The Herbrand Topos

The Herbrand T op os Benno v an den Berg April 18, 2013 Abstract W e define a new topos, the Herbr and top os , in sp i red by the mo dified realizabilit y top o s and our earl ier work on Herbrand realiza bility . W e also introduce the category of Herb ra nd assembli es and c h a racterise these as the ¬¬ -separated ob jects in t h e H erb ra nd topos. In addition, w e show that the category of sets is included as the category of ¬¬ -sheav es and prov e that the inclusion functor preserves and reflects v alidit y of first-order form ulas. 1 In tro duction In [2] t he author, together with Eyvind Br iseid and Pa vol Safar ik, hit up on a new realizability interpretation in an attempt to find computational con tent in arguments perfor m ed in nonstandard analysis. This new interpretation, which was a v ariant of mo dified realizability , was dubb ed Herbrand realizability . Our inv estig ations in [2] were entirely proo f-theoretic; the question w as whether it would b e possible to un derstand Herbrand realizability from a seman tic point of view a s well. This pap er sho ws that that is indeed the case. T o develop this semantics we use top os theory (for whic h see [12, 8, 9]). This choice w as motiv ated b y the fact that the notion of a top os is the mo st comprehensive notion of mo del for a constr uctiv e system we have av a ilable, in- corp orating top ological, sheaf and Kripke mo dels, as w ell as v arious realizability and functional interpretations. In addition, it shows that these interpretations can be made to work for full hig her-order ar ithm etic. The starting po in t fo r this pap er was the theory of realizability top oses (b eginning with [7] a nd survey ed in [15]): indeed, the topo s mo st closely related to t he top os w e will in tro duce here is the mo dified realizability top o s (for whic h, see [14, 1 5 ]). In order to arrive at the modified rea lizabilit y top os, o ne has to abstract considerably from Kreisel’s original definition [11]. First of all, one fixes the hereditarily effectiv e o p erations (HEO) as a mo del of G¨ odel’s T . Then a type gets iden tified with a ce r tain inhabited set o f co des and a s et of realizers of that 1 t yp e will simply b e subset of that set. The step that Grayson to ok in [6] was to take as truth v alues an y pair ( A 0 , A 1 ) where A 0 and A 1 are t wo sets of codes (often called the actual r ealizers and the potential realizer s, respe c tively) with A 0 ⊆ A 1 and A 1 containing a fixed element. One can build a tripo s around such pairs and in the asso ciated topo s the finite types will b e interpreted as the hereditarily effective op erations. In order to define the Her brand top os, we make a similar mov e. Recall from [2] that in or der to realize ( ∃ st n ∈ N ) ϕ ( n ) it suffices to supply a finite lis t of natural num b ers ( n 1 , . . . , n k ) such that ϕ ( n i ) is r ealized for some i ≤ k . Abstra cting a wa y from the de ta ils, this means that po ten tial re a lizers are finite lists of natural n umbers, while the actual realiz- ers are those finite lists ( n 1 , . . . , n k ) which con tain an n i which works (this is similar to th e idea of Herbra nd dis junctions in pro of theory; hence the name). Abstracting even f urther, w e say t hat truth v alues in the Herbrand topos ar e pairs of sets of co des ( A 0 , A 1 ) such that A 0 consists of finite sequences a ll who se elements b elong to A 1 and which is closed up wards (by this we mean that it is closed under super sets, if w e reg ard finite sequences a s r epresen tatives for their set of comp onen ts; see a lso Lemma 5.4 in [2]). This pap er sho ws that on the basis of these pair s one can construct a tripos, whose ass o ciated top os we will call the Her brand top os. The Her br and to p os turns out to have se veral fea tu res in co mmon with other realizability to poses. It ha s an interesting subca tegory consis tin g of the ¬¬ - separated ob jects (we will call these the Her br and assemblies) and the catego ry of sets is included as a subtop os via the ¬¬ -top ology . What is v ery unusual, how ever, is tha t this inclusion functor , whic h we will call ∇ , preser ves and reflects the v alidity o f first-o r der logic; in fact, ∇ pre serv es and reflects the structure of a lo cally cartesian closed pr etopos. In particular , ∇ 2 = 2 in the Herbrand topo s. This is a striking illustratio n of the fact that in the Her br and top os disjunc- tion has essentially no cons tr uctiv e con tent. Indeed, in or der for a disjunction ϕ ∨ ψ t o be realized it is sufficien t that one of the tw o disjuncts is realized; but a realizer f or ϕ ∨ ψ need not say whic h disjunct