Interpreting classical theories in constructive ones

Abstract A number of classical theories are interpreted in analogous theories that are based on intuitionistic logic. The classical theories considered include subsystems of first- and second-order arithmetic, bounded arithmetic, and admissible set theory.

a single inductive definition can be adapted to yield a remarkably straightforward reduction of the fragment of arithmetic 72/ to its intuitionistic version IL',. In [ 11 ], Coquand and Hofmann also present an interpretation of Buss' theory of bounded arithmetic S\, that is independent of (and somewhat different from) the one given here.
In Section 2,1 will describe a general framework for these interpretations, and in the sections that follow this framework will be instantiated in a number of different settings. In Section 3, I will use it to give a slightly different presentation of Coquand's interpretation, which shows that I'Li is conservative over its intuitionistic version for II2 formulae. I will then adapt the argument to S 2 , yielding VL^ conservation over its intuitionistic counterpart IS' 2 . Since IL) and IS' 2 have constructive realizability interpretations, the reductions enable one to extract constructive information from the classical theories as well. The analysis also extends to full classical arithmetic, PA, and the theory S 2 .
In Sections 4 and 5, I will consider the Kripke-Platek theory of admissible sets. Here the natural analogue of IL) is a variant of TCP in which foundation is restricted to formulae that are Si over the universe of sets. In Section 4,1 will use an idea due to Friedman [13] to interpret this theory in an "intensional" version that omits the axiom of extensionality, and, in Section 5, I will apply the framework to interpret the latter in its intuitionistic counterpart.
We will ultimately obtain interpretations for versions of KP with or without the axiom of infinity, and with either full or restricted foundation. The strongest theory analyzed in this way is KPco, which, by direct interpretation, encompasses a number of important classical theories. These include a theory of arithmetic inductive definitions, ID], and a subsystem of second-order arithmetic, Yl\-CA ~, which allows n j comprehension without parameters (for details, see [8]). On the other hand, the intensional intuitionistic version IKPco"" described below is contained in Aczel's constructive set theory, CZF; the latter can, in turn, be embedded in an appropriate version of Martin-L6f type theory (see [2,3]). In sum, the analysis below provides a net reduction of a number of classical theories to constructive ones.
In Section 6, I will show that the methods can also be applied in the context of subsystems of second-order arithmetic. In particular, we will see that the theory L\-AC, which includes arithmetic comprehension and an axiom of choice for arithmetic formulae, is reducible to its intuitionistic counterpart, L\-AC. I, 1 ,-AC is interesting in its own right, but also interprets ID), which allows a weak form of arithmetic inductive definition. On the other hand, one can interpret I. 1 ,-AC in an appropriate version of Martin-L6f type theory' so once again we have a net reduction of the classical theory to a constructive one.
I should emphasize that the proof-theoretic equivalences of the classical and intuitionistic theories discussed here are well-known. What is notable about this approach is that it applies uniformly to a wide range of theories, and does not require external "machinery." Because the approach involves effective translations of the classical theories to the constructive ones, we obtain conservation results in which the increase in the lengths of proofs can be bounded by a polynomial; as far as I know, when it comes to S' 2 and IS' 2 , this sharper form of the conservation result is new. §2. The framework. One thing that makes intuitionistic logic attractive is that the logical connectives have a constructive interpretation, commonly attributed to Brouwer, Heyting, and Kolmogorov. What makes negation, in this context, particularly ornery is that it obliterates any constructive information that a formula might otherwise have held: the interpretation of 9 -> _L tells us nothing beyond the fact that there cannot possibly be a proof of 8. The remedy offered by the Friedman-Dragalin translation is to insist that ±, along with the other atomic formulae, carry additional information; but the type of information these formulae carry is fixed in advance and remains static throughout the proof. The Buchholz-Coquand methods provide a more dynamic interpretation of J_, by reinterpreting implication as well.
