i, It is interpreted by J := λE, A . E, C := λE, A, i . Ai, G := λE, A, k, l . k and F := λE, A, k, l . l. ˜ ≤∗ Eg ∀b ˜ ≤∗ f ΦB (Ag; b) with x := g we get ˜ f ≤∗ k, l ∀x ˜ ≤∗ g ∃y Indeed, from the premise ∀g, ∗ ∗ ∗ ˜ h ≤∗ k, l ∀x ˜ f ≤ k, l ∃y ˜ ≤ Ex ∀b ˜ ≤ f ΦB (Ag; b), so we get the conclusion ∃Y ≤∗ E ∀i, ˜ ≤∗ ∀x, ˜ ≤∗ Y x ∀d ˜ ≤∗ h ΦB (Ai; d)) with Y := E. i ∃y st st ˜ ∀x ∃ y Φ → ∃ z ∀x ∃y ≤∗ z Φ. Its interpretation is equivalent to ˜st H, C, F ∀ ˜st e, a, i ∃ ˜ ≤∗ F eai ∀x ∃y ≤∗ e ∀b ˜ ≤∗ f ΦB (a; b) → (∀f ˜ ≤∗ g ΦB (Cea; d)). ˜ ≤∗ Hea ∀g ˜ ≤∗ i ∀x ∃y ≤∗ z ∀d ∃z It is interpreted by H := λe, a . e, C := λe, a . a and F := λe, a, i . i. ˜st z ∃x ∀y ≤∗ z φ → ∃x ∀st y φ. Its interpretation is ∀ ˜ ≤∗ Ac ∀z ˜st A ∀ ˜st c (∀a ˜ ≤∗ a ∃x ∀y ≤∗ c φ → ∃x ∀b ˜ ≤∗ c ∀y ˜ ≤∗ b φ). ∃ It is interpreted by A := λc . c. st ˜ ˜st y Ψ) → ∃ ˜st z (∀ ˜st x φ → ∃y ˜ ≤∗ z Ψ). Its interpretation is equivalent to (∀ x φ → ∃ ˜st J, C, H, G ∀ ˜st f, a, E, k ∃

 ˜ ≤∗ Gf aEk (∀e ˜ ≤∗ Eg ∀x ˜ ≤∗ f ∀b ˜ ≤∗ g ΨB (a; b)) ˜ ≤∗ e φ → ∃y ∀g

 ˜ ≤∗ Hf ai ∀x ˜ ≤∗ z ∀d ˜ ≤∗ i ΨB (Cf aE; d)) . ˜ ≤∗ Jf aE ∀i ˜ ≤∗ k (∀h ˜ ≤∗ h φ → ∃y ∃z It is interpreted by J := λf, a, E . f , C := λf, a, E . a, H := λf, a, E, i . Ei and G := λf, a, E, k . k. ˜st x φ → ψ) → ∃ ˜st y (∀x ˜ ≤∗ y φ → ψ). Its interpretation is equivalent to (∀ ˜ ≤∗ b ∀x ˜ ≤∗ Cb (∀x ˜ ≤∗ y φ → ψ)). ˜st C ∀ ˜st b ((∀a ˜ ≤∗ a φ → ψ) → (∃y ∃ It is interpreted by C := λb . b.

2

B. Dinis, J. Gaspar / Annals of Pure and Applied Logic 169 (2018) 392–412

406

Theorem 26 (characterisation theorem of B). For all formulas Φ of E-HAω st , we have ω ω ω ω ω ω ω B E-HAω ˜ st + M + BUD + MAJ  Φ ↔ Φ . st + mAC + R + I + IP∀

Proof. The proof is by induction on the length of Φ. The cases of internal atomic formulas and ∧ are easy, so let us focus on the remaining cases. We will sometimes use implicitly Lemma 23. st(t). We have st(t) ↔

(MAJω , st(y) ∧ x ≤∗ y → st(x))

˜st a (t ≤∗ a) ≡ ∃

(definition)

B

st(t) . ∨. We have Φ∨Ψ↔

(induction hypothesis)

˜st c ∀ ˜st a ∀ ˜st b ΦB (a; b) ∨ ∃ ˜st d ΨB (c; d) ↔ ∃

(logic)