it is tha t is actually realized. This fact explains many of the f eatures of the Herbrand top os: why it believes in the law of excluded middle for Π 0 1 -formulas, and wh y it does not b elieve in Chu rch’s thesis or in cont inuit y principles. How ever, ar ithm etic in the Herbra nd top os is not classica l. This is due to the fact that existential quantifiers still have s ome construc tive conten t. Admittedly , this cont ent is less than is usually the case, but it is still s trong enough to rule out Marko v’s principle. Finally , there ar e t wo other prop erties of the Herbra nd top os which a re worth men tioning. First of a ll, b ecause ∇ preserves and reflec t s fir st-order logic, ∇ A will be a nonstandard mo del of arithmetic in the Herbrand top os for 2 any nonstandard mo del A (actually , it will also be a nonstandard mo del when A is the standard model). This is in teresting, b ecause realiza bilit y top oses are unfav o ur able terrain for nonsta ndard mo dels of arithmetic (see [13]). A pro of-theoretic feature of the Herbrand top os which is w orth stressing is that in it the F an Theo r em holds (see Pr o position 7.10 below); for this it is not nece s sary to assume its v alidity in the metatheo r y . The pro of should lo ok very familiar to an yone who is aw are of the b ounded mo dified realiza bilit y int erpreta t ion and its pro perties (for which, s e e [5]). The cont ents of the pa per are as follows. In Section 2 w e will explain the notation that we will use, in so far as it is not standard. W e define the Herbrand trip os and top os in Section 3. Section 4 in tro duces the Herbrand assemblies and prov es that they form a lo cally cartes ia n clo sed regular categor y with finite stable colimits a nd a natural nu mbers ob ject. Then, in Section 5, we characterise these Herbrand assemblies a s the ¬¬ -separa t ed ob jects in the Herbrand top os; in addition, we sho w that the category of sets is included as the ¬¬ -sheaves and that the inclusion functor preserves and reflects the s tr ucture of a loca lly cartesian closed pretop o s. In Sec tio n 6 , we c haracteris e the pro jectives in the Herbrand assem blies and sho w that the natural n um b ers o b ject is pr o jectiv e in the category of Herbrand assemblies, but not in the Herbrand topos. Section 7 studies the lo g ical prop erties of the Herbra nd topo s and, fina lly , in Section 8 we discuss some fur ther developments. In this paper w e a s sume familiarit y with the theor y of tr ip oses and partial combinatro y a lgebras; for the necessar y background informatio n, we r ecommend [15]. W e w ould like to thank Jaa p v an Oos ten, W outer Stekelen burg and the referee for us e f ul comments. The author was supp orted by a gra n t from the Netherlands Organisatio n for Scientific Resear c h (NW O ). 2 Notation Throughout this pa per, P will be a fixed nontrivial partial combinatory algebra (pca ). It is well kno wn that w e can code finite sequences of elemen ts in P a s elements of P . If n is such a co de, then we write | n | for the length of the sequence it co des and n i for the i th pr o jection (where n 1 is the first ele m ent of the sequence and n | n | the last). W e will write h n 1 , . . . , n k i f or the code of the sequence ( n 1 , . . . , n k ) and hi fo r the co de of the empty sequence . Moreov er , ∗ will denote the op eration of conca tenation o f co ded sequences and a 1 ∗ a 2 ∗ . . . ∗ a k will stand for the result of conc a tenating a 1 till a k . If k = 0 (i.e., if we are taking the empt y conca tenation), then the result should b e the co de of the empt y seq uence hi . If A ⊆ P is some s ub set of P , then w e will wr ite ! A for the set of co des 3 of finite seq uences all whose elements b elong to A (note that this will alw ays include the empt y se q uence). F or our purp oses it will b e imp ortant that ! A carries a pr eorder structure with m  n if ( ∀ i ≤ | m | ) ( ∃ j ≤ | n | ) m i = n j . W e will write p for the pa iring o p erator and denote the n th Church numeral simply b y n . In addition, if A and B are subsets of P , we will write A & B = { p 0 a : a ∈ A } ∪ { p 1 b : b ∈ B } , A ⊗ B = { p ab : a ∈ A, b ∈ B } . Observe that there is an exp onen tial isomorphism !( A & B ) ∼ = ! A ⊗ ! B , where bo th the map itself and and its inv erse are or de r -preserving and repr e- sented by elements in the pca, independent o f the sp ecific A and B . W e will often implicitly use this iso morphism and reg ard elements of !( A & B ) as pa irs p xy with x ∈ ! A and y ∈ ! B . 