For our purposes it is convenient (but not necessary) to take intuitionistic logic to be given by a system of natural deduction, where derivations yield assertions of the form r =>• tp, i.e., "tp follows from the hypotheses in I \ " To describe the method in full generality, let L be any first-order language, and consider the following two-sorted "forcing language" Lf. One sort of Lf has variables corresponding to the universe of L, with the associated constants and functions. The other sort has variables p, q, r, ... ranging over "conditions," and there is a binary relation P ^ 1 C'/ 7 is stronger than q") between objects this sort. Finally, for every «-ary relation symbol R(x\,... , x") of L, there is a corresponding (« + l)-ary relation symbol R'{p,x\ ,x n )ofL t . Intuitively, R'{p,x\,... ,x n ) asserts that condition p "forces" R(x\,... ,x"). Then to every formula tp in the language of L, we can inductively associate a formula p lh tp in the language of Lf, as follows: p lh R{t\,... ,t") = R'{p, t\,... , t n ), for every relation symbol R As usual, -up is defined to be tp -> ±, and Vg •< pQ is shorthand for "iq{q < p -> 0). I. is taken to be a 0-ary relation, thereby covered by the first clause. If T is a set of formulae {y/\,... , y k .}, I will write T, tp for T U {tp}, and p lh T for the set of formulae {p lh y/\,... ,p lh y/k). The inscription lh tp is read "tp is forced," and means that every condition forces tp.
3. For every atomic formula 0, if p lh ± then p II-0.
Notice that these conditions are expressible in L t . PROPOSITION 2.3 (monotonicity). From the assumption that the forcing notion good, in intuitionistic logic one can prove monotonicity for all formulae in L.
PROOF. By induction on formulae n. Monotonicity for atomic formulae takes care of the base case, and the transitivity of ^ is used when n is of the form tp -> y/. H PROPOSITION 2.4. Let T be a set of formulae in L, and tp any formula in L. If tp is provable from T intuitionistically, then p lh tp is provable from p \\-T and the assumption that the forcing notion is good.
PROOF. Use induction on the length of the proof. We can assume that the only rule governing ± is "ex falso sequitur quodlibet" for atomic formulae; that is, the rule "From r => _L conclude T =>• 0" for atomic 0. The case in which this is the last inference of the proof is covered by clause 3 in Definition 2.2.
Otherwise, the only interesting cases occur when the last inference is either an introduction or elimination rule for ->. To handle ^-introduction, suppose the last rule of the proof yields T => <p -> y/ from T, tp =>• y/. Assume p lh V, q < p, and q lh tp; we need to show that q lh y/. By monotonicity, we have q lh T; but then q lh y/ follows from the inductive hypothesis.
Dealing with ^-elimination is no more difficult. H NOTES. The forcing clauses above formalize the usual Kripke semantics, provided ± is treated as a propositional variable and there is a fixed universe for all the possible worlds. See, for example, [23].
If X is defined from a commutative, idempotent "meet" operation A by then the clause for implication is equivalent to If •< has a greatest element 0, then lh <p is equivalent to 0 lh <p. If one focuses one's attention on the negative fragment of intuitionistic logic, which involves only the connectives A, ->, and V, the clauses above are consistent with the intuition that p lh >p means, in some sense, that there is a proof of ip from p. We will see in Section 5 that the forcing relation behaves particularly well with respect to these connectives.
The Friedman-Dragalin translation arises from the clauses above in the special case where the partial order is trivial (i.e., has a single element) and there is a fixed formula y/ such that for every atomic formula 6, p lh 0 is just 0 V y/.
To apply the framework above one need only find suitable interpretations for the conditions and atomic forcing relations. COROLLARY 2.5. Let T be an intuitionistic theory given by a set of axioms. Suppose one defines a good forcing notion in another theory T', in such a way that T' proves that every axiom of'T is forced. Then whenever T proves a formula tp, V proves that tp is forced.
PROOF. If T proves ip intuitionistically, then there is a set of axioms r of T such that r => ip is provable intuitionistically. If p is any condition, the hypothesis and Proposition 2.4 imply that V proves p\VY and hence p \\-<p.
-\ §3. Arithmetic. In this section I will show that 121 is conservative over its intuitionistic analogue 12). The interpretation is essentially that of Coquand and Hofmann [11]; the only difference is that here I will use first-order forcing conditions instead of second-order ones, and divide the interpretation into two steps. In this form, it is easy to extend the results to S' 2 .
121 denotes the fragment of classical arithmetic in which the schema of induction is restricted to 2\ formulae. Because any primitive recursive function can be introduced in a definitional extension of 12/, we can conveniently blur the distinction between this theory and I2i(PRA), which has symbols denoting such functions in its language. Identifying primitive recursive relations with their characteristic functions, we can then take the S| formulae to be of the form 3xA(x), where A is primitive recursive, possibly with free variables other than x. The axioms of 121 consist of 1. Quantifier-free defining equations for the primitive recursive functions.
denotes the successor of x. 12) denotes the corresponding theory based on intuitionistic logic.