˜st b ΦB (a; b) ∨ ∀ ˜st d ΨB (c; d)) ↔ ˜st a, c (∀ ∃

(BUDω )

˜ ≤∗ e ΦB (a; b) ∨ ∀d ˜ ≤∗ f ΨB (c; d)) ≡ ˜st a, c ∀ ˜st e, f (∀b ∃

(definition)

(Φ ∨ Ψ)B . →. We have (Φ → Ψ) ↔

(induction hypothesis)

˜st c ∀ ˜st a ∀ ˜st b ΦB (a; b) → ∃ ˜st d ΨB (c; d)) ↔ (∃

(logic)

˜st

˜st

˜st

˜st

∀ a (∀ b ΦB (a; b) → ∃ c ∀ d ΨB (c; d)) ↔

(IPω ˜ st ) ∀

˜st b ΦB (a; b) → ∃ ˜st c˜ ≤∗ c ∀ ˜st a ∃ ˜st c (∀ ˜st d ΨB (˜ c; d)) ↔ ∀

(monotonicity)

˜st b ΦB (a; b) → ∀ ˜st d ΨB (c; d)) ↔ ˜st a ∃ ˜st c (∀ ∀

(logic)

˜st

˜st

˜st

˜st

∀ a ∃ c ∀ d (∀ b ΦB (a; b) → ΨB (c; d)) ↔

(Mω )

˜st˜b ≤∗ b ΦB (a; b) → ΨB (c; d)) ↔ ˜st a ∃ ˜st c ∀ ˜st d ∃ ˜st˜b (∀ ∀

(mACω )

˜st C, B ∀ ˜st a ∃ ˜st c ≤∗ Ca ∀ ˜st d ∃ ˜st˜b ≤∗ Bad ∃ ˜st b ≤∗ ˜b ΦB (a; b) → ΨB (c; d)) ↔ (∀

(logic, monotonicity)

˜st b ≤∗ Bad ΦB (a; b) → ΨB (Ca; d)) ≡ ˜st a, d (∀ ˜st C, B ∀ ∃

(definition)

(Φ → Ψ) . B

∀. We have ∀x Φ ↔

(induction hypothesis)

∀x ∃ a ∀ b ΦB (a; b) ↔

(Rω )

˜a ≤∗ a ∀ ˜st a ∀x ∃˜ ˜st b ΦB (˜ a; b) ↔ ∃

(monotonicity)

˜ st b ΦB (a; b) ↔ ˜st a ∀x ∀ ∃

(logic)

˜ st

˜st

B. Dinis, J. Gaspar / Annals of Pure and Applied Logic 169 (2018) 392–412

˜st a ∀ ˜st b ∀x ΦB (a; b) ≡ ∃

407

(definition)

B

(∀x Φ) . ∃. We have ∃x Φ ↔

(induction hypothesis)

∃x ∃ a ∀ b ΦB (a; b) ↔

(logic)

˜ st b ΦB (a; b) ↔ ˜st a ∃x ∀ ∃

(Iω )

˜ ≤∗ c ΦB (a; b) ≡ ˜st a ∀ ˜st c ∃x ∀b ∃

(definition)

˜ st

˜st

(∃x Φ)B .