3 The Herbrand trip os The Herbr and to pos will b e o btained fr om the Herbr and trip os . T o construc t this tripo s first put Σ = { ( A 0 , A 1 ) : A 0 , A 1 ⊆ P , A 0 ⊆ ! A 1 and A 0 is up wards closed in ! A 1 } , where A 0 being upw ards closed in ! A 1 means that m ∈ A 0 , n ∈ ! A 1 and m  n imply n ∈ A 0 . F or X ∈ Σ, we will denote the res ult of pro jecting on the first and second co ordinate by X 0 and X 1 . As for mo dified rea lizabilit y , we will refer to the e lemen ts of X 0 as the actual r e alizers and the elemen ts of ! X 1 as the p otential re alizers . If X is any s et, then we preo rder Σ X as follo ws: ( φ : X → Σ) ≤ ( ψ : X → Σ) if there is an element r ∈ P such that for all x ∈ X and n ∈ P if n ∈ ! φ ( x ) 1 , then r · n ↓ and r · n ∈ ! ψ ( x ) 1 and if n ∈ φ ( x ) 0 , then r · n ∈ ψ ( x ) 0 . Finally , if f : X → Y is a n y function, then reindexing f ∗ : Σ Y → Σ X is giv en simply b y precomp osition. Theorem 3.1. The indexe d pr e or der Σ X define d ab ove is a trip os. 4 The a s sociated top os we will call the Herbrand topo s and deno te by HT [ P ]. Pr o of. W e first verify tha t Σ ha s the structure of a Heyting pr ealgebra. The top and b ottom element are (! P , P ) a nd ( ∅ , ∅ ), resp ectiv ely . The c o njun ction ( C 0 , C 1 ) = ( A 0 , A 1 ) ∧ ( B 0 , B 1 ) is given b y C 1 = A 1 & B 1 , C 0 = { p ab : a ∈ A 0 and b ∈ B 0 } (making use of the exp onen tial is omorphism). The disjunction ( C 0 , C 1 ) = ( A 0 , A 1 ) ∨ ( B 0 , B 1 ) is giv en b y C 1 = A 1 & B 1 , C 0 = { p ab : a ∈ A 0 or b ∈ B 0 } (again making use of the exp onen tial isomorphism). T o see that this works, suppo se that ( A 0 , A 1 ) ≤ ( D 0 , D 1 ) is tra c ked by r and ( B 0 , B 1 ) ≤ ( D 0 , D 1 ) is track ed by s . Then ( C 0 , C 1 ) ≤ ( D 0 , D 1 ) is track ed by the following map t : t ( p ab ) = r a ∗ sb . This works, beca use for p ab ∈ C 0 we ha ve a ∈ A 0 or b ∈ B 0 : in the for mer case , we hav e ra ∈ D 0 , in the la tt er sb ∈ D 0 . In either case we hav e ra ∗ sb bec ause D 0 is up wards closed. The implica tion ( C 0 , C 1 ) = ( A 0 , A 1 ) → ( B 0 , B 1 ) is giv en b y C 1 = { c ∈ P : ( ∀ m ∈ ! A 1 ) c · m ↓ and c · m ∈ ! B 1 } , C 0 = ↑ ! C 1 {h c i : c ∈ C 1 and ( ∀ m ∈ A 0 ) c · m ∈ B 0 } . T o se e that this works, note that the ev alua tion map ( C 0 , C 1 ) ∧ ( A 0 , A 1 ) → ( B 0 , B 1 ) is tra c ked by the function which maps p ( h c 1 , . . . , c k i , a ) to c 1 ( a ) ∗ . . . ∗ c k ( a ) . It is clea r that this Heyting pr e algebra structure lifts to ea c h Σ X . As said, the re ind exing functors a re given by precomp osition. W e now chec k that these hav e both a djoin ts sa tis fying the Beck-Chev alley condition. So sup- po se we have φ : X → Σ , χ : Y → Σ and f : X → Y . The existential qua n tifier can be defined as follo ws: ∃ f ( φ )( y ) 1 = [ x ∈ f − 1 ( y ) ! φ ( x ) 1 , ∃ f ( φ )( y ) 0 = ↑ ! ∃ f ( φ )( y ) 1 {h n i : ( ∃ x ∈ f − 1 ( y )) n ∈ φ ( x ) 0 } . T o see this, note that 1. we hav e ∃ f ( φ ) ≤ χ if there is an r ∈ P sending for each y ∈ Y elements from ! ∃ f ( φ )( y ) 1 to ele ments in ! χ ( y ) 1 , in such a way that if m ∈ ∃ f ( φ )( y ) 0 , then r · m ∈ χ ( y ) 0 . 5 2. And that w e hav e φ ≤ f ∗ ( χ ), if there is an s ∈ P sending for eac h x ∈ X elements in ! φ ( x ) 1 to element s in ! χ ( f x ) 1 , in s uc h a wa y that if m ∈ φ ( x ) 0 , then s · m ∈ χ ( f x ) 0 . T o construct such an s from an r , put s ( m ) = r ( h m i ) and to c o nstruct such an r fro m an s , put r ( h m 1 , . . . , m k i ) = s ( m 1 ) ∗ . . . s ( m k ) . The universal quan tifier is constructed as follows: ∀ f ( φ )( y ) 1 = { a ∈ P : ( ∀ x ∈ f − 1 ( y )) ( ∀ b ∈ P ) a · b ↓ and a · b ∈ ! φ ( x ) 1 } , ∀ f ( φ )( y ) 0 = ↑ ! ∀ f ( φ )( y ) 1 {h a i : a ∈ ∀ f ( φ )( y ) 1 and ( ∀ x ∈ f − 1 ( y )) ( ∀ b ∈ P ) a · b ∈ φ ( x ) 0 } . T o see this, note that 1. we ha ve f ∗ ( χ ) ≤ φ if there is an r ∈ P sending for each x ∈ X elements from ! χ ( f x ) 1 to element s in ! φ ( x ) 1 , in such a way that if m ∈ χ ( f x ) 0 , then r · m ∈ φ ( x ) 0 . 2. And that we hav e χ ≤ ∀ f ( φ ), if there is an s ∈ P sending for ea c h y ∈ Y elements in ! χ ( y ) 1 to elements in ! ∀ f ( φ )( y ) 1 , in s uch a wa y that if m ∈ χ ( y ) 0 , then s · m ∈ ∀ f ( φ )( y ) 0 . T o construct such an s from an r , put s = λp. h λq .r · p i and to c o nstruct such an r fro m an s , put r = λp.  ( s · p ) 0 ( k ) ∗ . . . ∗ ( s · p ) | s · p | ( k )  . It is easy to see that the universal quan tifier, a nd hence also the the existen tial quantifier, satisfies Beck-Chev alley . Finally we need to construct a generic e le ment; but that can quite straight- forwardly b e taken to b e the identit y map id : Σ → Σ. 4 Herbrand assem blies In this sectio n w e in tro duce the Herbrand a ssem blies and prov e that they for m a lo cally cartesia n closed r egular categ o ry with stable c olimits. La ter we will 6 characterise them a s the ¬¬ -separ ated ob jects in the Herbrand top os. Of course, it follows from this that the categ ory is a regular lo cally ca r tesian closed category with stable co limit s, but w e need an explicit descriptio n of this s t ructure later; in addition, s uch a des cription is, w e be liev e, of indepe ndent in terest. Definition 4. 