In I2\ one can use induction to prove that equality is decidable, and hence that the law of the excluded middle holds for quantifier-free formulae. Markov's principle for primitive recursive predicates is given by where A is primitive recursive. The interpretation of 72/ in 12) will proceed in two steps: first we will interpret 121 in 12.', + (MP pr ), and then we will interpret the latter theory in 12',.
For the first interpretation, the double-negation interpretation suffices. In this context, the interpretation takes (ip V y/) N to ->(-*(p N A ^y/ N ), takes (3xip) N to ->\/x-np N , fixes atomic formulae, and commutes with A, ->, and V. PROOF. Using (MP pr ) and the law of the excluded middle for atomic formulae, 12', proves that quantifier-free and 2\ formulae are equivalent to their N-translations. As a result, the doubly-negated axioms of 12 1 are equivalent to themselves in 12) + (MPpr).
H For the remainder of this paper I will say that a formula is "negative" if it is part of the negative fragment of intuitionistic logic, that is, it does not contain any instances of the connectives V or 3. 2 Let & pr denote the set of "almost negative" formulae, that is, the smallest set containing the Si formulae and closed under conjunction, implication, and universal quantification. Since 12) + (MP pr ) proves that any formula in % pr is equivalent to its N-translation, we have THEOREM 3.2. 72/ is conservative over 72' ; + (MP pr ) for formulae in W pr . Let us now apply the framework of Section 2 to interpret 72, + (MP pr ) in 72' ; . It turns out that the appropriate conditions are finite sets of 111 sentences with parameters, i.e., Ill formulae together with assignments to their free variables. 3 The ordering -< between conditions is defined to be the set containment relation, 3 . Fixing some reasonable encoding, if \/xA(x) is a Oi formula with free variables / , let r \fxA ( x p denote the function of/that returns the code of the corresponding sentence with parameters, and let the variables p,q,r ... range over finite (coded) sets of such sentences. I will usually write p, q instead of p U q and p, tp instead of p U {tp}. Since q D p is equivalent to q U p = q, p\\-(tp -> y/) is equivalent to which is the characterization I will use to verify the interpretations. Given any proof in 72y + (MP pr ), choose m large enough so that all the primitive recursive relations mentioned have complexity less than m, and let Tr™ denote a primitive recursive truth predicate for such relations. (I will come back to this issue below.) If p codes a set of the form (1), define k Tr(p,u) = /\Ai{u), where Tr™ r is used to express the right-hand-side; this asserts that the formulae in p are true at least as far as u is concerned. Define for arbitrary formulae tp. Intuitively, the witness u in (2) can be interpreted as a "proof" that tp follows from the conjunction of the universal sentences in p, since (2) asserts that ip follows more specifically from their instantiations at u. As one might expect, this "provability" relation is monotone in the first argument: if q and p are conditions such that q D p, then Tr(<7, u) implies Tr(/>, u) and hence p h tp implies qh tp. Define p lh 0 to be p h 0 when 8 is atomic, and extend the forcing relation to arbitrary formulae in the language of arithmetic as in Section 2. It is not difficult to verify that the forcing notion is a good one, according to Definition 2.2. The following lemma shows that when it comes to quantifier-free formulae, the relations lh and h coincide. LEMMA 3.3. Iftp is any quantifier-free formula, then 72, proves P^<P iff P h <P- 3 Alternatively one can take these to be IT] sentences in an expanded language that has a name for every element of the universe. Of course, in the context of arithmetic these names are not necessary, since every number is denoted by the corresponding numeral; but we will need this more general formulation in Sections 5 and 6 below.
PROOF. The proof of this lemma relies heavily on the fact that TL) proves the law of the excluded middle for quantifier-free formulae. In particular, this implies that for such formulae tp, p h tp is equivalent to 3w(-Tr(/?, u) V tp), as well as The proof proceeds by induction on the complexity of tp. The cases in which tp is atomic, of the form 0 A y/, or of the form 0 V y/ are readily dealt with, using the observations in the preceeding paragraph. When tp is of the form 0 -> y/ we have We need to show that this is equivalent to p h {0 -> y/). In the forwards direction, assume the last line of the equivalence holds. Since 0 implies 0 h 0, we have 0 -> p h y/. This is equivalent to -10 V 3M-iTr(/>, w) V y/, which is equivalent to p h (0 -* y/). For the other direction, suppose p\-(0 -> y/) and ^ h 0; we need to show /?, g h y/. From the assumption, we have 3w-Tr(/>, u) V (0 -> ^) and 3i>-.Tr(#, v) V 0.