2

In the next result we give some applications of B: 1. the first, second, third, fourth, fifth, seventh and eight points are similar to the applications of b; ˜st x φ → ∃st z ψ) → ∃ ˜st y (∀x ˜ ≤∗ y φ → 2. the sixth point is a term-and-rule variant of a generalisation (∀ st ω ω ∃ z ψ) of M (since M is the particular case in which the tuple z is empty). ω ω ω ω ω ω ω ω Application 27. Let E-HAω ˜ st + M + BUD + MAJ . st + P := E-HAst + mAC + R + I + IP∀ ω 1. The theories E-HAω st + P and E-HAst are equiconsistent. ω st 2. If E-HAω st +P  ∃ x Φ, then there are closed monotone terms t of appropriate types such that E-HAst +P  ω st ∗ ∃ x ≤ t Φ. Moreover, if Φ is internal, then in the conclusion we can replace E-HAst + P by E-HAω st . Analogously, if all quantifications of x are monotone. st st 3. If E-HAω st + P  ∀ x ∃ y Φ, then there are closed monotone terms t of appropriate types such that ω st ∗ ˜ z ∀x ≤ z ∃st y ≤∗ tz Φ. Also, in the conclusion we can replace ∀ ˜st z ∀x ≤∗ z ∃st y ≤∗ tz Φ E-HAst + P  ∀ ˜st x ∃st y ≤∗ tx Φ. Moreover, if Φ is internal, then in the conclusion we can replace E-HAω + P by by ∀ st E-HAω st . Analogously, if all quantifications of x are monotone, or if all quantifications of y are monotone, or if both. ω st 4. If E-HAω st +P  ∀x ∃ y Φ, then there are closed monotone terms t of appropriate types such that E-HAst + ω P  ∀x ∃st y ≤∗ t Φ. Moreover, if Φ is internal, then in the conclusion we can replace E-HAω st +P by E-HAst . Analogously, if all quantifications of x are monotone, or if all quantifications of y are monotone, or if both. st st 5. If E-HAω st + P  ∀ x φ → ∃ y Ψ, then there are closed monotone terms t of appropriate types such that ω st st E-HAst + P  ∀ x φ → ∃ y ≤∗ t Ψ. Moreover, if Ψ is internal, then in the conclusion we can replace ω E-HAω st + P by E-HAst . Analogously, if all quantifications of x are monotone, or if all quantifications of y are monotone, or if both. st st 6. If E-HAω st + P  ∀ x φ → ∃ y ψ, then there are closed monotone terms s, t of appropriate types such ω that E-HAst  ∀st x ≤∗ s φ → ∃st y ≤∗ t ψ. Analogously, if all quantifications of x are monotone, or if all quantifications of y are monotone, or if both. ω ˜st st ˜st st 7. If E-HAω st + P  ∀ x ∃ y φ, then E-HAst  ∀ x ∃ y φ. Analogously, if all quantifications of y are monotone. ω ω 8. If E-HAω st +P is consistent, then there is an instance Φ of LEM such that E-HAst +P  Φ and E-HAst +P  ¬Φ.

Proof. The “Moreover, . . . ” parts are proved analogously to their preceding parts but using Lemma 22 ω instead of the characterisation theorem to remain in E-HAω st instead of going to E-HAst + P. The “Analogously, . . . ” parts are proved analogously to their proceedings parts and sometimes are also a corollary to

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their preceding parts. We will sometimes use implicitly Lemma 23. We will sometimes use the fact that Iω ˜st z ∃x ≤∗ t ∀y ˜ ≤∗ z φ˜ → ∃x ≤∗ t ∀ ˜st y φ˜ (proof sketch: rewrite the premise as ∀ ˜st z ∃x ∀y ≤∗ z (x ≤∗ implies ∀ ∗ ω ∗ ∗ st ∗ ˜ apply I with φ :≡ x ≤ t ∧ (y ≤ y → φ) ˜ getting ∃x ∀ y (x ≤ t ∧ (y ≤∗ y → φ)), ˜ and t ∧ (y ≤ y → φ)), then rewrite this as the conclusion). We will sometimes use implicitly that closed terms are standard. 1. We have E-HAω st + P  0 =0 1 ⇒

(soundness)

E-HAω st  (0 =0 1)B ⇒

(definition)

E-HAω st

 0 =0 1.

ω Trivially, E-HAω st  0 =0 1 ⇒ E-HAst + P  0 =0 1. 2. We have st E-HAω st + P  ∃ x Φ ⇒

(soundness)

st E-HAω st  (∃ x Φ)B (t, s) ⇒

(Lemma 20)

∗

˜st

∗

˜ ≤ d ΦB (s; b) ⇒  ∀ d ∃x ≤ t ∀b

(Iω )

ω ∗ ˜ E-HAω st + I  ∃x ≤ t ∀b ΦB (s; b) ⇒

(t closed)

ω st ∗ ˜ E-HAω st + I  ∃ x ≤ t ∀b ΦB (s; b) ⇒

(s closed monotone)

E-HAω st

E-HAω st

∗

+ I  ∃ x ≤ tΦ ω

st

B

⇒

(characterisation)

st ∗ E-HAω st + P  ∃ x ≤ t Φ.