1. A Herbr and assembly ov er P is a triple ( A, A , α ) in whic h • A is a set, • A is a subset of P , and • α : A → Pow upcl i (! A ) is a function whose co domain Po w upcl i (! A ) consists o f the subsets X of ! A that are inhabited and upw ards closed in ! A . A morphism f : ( B , B , β ) → ( A, A , α ) of Herbr and asse m blies is a function f : B → A for whic h there is an n ∈ P such that • for all m ∈ ! B , the expression n · m is de fined and its v alue b elongs to ! A , • and if b ∈ B and m ∈ β ( b ) , then n · m ∈ α ( f b ). W e will say tha t such an n t r acks f or is a tr acking of f . This clearly defines a category : we will denote it by HA sm [ P ]. Lemma 4. 2. The c ate gory H A sm [ P ] has finite limits. Pr o of. The terminal o b ject is (1 , C , γ ) with C = P and γ ( ∗ ) =! P . Equalizers as in ( C, C , γ ) / / / / ( B , B , β ) f / / g / / ( A, A , α ) can be computed by putting C = { b ∈ B : f b = g b } , C = B and γ = β ↾ C . A pro duct ( C, C , γ ) = ( A, A , α ) × ( B , B , β ) can be constructed by putting C = A × B , C = A & B , γ ( a, b ) = { p a b : a ∈ α ( a ) , b ∈ β ( b ) } . Lemma 4. 3. The c ate gory H A sm [ P ] is r e gu lar. Pr o of. W e claim that a map f : ( B , B , β ) → ( A, A , α ) is monic pr e cisely when the underlying function f is injectiv e (this sho uld b e cle ar), and is a co ver precisely when there is a n element r ∈ P suc h that 7 • for all n ∈ ! A the expr ession r · n is defined a nd b elongs to ! B , and • there is for any a ∈ A and n ∈ α ( a ) an element b ∈ f − 1 ( a ) with r · n ∈ β ( b ). Let us call maps which hav e these tw o prop erties sup er epis . It is not hard to chec k that sup er epis are c o vers and that they are stable under pullbac k, so the proof will b e finishe d once w e show that every map can b e factor ed as a sup e r epi fo llo wed by a mono. But if f : ( B , B , β ) → ( A, A , α ) is any map, then it factor s thro ug h ( C, C , γ ) with C = Im( f ), C = B and γ ( a ) = S b ∈ f − 1 ( a ) β ( b ). Moreov er, the o bvious ma ps ( B , B , β ) → ( C, C , γ ) and ( C, C , γ ) → ( A, A , α ) ar e a super epi and a mo no, resp ectiv ely . Lemma 4. 4. The c ate gory H A sm [ P ] has stable sums and c o e qualizers. Pr o of. The initial ob ject is (0 , P , γ ) with γ the empt y map. The sum ( C , C , γ ) = ( A, A , α ) + ( B , B , β ) is giv en by C = A + B , C = A & B , with γ ( a ) = { p xy : x ∈ α ( a ) , y ∈ ! B } and γ ( b ) = { p xy : x ∈ ! A , y ∈ β ( b ) } . A co equalizer ( B , B , β ) f / / g / / ( A, A , α ) q / / / / ( C, C , γ ) is computed by letting C be the co equalizer in the catego r y of sets, C = A and γ ( c ) = S a ∈ q − 1 ( c ) α ( a ). Lemma 4. 5. The c ate gory H A sm [ P ] is lo c al ly c artesian close d. Pr o of. Assume f : ( B , B , β ) → ( A, A , α ) a nd g : ( S, S , σ ) → ( B , B , β ) are tw o morphisms of Herbrand asse mblies. The ob ject Q f ( g ) = ( T , T , τ ) is computed as follo ws: T = { ( a ∈ A, t : B a → S ) : g t = id B a and t is tra c ked } , T = A & { n ∈ P : ( ∀ m ∈ ! B ) n · m ↓ a nd n · m ∈ ! S } , τ ( a, t ) = { p ( m, h n 1 , . . . , n k i ) : m ∈ α ( a ) and there is an n i tracking t } . The ev aluation ma p f ∗ Q f ( g ) → g : (( a, t ) , b ) 7→ t ( b ) is tracked b y a co de for the function L defined by L ( p ( h n 1 , . . . , n k i , m )) = n 1 ( m ) ∗ . . . ∗ n k ( m ) . Lemma 4. 6. The c ate gory H A sm [ P ] has a natura l numb ers obje ct (n no). 8 Pr o of. The nno is given by ( N , N , ν ) with N the collection of Ch urch numerals in P and ν ( n ) = ↑ ! N h n i . F or supp ose a structure 1 → ( A, A , α ) → ( A, A , α ) is giv en a nd the first map is track ed by p and the second by q . Define s by recursion as s 0 = p 0 and s ( n + 1) = q ( s ( n )), and t as t ( n ) = s ( n 0 ) ∗ . . . ∗ s ( n | n | ). Then the canonical map N → A will b e tra c ked by t . T o summarise: Theorem 4.7. The c ate gory HA sm [ P ] is a lo c al ly c artesian close d r e gu lar c at- e gory with nn o and st a ble c olimits. 