For 2, suppose p II-'Vx/l(x). Then we have In particular, 1 implies /?, VxA(x) Ih ±. In other words, there is an element u such that Tr(p,u) AA(u) -> 1 and hence

. HL) proves that each axiom ofrL) + (MP pr ) is forced.
PROOF. By Lemma 3.3, a quantifier-free formula is forced iff it is true, so the quantifier-free axioms are reduced to themselves. Lemma 3.4 shows that if p forces the antecedent of (MP pr ), then it forces the conclusion as well; so (MP pr ) is forced.
Finally, to handle induction, suppose /»H-y(0) and p\\-Vx(y/(x)^y/(x')). There is a correspondence between intermediate theories as well. Suppose we start with a n" formula Vxi 3x 2 . •. Qx n <p, where n > 2 and <p is quantifier-free. Then its double-negation translation is intuitionistically equivalent to Vxi-i3x2... Q'x n -n<p N , where Q' is 3 if Q is V, and vice-versa. By definition, the assertion that p forces this latter formula is equivalent to and this formula is equivalent (over, say H,\) to one of the form V(S" -> Si). For each n > 2, C n be the set of formulas of this form. Since from a classical point of view formulas in C" are n " , and since n" and S" induction are equivalent in the classical setting, we have THEOREM 3.9. For n > 2, 7S" is a II2 conservative extension ofIC' n .
More satisfying results involving the correspondence between classical and intuitionistic fragments of arithmetic are available; see Burr [9].
The argument we have just carried out required a primitive recursive truth predicate Tr™ r for primitive recursive relations of complexity less than or equal to m. If we measure the complexity of a primitive recursive function by the number of instances of composition and primitive recursion employed in its definition, we can obtain such a predicate as follows. For each m, we define a primitive recursive function Eval m (/, s), which evaluates a function of complexity at most m, coded by / , at the list of parameters coded by s. Eval 0 is defined so that it computes the result of applying either a constant, successor, or projection function to its arguments, and then for each m, Eval m+1 is defined by cases using Eval" 1 . (A similar construction can be found in [22,Section 1.5].) We can use Eval m to define Tr^., and derive the necessary properties in IY.).
Alternatively, we can use a Si evaluation predicate for the primitive recursive functions (also described in [22]) and take p h 6 to assert that "there is a u and a computation sequence c for A\{u),... ^^{u), such that if the computation determines that these are all true, then 6." Choosing this method means that we no longer need to use a separate interpretation for each fixed complexity level m.
For the simplest method of all, note that in any given proof only finitely many relations A i,... , At are mentioned, in which case the "truth predicate" need only work for rii sentences involving these relations. We can represent every such sentence as a pair (i,s), where / is a value between 1 and k and s codes an assignment to the free variables of MxA^x). With this representation, Tr™ can be defined as a straightforward disjunction of length k, and to carry out the interpretation we only need a minimal theory of finite sets and sequences in TL).
In the applications that follow, one may be able to use variations of each of these three options. Since the last one requires the least effort and is the most clearly applicable in all cases, let us, for concreteness, adopt this way of interpreting the various forms of Tr m referred to in the sequel. Though I will neglect to qualify the statements of the lemmata below with the condition that certain formulae mentioned must have complexity less than m, the reader can readily supply the additional details.
With these considerations in hand the interpretation above can be adapted to Buss' theory of bounded arithmetic S' 2 , and its constructive counterpart, IS' 2 . I will rely on the presentation in Cook and Urquhart [10], which provides a nice account of these two theories and their properties. In fact, it will be more convenient to work with the theories CPV and IPV, which are definitional extensions of S 2 and IS2 respectively; these extensions include the terms of Cook's theory PV, which denote polynomial time computable functions. Bounded quantification is defined by In analogy to the Si formulae above, a formula is said to be NP if it is of the form 3x < tA{x), where A is a PV relation and t is a PV term. CPV is the classical first-order theory given by 1. Quantifier-free axioms defining the basic symbols of the language.
2. NP induction: where ip is NP. IPV is simply CPV based on intuitionistic logic. From [10, Theorem 4.5] we know that IPV proves the law of the excluded middle for quantifier-free formulae (and, in fact, formulae in which every quantifier is sharply bounded). In particular, we can take the double-negation translation to fix atomic formulae, and show that the translation of a bounded formula is equivalent to a bounded formula.