3. We have st st E-HAω st + P  ∀ x ∃ y Φ ⇒

(soundness)

 (∀ x ∃ y Φ)B (t, s) ⇒

(Lemma 20)

∗ ∗ ∗ ˜st ˜ E-HAω st  ∀ z, d ∀x ≤ z ∃y ≤ tz ∀b ≤ d ΦB (sz; d) ⇒

(Iω )

ω ∗ ∗ ˜st ˜st E-HAω st + I  ∀ z ∀x ≤ z ∃y ≤ tz ∀ b ΦB (sz; d) ⇒

(t closed)

E-HAω st

E-HAω st

˜st

∗

st

st

∗

˜st

+ I  ∀ z ∀x ≤ z ∃ y ≤ tz ∀ b ΦB (sz; d) ⇒

(sz closed monotone)

ω ∗ st ∗ B ˜st ⇒ E-HAω st + I  ∀ z ∀x ≤ z ∃ y ≤ tz Φ

(characterisation)

ω

E-HAω st

st

˜st

∗

∗

+ P  ∀ z ∀x ≤ z ∃ y ≤ tz Φ. st

ω ∗ st ∗ ∗ ˜st ˜st st Also, E-HAω st + P  ∀ z ∀x ≤ z ∃ y ≤ tz Φ ⇒ E-HAst + P  ∀ x ∃ y ≤ tx Φ (by taking x := z). 4. We have

˜st

st E-HAω st + P  ∀x ∃ y Φ ⇒

(soundness)

st E-HAω st  (∀x ∃ y Φ)B (t, s) ⇒

(Lemma 20)

∗

∗

˜ ≤ d ΦB (s; b) ⇒  ∀ d ∀x ∃y ≤ t ∀b

(Iω )

ω ∗ ˜ st E-HAω st + I  ∀x ∃y ≤ t ∀ b ΦB (s; b) ⇒

(t closed)

ω st ∗ ˜ st E-HAω st + I  ∀x ∃ y ≤ t ∀ b ΦB (s; b) ⇒

(s closed monotone)

E-HAω st

B. Dinis, J. Gaspar / Annals of Pure and Applied Logic 169 (2018) 392–412

ω st ∗ B E-HAω ⇒ st + I  ∀x ∃ y ≤ t Φ

E-HAω st

409

(characterisation)

∗

+ P  ∀x ∃ y ≤ t Φ. st

5. We have st st E-HAω st + P  ∀ x φ → ∃ y Ψ ⇒

(soundness)

st st E-HAω st  (∀ x φ → ∃ y Ψ)B (t, r, s) ⇒

(Lemma 20)

∗ ∗ ∗ ˜ ∗ ˜st ˜ E-HAω st  ∀ e (∀c ≤ se ∀x ≤ c φ → ∃y ≤ t ∀b ≤ e ΨB (r; b)) ⇒

(se standard)

st ∗ ˜ ∗ ˜st E-HAω st  ∀ x φ → ∀ e ∃y ≤ t ∀b ≤ e ΨB (r; b)) ⇒

(Iω )

ω st ∗ ˜ st E-HAω st + I  ∀ x φ → ∃y ≤ t ∀ b ΨB (r; b)) ⇒

(t closed)

ω st st ∗ ˜ st E-HAω st + I  ∀ x φ → ∃ y ≤ t ∀ b ΨB (r; b)) ⇒

(r closed monotone)

ω st st ∗ B E-HAω ⇒ st + I  ∀ x φ → ∃ y ≤ t Ψ

(characterisation)

st st ∗ E-HAω st + P  ∀ x φ → ∃ y ≤ t Ψ.

6. We have st st E-HAω st + P  ∀ x φ → ∃ y ψ ⇒

(soundness)

st E-HAω st  (∀ x φ → ψ)B (t, s; ) ⇒

(Lemma 20)

∗ ∗ ∗ ˜ E-HAω st  ∀a ≤ s ∀x ≤ a φ → ∃y ≤ t ψ ⇒

(s, t closed monotone)

st ∗ st ∗ E-HAω st  ∀ x ≤ s φ → ∃ y ≤ t ψ.