5 Gamma and nabla In this s e c t ion we show that s ome of the theory developed for mo dified rea liz- ability also applies to Herbrand realiza bility . In particular, we show that the facts es tablished on pages 281 a nd 28 2 of V an Oos ten’s paper on the mo dified realizability to pos [14] ho ld for the Herbr and top os as well. (W a rning: W e follow the notation of that pap er, r ather than that of [15].) First of all, note that HA sm [ P ] is a full sub c a tegory of H T [ P ], b ecause every triple ( A, A , α ) can b e considered as an ob ject ( A, =) of HT [ P ] as follows: J a = a ′ K =  ( α ( a ) , A ) if a = a ′ ( ∅ , A ) otherwise In addition, w e hav e a functor ∇ : S e ts → HA sm [ P ] w hich sends a set X to the Herbr and assembly ( X, P , φ ) with φ ( x ) =! P for all x ∈ X . T a k ing the comp osition o f these tw o functors we obtain a functor S ets → HT [ P ] which we will also deno te by ∇ . Remark 5. 1. This is n o t the cons ta n t o b jects functor as defined in [15] (and denoted ∇ there). Prop osition 5 .2. The fun ct o r ∇ has a finite limit pr eserving left adjoint Γ: HT [ P ] → S ets. Mor e over, Γ ∇ ∼ = 1 . Pr o of. As usual, w e send an ob ject ( X , =) to X 0 / ∼ , w her e X 0 = { x ∈ X : J x = x K 0 inhabited } with x ∼ x ′ if J x = x ′ K 0 inhabited. In additio n, the transp ose of a function f : Γ( X , = ) → Y is the function ( X , =) → ∇ Y represented by: F ( x, y ) =  J x = x K if x ∈ X 0 and f ([ x ]) = y , ( ∅ , J x = x K 1 ) other wise. Note that ther e fo re the unit η : ( X , =) → ∇ Γ( X , =) is represented by H : X × X 0 → Σ with H ( x, [ x ′ ]) = J x = x K if x ∈ [ x ′ ], and ( ∅ , J x = x K 1 ) otherwise. 9 Lemma 5. 3. Le t ( A 0 , A 1 ) ∈ Σ . Then A 0 is inhabite d iff ( ¬¬ ( A 0 , A 1 )) 0 is inhabite d; in which c ase, h λp. hii ∈ ( ¬¬ ( A 0 , A 1 )) 0 . Pr o of. This follows from the fact that ¬ ( A 0 , A 1 ) = ( C 0 , C 1 ), where C 1 is the set of co des o f functions which map elements fr om ! A 1 to the empty s equence, and C 0 =! C 1 − { hi} if A 0 is empt y , and C 0 = ∅ otherwise. Hence λp. hi ∈ C 1 alwa ys and h λp. hii ∈ C 0 if A 0 is empt y . Prop osition 5.4. F or an obje ct ( X , =) in HT [ P ] the fol lo wing ar e e quivalent: 1. η ( X, =) is a monomorphism. 2. ( X , = ) is ¬¬ -sep ar ate d. 3. ( X , = ) is isomorphic t o a Herbr and assembly. Pr o of. 1 ⇒ 2: Supp ose η is mono; so H ( x, [ z ]) ∧ H ( x ′ , [ z ]) → x = x ′ holds. Suppo se h a 1 , . . . , a n i is an a ctual r ealizer for this. F urthermo r e, suppos e b ∈ J x = x K 0 , c ∈ J x ′ = x ′ K 0 and d ∈ ¬¬ J x = x ′ K 0 . T he n b ∈ H ( x, [ x ]) 0 , c ∈ H ( x ′ , [ x ′ ]) 0 and [ x ] = [ x ′ ] b y the pr evious lemma, so a 1 ( p bc ) ∗ . . . ∗ a n ( p bc ) ∈ J x = x ′ K 0 ; similar for potential rea lizers. So x = x ∧ x ′ = x ′ ∧ ¬¬ ( x = x ′ ) → x = x ′ holds and ( X, =) is ¬¬ -separated. 2 ⇒ 3: Supp ose t is a n actual r ealizer of x = x ∧ x ′ = x ′ ∧ ¬¬ ( x = x ′ ) → x = x ′ . Define ( A, A , α ) b y A = X 0 , A = [ x,x ′ ∈ X ! J x = x ′ K 1 , α ( a ) = ↑ ! A {h s i : ( ∃ x, x ′ ∈ a ) s ∈ J x = x ′ K 0 } . One easily s ees that there is a map F : ( X , =) → ( A, A , α ) defined b y F ( x, a ) =  J x = x K ∧ ( α ( a ) , A ) if x ∈ a, J x = x K ∧ ( ∅ , A ) otherwise. T o see that F ( x, a ) ∧ F ( x ′ , a ) → x = x ′ is r ealized (i.e., that F is monic), one uses that if ther e a re actua l for r ealizers F ( x, a ) of F ( x ′ , a ), then J x = x ′ K 0 m ust be inhabited; but then one can compute an element in this set using t and the previous lemma. 10 T o show that a = a → ( ∃ x ∈ X ) F ( x, a ) is r ealized (i.e., to show that F is epic), one needs an elemen t s ∈ P which uniformly in a ∈ A sends elements from ! A to elements in ! [ x ∈ X ! F ( x, a ) , in suc h a way that if m ∈ α ( a ), then s · m ∈↑ {h n i : ( ∃ x ∈ X ) n ∈ F ( x, a ) 0 } . This can b e done: for if k = h k 1 , . . . , k n i ∈ ! A , then each k i belo ngs to some ! J x = x ′ K 1 . Since this eq ualit y is symmetric and tra nsitiv e , we can compute from k i an element tk i ∈ ! J x = x K 1 . Then p ( tk i , k ) ∈ ! F ( x, a ) 1 and hence h p ( tk i , k ) i i ∈ ! S x ∈ X ! F ( x, a ). If h k 1 , . . . , k n i a lso b elongs to α ( a ), then some k i belo ngs to J x = x ′ K 0 with x, x ′ ∈ a . Then p ( tk i , k ) ∈ F ( x, a ) 0 and hence s · m ∈↑ {h n i : ( ∃ x ∈ X ) n ∈ F ( x, a ) 0 } , as desir ed. The implica tion 3 ⇒ 1 is left to the reader. Prop osition 5.5. F or an obje ct ( X , =) in HT [ P ] the fol lo wing ar e e quivalent: 1. η ( X, =) is an isomorphism. 2. ( X , = ) is isomorphic t o an obje ct of the form ∇ Z . 3. ( X , = ) is a ¬¬ -she af. Pr o of. 1 ⇒ 2 is trivial. 2 ⇒ 3: One easily chec ks by hand that if f : ( X , =) → ( Y , =) is a dense mono and g : ( X , =) → ∇ Z is any map, then the rela tio n g ◦ f − 1 is a ctually a function. 