A bounded analogue of Markov's principle is given by where A is a PV relation. Finally, in analogy to the almost negative formulae, define Wb to be the smallest set of formulae containing the NP formulae and closed under conjunction, implication, and universal quantification. The double-negation interpretation yields and Finally, define p Ih 6 to be p h 0 for atomic 0, and then extend the forcing relation to the entire language of IPV.
The following lemma asserts that bounded quantification commutes with the lh operator.
PROOF. As in the proof of Lemma 3.3, if 0 is atomic then p Ih 0 is equivalent to 8 V p Ih _L As a result, we have We need to show that this is equivalent to 3x < tp Ih tp. One direction is easy; for the other direction, suppose the last line of the equivalence above holds.  LEMMA

IPV proves that each axiom oflPV + (MPh) is forced.
Let 31 h be the smallest set of formulae in the language of IPV containing the quantifier-free ones and closed under conjunction, disjunction, and universal and existential quantification.
Using the analysis in [10], which shows that CPV and IPV are definitional extensions of S 2 and IS l 2 , respectively, this yields THEOREM 3.17. S 2 is conservative over IS 2 for \FL\ formulae. Let Si and IS2 denote the extensions of S 2 and IS 2 in which induction is allowed for arbitrary bounded formulae. Using Lemma 3.11, we see that Lemma 3.14 still holds for this strengthened form of induction, so we have THEOREM 3.18. S2 is conservative over ISjfor VEj formulae. In [11], Coquand and Hofmann interpret CPV in a second-order version of IPV, and then invoke a result from [10] which reduces this to first-order IPV. Their methods yield a strengthening of Theorem 3.17, where the class of formulas conserved include those of the form V32p On the other hand, because the conservation result from [10] uses a normalization argument, there is the possibility of a superexponential increase in the lengths of proofs. Alternatively, one can derive the stronger conclusion from Theorem 3.17 using Parikh's theorem [19], but once again this allows for superexponential growth. As far as I know, it is still an open question as to whether one can obtain the strengthened version of Theorem 3.17 with a polynomial bound on the increase in the lengths of proofs. §4. Extensionality in admissible set theory. In the context of set theory, the proper analogue of /£/ is the Kripke-Platek theory of admissible sets, without the axiom of infinity, and with foundation restricted to Si formulae: in addition to the fact that the two theories can be interpreted in one another, one finds additional structural similarities in the work of Rathjen [20].
I will take the language of set theory to contain only a single binary relation symbol £, and take equality to be defined by Here bounded quantification is given by 3y e x(p = 3y(y £ x A <p).
I have written Ao separation to emphasize that it is 2): this axiom can be used to derive the more usual forms of pair and union. The foundation axiom as presented here is sometimes also called "set induction," and is equivalent to the assertion that every nonempty definable class of sets has an G-least element. I will use KP\ to denote the theory in which foundation is restricted to Si formulae.
The definition of equality given above corresponds to the usual notion of extensional equality between sets. It is easy to prove that this relation is reflexive, symmetric, and transitive; and from axiom 1 we can derive x = y -> (<^(x) <-+ <p{y)) for every formula ip. Alternatively, we could have taken equality to be a basic logical symbol having these properties, and then replaced axiom 1 with what was previously the definition. The two approaches are equivalent, and the first is more convenient for our purposes.
We would like to interpret KP and KP \ in intuitionistic versions. One problem that we will encounter is that extensionality is not well-behaved under the doublenegation translation. So let us take KP"" and KP"" \ to be "intensional" versions, in which the axiom of extensionality is omitted, and let us consider what life in an intensional universe might be like. One can think of such a universe as consisting of "names" for sets, where, in particular, there may be many names for the empty set; i.e., there may be two sets x and y satisfying Mz{z 0 x) and Wz(z £ y), while for some w we have x e w but y £ w. Also, taking x = {y, z} to abbreviate (3) yexAzexA Vw e x(w = y V w = z) is misleading, since it is consistent that z = {x, y} and z' = {x, y} while z ^= z': z and z' may contain different names for x and y. Friedman [13] (see also Chapter VIII of [6]) has found an elegant way of interpreting extensionality in an intensional universe: declare all the empty sets to be "isomorphic" to each other, and, more generally, call two sets isomorphic if (inductively) they have isomorphic elements; then replace elementhood by elementhood up to isomorphism. Here I will show that this approach can be implemented A formula is said to be A\ (relative to a theory) if it is provably equivalent to both Si and IIi formulae. The next lemma is standard in admissible set theory, and does not require extensionality. PROOF. One obtains S) collection by pairing the existentially quantified variables. For Ai separation, note that if ip(y) is equivalent to a Si formula 3uy/(y, u) as well as to a IIi formula \/ud(y, u), then (classically) we have Vy3w(^(j, u) V -<6(y,«)); one can then reduce separation for <p to an instance of Ao separation by first using collection to gather a sufficiently large set of witnesses. The last claim is proved by induction on formulae, again using collection. For details, see [5]. H Ignoring the caveat above and using x = {y, z} to denote (3), I will write "x is an unordered pair" for 3y e x, z £ x{x = {y, z}), and {y, z} £ w for 3x £ w(x = {y, z}). We can think of a symmetric relation R as given by a set of unordered pairs, allowing {x, x} as a degenerate case. With this in mind, let us write y ~« z for {y, z} £ R, and y £field{R) for 3x £ R3z £ x(x = {y, z}). Call such a relation R an isomorphism relation if, for every y and z in the field of R, we have The definition implies that the field of any isomorphism relation R is transitively closed, i.e., y e field(R) and w £ y imply w £field(R). Using foundation with Ao formulae one can also show that any isomorphism relation is an equivalence relation on its field. The global isomorphism relation we are looking for is given by y ~ z = 3R{"R is an isomorphism relation and y ~« z").