7. Follows from point 3. ˜ be as in point 8 in the proof of Application 18. 8. Let φ, Φ and Φ E-HAω st + P  Φ. We have E-HAω st + P  Φ ⇒

(logic)

˜ E-HAω st + P  ∀x, y Φ ⇒

(soundness)

˜st ˜, y˜, z˜ (∀st x, y Φ) ˜ B (t; x E-HAω ˜, y˜, z˜) ⇒ st  ∀ x

(Lemma 9)

˜x, y˜, z˜ ∀x, y ≤0 x E-HAω ˜, y˜ st  ∀˜ (∃z ≤0 t˜ xy˜ φ(x, y, z) ∨ ∀z ≤0 z˜ ¬φ(x, y, z)) ⇒

(logic, arithmetic)

st st st E-HAω st  ∀ x, y (∃ z ≤0 txy φ(x, y, z) ∨ ∀ z ¬φ(x, y, z)).

If E-HAω st + P  Φ, then there is a term t as above and t induces a computable function that can be used to solve the halting problem, a contradiction. ω ω E-HAω + P  ¬Φ. If E-HAω st st + P  ¬Φ, then E-HAst + P is inconsistent because E-HAst  ¬¬Φ. 2 5. Transfer Nonstandard theories usually include a transfer principle for two reasons:

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410

1. transfer connects the standard and the nonstandard universes [4]; 2. transfer implies that all uniquely defined objects are standard [12, page 1166]. The price to pay for transfer is: 1. we can only apply transfer to internal assertions with standard parameters; 2. transfer may lead to nonconservativity and inconsistency. The following two principles are inspired by or even taken from Nelson’s article [12]. Definition 28. We define the following principles: 1. the universal transfer principle T∀ is all instances of ∀st f (∀st x φ → ∀x φ); 2. the existential transfer principle T∃ is all instances of ∀st f (∃x φ → ∃st x φ); where f are all free variables in the internal formula φ. Let us see that transfer may lead to nonconservativeness or inconsistency (one solution is to consider st φ instead the rules ∀∀xxφφ and ∃∃x st x φ [2, proposition 5.12] [11]). This intuitionistic result (and part of its proof) is an intuitionistic variant of Ferreira and Gaspar’s similar classical result (and part of its proof) [6, appendix A]. We refer to Van den Berg, Briseid and Safarik’s article [2] for the notation E-HAω∗ st , R, HGMPst and US0 used below. Theorem 29. st 1. Adding T∀ or T∃ to E-HAω∗ st + R + HGMP leads to nonconservativity over HA. ω 2. Adding T∀ or T∃ to E-HAst leads to inconsistency.

Proof. st ω∗ 1. The theory E-HAω∗ st + R + HGMP proves US0 [2, proposition 5.11]. The theory E-HAst + US0 plus T∀ or T∃ is nonconservativeover HA [1] [2, page 1973]. <0 n 2. Let ¬st(n0 ) and t1 x := 10 ifif xx ≥ . It follows from st(λx0 . 1) and t ≤∗ λx0 . 1 that st(t). The internal 0 n

formula φ :≡ tx =0 0 is such that ∀st x φ and ¬∀x φ, contradicting T∀ . The internal formula ψ :≡ (tx =0 1) ∧ ∀y < x (ty =0 0) (notice ψ ↔ x =0 n) is such that ∃x ψ and ¬∃st x ψ, contradicting T∃ . 2 6. Recovering standard interpretations As mentioned in the introduction, if we restrict our realisability b to the “purely external fragment” (where there are only quantifiers of the form ∃st and ∀st ), then we recover Ferreira and Nunes’s bounded modified realisability, and if we restrict our functional interpretation B to the “purely external fragment”, then we recover Ferreira and Oliva’s bounded functional interpretation. In this section we sketch a proof of these results, actually focusing only on the second one because it is the more difficult one and the first result is perfectly analogous. Our results and proof are similar to Ferreira and Gaspar’s result and proof [6, section 4]. For brevity, we need to assume familiarity with Ferreira and Oliva’s theory HAω  [8, definition 5] and Ferreira and Oliva’s bounded functional interpretation B [8, definition 4]. To state the results, we need to introduce our variant of Ferreira and Gaspar’s diamond translation [6, page 707].