3 ⇒ 1 : If ( X , =) is a ¬¬ -sheaf, then it is cer tainly se pa rated. Hence η ( X, =) is monic, b y the previo us propos ition; as it is also dense, it follows that it has a left inv erse. But then it is no t hard to see that it must b e an isomorphis m. W e take a closer loo k at the o b jects in the image of ∇ . Lemma 5.6 . If ( A, A , α ) is a Herbr and assembly and ther e is an element e ∈ P such that e ∈ A and h e i ∈ α ( a ) for al l a ∈ A , then ( A, A , α ) ∼ = ∇ A . Pr o of. Obvious. Theorem 5.7. The functor ∇ : S ets → HT [ P ] pr eserves and r efle cts the st ruc- tur e of a lo c al ly c artesian close d pr etop os. In p articular, it pr eserves and r efle cts validity of first-or der formulas. 11 Pr o of. One e a sily c hecks by hand that ∇ prese r v e s a nd reflects quotients of equiv alence relations. Ther ef ore it suffices to pr o ve that the functor ∇ : S e ts → HA sm [ P ] preserves and r e flects the str uc tur e of a lo cally car tesian c lo sed regular category with sums, beca use this structure is pres erv ed and reflected by the inclusion HA sm [ P ] → H T [ P ]. B ut then the result follows immediately from the constructions we gav e in Section 4 and the previous lemma. W e have just seen tha t ∇ : S ets → H T [ P ] preserves a nd reflects first-o rder logic. But it does not preserve the natural n um b ers ob ject, because: Prop osition 5.8. The Herbr and assembly ( N , N , ν ) is the natur al numb ers obje ct in the Herbr and top os. Pr o of. Since the succ e ssor ma p s : ( N , N , ν ) → ( N , N , ν ) is monic in the Her- brand topos , the nno ther e will be the smallest s ubob ject of ( N , N , ν ) containing 0 and closed under this map (see, for example, [9, Cor ollary D5.1 .3]). Because any sub ob ject of a separ ated ob ject is automatically separated, this smallest sub o b ject is the ob ject ( N , N , ν ) itself. Therefore ∇ N is a nonstandard mo de l of arithmetic in the Herbr and top os. It is this mo del which underlies the realizability interpretation of no nstandard arithmetic defined in Section 5 of [2]. 6 Pro jectiv es In this section we study the pro jective ob jects in the Herbrand assemblies and the Herbrand topos. Definition 6.1. W e ca ll a Herbr and a ssem bly ( A, A , α ) p artitione d , if for every a ∈ A there is an element g a ∈ P suc h that α ( a ) = ↑ ! A h g a i . W e first note that: Lemma 6. 2. We have: 1. The natur al nu mb ers obje ct ( N , N , ν ) is p artitione d. 2. Every obje ct of t h e form ∇ Z is isomorph ic t o a p artione d Herbr and as- sembly and henc e every Herbr and assembly is a sub obje ct of a p artitione d one. 3. Partitione d Herbr and assemblies ar e close d under fi n i te limits. 4. Every Herbr and assembly c an b e c over e d by a p artitione d one. 12 5. Every re tr act of a p artitione d Herbr and assembly is isomorphic t o a p ar- titione d Herbr and assembly. Pr o of. Items 1 a nd 2 are o b vio us, so w e concen trate on the others. It is cle a r that the terminal ob ject in the ca tegory of Herbrand a s sem blies is partitioned and tha t pa r titioned Herbra nd a s sem blies ar e closed under equa l- izers. Moreover, if ( A, A , α ) and ( B , B , β ) are par tit ioned Herbrand assemblies, then the following par titio ned Herbr and assembly is a pro duct in the ca teg ory of Herbrand a ssem blie s : C = A × B , C = A ⊗ B , γ ( a, b ) = ↑ !( A⊗B ) h p g a g b i . A Herbrand a ssem bly ( A, A , α ) can b e cov ered by the partione d Herbrand assembly ( A ′ , A ′ , α ′ ), where A ′ = { ( a, n ) ∈ A × P : a ∈ A, n ∈ α ( a ) } , A ′ = ! A and α ′ ( a, n ) = ↑ !! A h n i , via the pr o jection. The easy details ar e left to the reader. If ( A, A , α ) is a r etract o f the par tit ioned Herbr and ass em bly ( B , B , β ) via maps f : A → B and g : B → A with g f = id , then ( A, A , α ) is isomo r phic to the partioned assembly ( A, B , β f ). Prop osition 6. 3. Partitione d H erb r and assemblies ar e pr oje ctive in the c ate- gory of Herbr and assemblies, b oth internal ly and ext e rnal ly. Pr o of. W e firs t show that partitio ned Herbr and assemblies are exter nally pro- jective. So supp ose p : ( B , B , β ) → ( A, A , α ) is a cov e r b et ween Herbr and a ssem- blies a nd ( A, A , α ) is partitioned. Then there is an element r ∈ P such that r is defined o n all elements of ! A and then y ie lds v alues in ! B , and ( ∀ a ∈ A ) ( ∀ n ∈ α ( a )) ( ∃ b ∈ p − 1 ( a )) r · n ∈ β ( b ) . In particular, ( ∀ a ∈ A ) ( ∃ b ∈ p − 1 ( n )) r · h g a i ∈ β ( b ) . Using choice in the metatheory , this means that there is a function f : A → B such that pf = id and ( ∀ a ∈ A ) r · h g a i ∈ β ( f a ) . Therefore the function f is a sectio n of p track ed by s · h m 1 , . . . , m k i = r · h m 1 i ∗ . . . ∗ r · h m k i . 