B. Dinis, J. Gaspar / Annals of Pure and Applied Logic 169 (2018) 392–412

411

ω  Definition 30. The diamond translation  assigns to each formula φ of HAω  the formula φ of E-HAst accordingly to the following clauses. For atomic formulas, we define:

1. (s =0 t) :≡ s =0 t; 2. (s σ t) :≡ s ≤∗σ t. For the remaining formulas, we define: 3. (φ ◦ ψ) :≡ φ ◦ ψ  for ◦ ∈ {∨, ∧, →}; 4. (x  t φ) :≡ x ≤∗ t φ for  ∈ {∀, ∃} ; 5. (x φ) :≡ st x φ for  ∈ {∀, ∃} . Now we state and sketch the proof of our factorisation B  =  B along the lines of Ferreira and Gaspar’s similar factorisation [6, proposition 1]. The factorisation has two formulations: the first one is less clear but  (essentially) informs that the terms witnessing φB and (φ )B are exactly the same, and the second is more clear but less informative. Theorem 31. For all formulas φ of HAω  , we have:   1. E-HAω st  φB (a; b) ↔ (φ )B (a; b);  ω 2. E-HAst  (φB ) ↔ (φ )B .

Proof. 1. The proof is almost a simple induction on the length of φ. For example, in the case of ∃ (the most difficult one), we have (∃x φ)B (c, a; d) ≡

(definition of B )

˜  d φB (a; b)) ≡ (∃x  c ∀b

(definition of )

˜ ≤∗ d φB (a; b) ↔ ∃x ≤∗ c ∀b

(induction hypothesis)

∗

∗



˜ ≤ d (φ )B (a; b) ↔ ∃x ≤ c ∀b

(Lemma 20)

(∃st x φ )B (c, a; d) ≡

(definition of )



((∃x φ) )B (c, a; d). 2. We have 

(φB ) ≡

(definition of B )

˜ φB (a; b)) ≡ ˜ ∀b (∃a

(definition of )

˜st a ∀ ˜st b φB (a; b) ↔ ∃

(point 1)

˜st

˜st



∃ a ∀ b (φ )B (a; b) ↔ (φ )B .

(definition of B) 2

References [1] Jeremy Avigad, Jeffrey Helzner, Transfer principles in nonstandard intuitionistic arithmetic, Arch. Math. Logic 41 (6) (August 2002) 581–602.

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[2] Benno van den Berg, Eyvind Briseid, Pavol Safarik, A functional interpretation for nonstandard arithmetic, Ann. Pure Appl. Logic 163 (12) (December 2012) 1962–1994. [3] Marc A. Bezem, Strongly majorizable functionals of finite type: a model for bar recursion containing discontinuous functionals, J. Symbolic Logic 50 (3) (September 1985) 652–660. [4] Francine Diener, Marc Diener (Eds.), Nonstandard Analysis in Practice, Universitext, Springer-Verlag, Berlin, Germany, 1995. [5] Bruno Dinis, Fernando Ferreira, Interpreting weak Kőnig’s lemma in theories of nonstandard arithmetic, MLQ Math. Log. Q. 63 (1–2) (April 2017) 114–123. [6] Fernando Ferreira, Jaime Gaspar, Nonstandardness and the bounded functional interpretation, Ann. Pure Appl. Logic 166 (6) (June 2015) 701–712. [7] Fernando Ferreira, Ana Nunes, Bounded modified realizability, J. Symbolic Logic 71 (1) (March 2006) 329–346. [8] Fernando Ferreira, Paulo Oliva, Bounded functional interpretation, Ann. Pure Appl. Logic 135 (1–3) (September 2005) 73–112. [9] William A. Howard, Hereditarily majorizable functionals of finite type, in: Anne S. Troelstra (Ed.), Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, in: Lecture Notes in Mathematics, vol. 344, Springer-Verlag, Berlin, Germany, August 1973, pp. 454–461. [10] Richard Kaye, Models of Peano Arithmetic, Oxford Logic Guides, vol. 15, Oxford University Press / Clarendon Press, January 1991. [11] Ieke Moerdijk, A model for intuitionistic non-standard arithmetic, Ann. Pure Appl. Logic 73 (1) (May 1995) 37–51. [12] Edward Nelson, Internal set theory: a new approach to nonstandard analysis, Bull. Amer. Math. Soc. 83 (6) (November 1977) 1165–1198. [13] Edward Nelson, The syntax of nonstandard analysis, Ann. Pure Appl. Logic 38 (2) (May 1988) 123–134.