13 Since the terminal ob ject is partitioned (item 3 of the previo us lemma) and therefore externally pro jective, this implies that externally pr o jective ob jects are also internally pro jective. Corollary 6.4. Up t o isomorphism, the p artitione d Herbr and assemblies ar e the pr oje ctive obje cts in the c ate gory of Herbr and assemblies. In p articular, t hi s c ate gory is e quivalent to the r e g/lex-c ompletion of its ful l sub c ate gory on the p artitione d H erbr and assemblies. Pr o of. The first statement follows from the previous pr o position and items 4 and 5 of the previous lemma; the se cond follows from the characterisation of reg/lex- completions in [4]. Corollary 6.5. The natur al numb ers obje ct is b oth internal ly and ext e rnal ly pr oje ctive in the c ate gory of H erb r and assemblies. In contrast, it is no t gene r ally true that the na tu ral num b ers ob ject is ex- ternally pro jectiv e in the entire top os. Indeed, this fails if we take as our p ca P Kleene’s first p ca K 1 . T o define this p ca we need to fix an enumeration of the partial recur siv e functions satisfying some pr operties (for the precise prop erties that one needs, see [15, pages ix a nd 15]). Then K 1 has as underlying s e t the natural n umbers and the partial a pplication n · m is defined to be the result of applying the n th par tial recurs ive function to the natural num b er m . Prop osition 6.6. If P = K 1 , then the natur al numb ers obje ct is not external ly pr oje ctive in the H e rbr and top os. Pr o of. Let A 0 and A 1 be tw o recurs iv ely inseparable subsets of N and let ( N × { 0 , 1 } , E ) b e the ob ject of the Herbrand top os which has no actual or potential realizers for non triv al e q ualities, and whose exis tence pr edicate E = J − = − K is given by: E ( n, i ) 1 = N ⊗ { i } , E ( n, i ) 0 =  ∅ if n ∈ A 1 − i , ↑ ! E ( n,i ) 1 h p ni i otherwise . Now consider the pro jection p : ( N × { 0 , 1 } , E ) → ( N , N , ν ) given by p (( n, i ) , m ) =  E ( n, i ) ∧ ( ν ( m ) , N ) if n = m, ∅ otherwise. This map is surjective, b ecause its surjectivity is realized b y the element s ∈ P satisfying s · h n 1 , . . . , n k i = hh p n 1 0 i , h p n 1 1 i , . . . , h p n k 0 i , h p n k 1 ii . But, on the other hand, this ma p do es not have a sec tio n, b ecause a rea lizer r for such a map w ould a llow one to recursively s eparate the sets A 0 and A 1 (for n ∈ N compute the second pro jection of ( r · h n i ) 1 ; this alwa ys yields either 0 o r 1 and for n ∈ A i it yields i ) . 14 Corollary 6.7. If P = K 1 , then not every obje ct in the Herbr and top os c an b e c over e d by a Herbr and assemby. In p articular, the Herbr and top os is not t h e ex/r e g-c ompletion of the c ate gory of Herbr and assemblies. 7 Logical features of the Herbrand top os In this section we inv es t igate the v alidity in the Herbrand realizability of some significant logical principles. Suc h questions a r e p ca dependent and here we restrict atten tion to the Herbra nd top os based on the p ca K 1 . More informa tion on the princ iple s we co nsider can be found in [17, Chapter 4 ]. Lemma 7.1. A function f : N → N is tr acke d (as a morphism ( N , N , ν ) → ( N , N , ν ) in the Herbr and top os) iff it is b ounde d by a c omputable function. Inde e d, fr om a tr acking one c an c ompute a b oun d and vic e versa. Pr o of. Note that it is nec essary a nd sufficient for a function f : ( N , N , ν ) → ( N , N , ν ) to be track ed that one c an co mput e for every n a sequence r ( n ) = h n 1 , . . . , n k i such that f ( n ) = n i for so me i ≤ k . Since one can compute maxima, this implies that f is bounded by a co mputable function g . If, on the other hand, g is a co mputable function b ounding f , then r ( n ) = h 0 , . . . , g ( n ) i shows that f is track ed. Corollary 7.2. In the Herbr and top os the fol lowing b oun d e d form of Chur ch’s Thesis holds: ( ∀ x ∈ N ) ( ∃ y ∈ N ) ϕ ( x, y ) → ( ∃ e ∈ N ) ( ∀ x ∈ N )  e · x ↓ ∧ ( ∃ y ≤ e · x ) ϕ ( x, y )  . Pr o of. F ollows from the previo us lemma and Co rollary 6.5. Theorem 7.3. In the H e rbr and top os, the we ak law of exclude d midd le ¬ ϕ ∨ ¬¬ ϕ is valid. Henc e the De Mor gan laws hold, as do es: ( ∀ x ∈ N ) ( P ( x ) ∨ ¬ P ( x ) ) → ( ∀ x ∈ N ) P ( x ) ∨ ¬ ( ∀ x ∈ N ) P ( x ) . (1) In p articular, the law of exclude d midd le is valid for Π 0 1 -formulas. But Markov’s principle fails and so do es Chur ch’s t h esis. Pr o of. It follo ws from the pro of of Lemma 5.3 that h λp. hii is an actual realizer of ¬ ϕ o r of ¬¬ ϕ (dep ending on whether ϕ has an actual rea liz e r or no t). Hence p ( h λp. hii , h λp. hii ) is an actual rea lizer of ¬ ϕ ∨ ¬¬ ϕ . The De Morga n laws (in particular, ¬ ( ϕ ∧ ψ ) → ¬ ϕ ∨ ¬ ψ ), the principle in (1) and the law of excluded middle for Π 0 1 -formulas are immediate consequences of this. 15 Nevertheless, not all c la ssical arithmetic is v alid in the Her brand top os: this follows from the previo us corollar y . Therefore Marko v’s Princ iple must fail (beca use Marko v’s Principle tog ether with (1) implies full classic a l arithmetic). Alternatively , one can argue dire ctly for the failure of Markov’s Pr inciple by showing that a rea lizer for it would allow one to so lv e the halting problem (along the same lines as for modified realiza bilit y; see [2, Pr oposition 5.9]). Chu rch’s Thesis in the for m CT ∨ 0 is inco mpatible with the law of excluded middle for Π 0 1 -formulas (again, use tw o recursively insepar able r.e. subsets o f N , or see [1 7 , Section 4.3]). Also CT fails: in fact, it follows from Lemma 7.1 that every b o unded function f : N → N is tr ac ked. Corollary 7.4. Continuity principles, like every function f : R → R is c ontin- uous, fail in the Herbr and top os. Pr o of. Because thes e are incompatible with the law of excluded middle for Π 0 1 - formulas (see, for example, [1 7 , Prop osition 4.6 .4]). Corollary 7. 5. The Herbr and top os is not e quivalent t o a r e alizabil ity top os over an (or der-)p c a. Pr o of. Because in suc h to poses Marko v’s Principle holds. Corollary 7.6. The H erb r and top os is t wo -value d, but not b o ole an. Pr o of. The terminal ob ject has only t wo sub ob jects in the categ ory of Herbrand assemblies; since the terminal ob ject is sepa rated and separ a ted ob jects are closed under subob jects, the sa me applies to the Her br and top os. Hence the Herbrand top os is tw o - v alued. Nevertheless, it is no t b oolea n, s inc e Mar k ov’s Principle fails. Definition 7.7. (See [17, pag e 186].) In the following we will mean by a tr e e an inhabited and decidable set of finite sequences of natural num ber s, clo sed under predecessors. A tree w ill b e called fin i tely br anching if ( ∀ n ∈ T ) ( ∃ z ∈ N ) ( ∀ x ∈ N ) ( n ∗ h x i ∈ T → x ≤ z ) . Finally , b y an infinite p ath in a tree T w e will mean a function α : N → N such that αn = h α 0 , . . . , α ( n − 1) i ∈ T for all n ∈ N . Lemma 7.8. The c ol le ction of infinite p aths in a fi n i tely br anching tr e e is a uniform obje ct. Mor e pr e cisely, if T is a fin i tely br anching t r e e, then one c an c ompute fr om a r e alizer for t he statement that T is a finitely br anching tr e e an element n ∈ P such that h n i is a c ommon r e alizer fo r al l infinite p aths in the tr e e. Pr o of. F ro m a realizer for the sta tement tha t T is finitely br anc hing tre e one can compute a b ound f ( n ) on the v alue of α ( n ) for every infinite path α . This is sufficien t in v iew of Lemma 7.1. 16 Prop osition 7 .9. In t h e Herbr and top os K¨ onig’s L emma holds. Pr o of. Recall tha t K¨ onig’s Lemma says that every finitely bra nc hing tree which is infinite contains an infinite path. Of cour se, K¨ onig’s Lemma is true, but the question is whether we ca n compute a r ealizer for an infinite path. The prev io us lemma, how ever, guara n tees that we can. Prop osition 7 .10. In the Herbr and top os t h e F an The or em holds. Pr o of. Suppos e we hav e realize r s for the statements that T is a finitely branching tree, that A ( x ) is a pro perty o f its no des, inherited by successors, and that for every infinite path α there is a na tural num b er n ∈ N such that A ( α n ) holds. F ro m a realize r for the first statemen t we co mput e a common realizer fo r all infinite paths. Then fr o m this and a realizer for the third statement we co mpute a finite sequence h n 1 , . . . , n k i suc h that for e a c h infinite path there is an i such that A ( αn i ) holds. But as A is inherited b y successors , we must hav e A ( αn ) for all infinite paths α , if n = max( n 1 , . . . , n k ). 8 Conclusion W e hav e introduced a new top os and established some of its basic pro perties. Since this pap er was written, a num b er o f connections to other top oses have been uncov ered: J aap v an Oosten has shown that the Herbrand top os is a subtop o s o f the corresp onding modified rea lizabilit y top os, while Peter Johnstone has shown that it is the Gleason cover of the corres p onding realizability topos [10]. In addition, we have sho wn [1] that there is a top os for the nonstanda rd functional int erpreta t ion develop ed in [2] w hich is r elated to the mo dified Diller- N ahm top os (see [16] a nd [3]) in the same way as the Herbrand top o s is related to the mo dified rea liz abilit y topo s . But we ex p ect that m uch mo re can be said. References [1] B. v an den Berg. 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