Each of the following four lemmata is provable in KP"" \. The last shows that y ~ z has an equivalent 111 definition, and is hence A]. LEMMA

KP'"' \ proves that for every u and v, there is a set of unordered pairs of elements from u andv; that is, for every u andv, 3wMy £ u,z £ v({y, z} £ w).
PROOF. Fix u and v. For any given y, we have Vz £ v3r(r = {y, z}).
Use Ao collection to obtain

3.?Vz £ v({y, z} £ s).
In particular, this is true for every y in w; use A 0 collection again to show
-\ Say that an isomorphism relation is good for a set x if every element of x is in the field ofR. LEMMA

For every set x, there is an isomorphism relation that is good for x.
PROOF. Use 1\ induction on x. Suppose the claim is true for every element of x; in other words, for every y in x there is an isomorphism relation R v good for y. Using collection and union, we can define R to be the union of the R } . Using Lemma 4.2 and Ao separation, let S contain unordered pairs from x satisfying the right side of (4), with R in place of R; and let R be R U S. Then R is an isomorphism relation that is good for x. -\

. y ~ z is equivalent to the assertion \/R(("J? an isomorphism relation" A y efield(R) A z &field(R)) -> y ~R z\.
PROOF. Fix y and z. The pairing axiom and Lemma 4.3 imply that there is an isomorphism relation with y and z in its field, and Lemma 4.4 implies that any two such relations must agree. H We have established that ~ is a Ai relation. Using the definition and lemmata above one can show that ~ satisfies and if y> is any formula in the language of set theory, let tp* denote the formula obtained by replacing G by G*. Observe that (x = y)* is given by LEMMA 4.6. Le£ <p be any formula. The following are provable in KP" 1 ' \: 1. x G Z -> X G* Z.
PROOF. Clause 1 follows from the fact that ~ is reflexive. Clause 2 follows from the definition of e*, and 3 follows from 2 together with equivalences (5) and (6) above. Clause 4 is proved using induction on tp, with 3 as the base case. The forwards direction of 5 is easy, using 1. For the other direction, suppose Vx G zip* and x €* z. The latter means that there is an xo G z such that x 0 ~ x; but then ip*(xo), and hence ip*(x) by 3. The proof of 6 is similar to that of 5. Clause 7 follows from 5, 1, and the definition of equality. In the next section we will consider versions of admissible set theory with an axiom of infinity: we can use Lemma 4.6 to show that Theorem 4.9 still holds with this addition. One may also wish to consider versions of Kripke-Platek set theory in which one has a set N containing the natural numbers as urelements, as well as the primitive recursive functions and a built-in notion of equality on that set. Once again, Theorem 4.9 still holds for these theories, provided that in the intensional versions we have the usual axioms governing equality on N. The modifications necessary for this interpretation are well described in [13,6], and pose no additional problems in the present setting. §5. Interpreting intensional KP. Having dealt with extensionality, we can now restrict our attention to the interpretation ofKP int and KP int \. Let IKP inl and IKP im \ denote the corresponding theories where the underlying logic is intuitionistic. Our goal is to show that the classical theories are conservative over the intuitionistic ones for a certain class of formulae; for the moment, we will focus our attention on KP"" \. The argument below is modeled after the one in Section 3, but is more delicate because in IKP"" \ one can not, in general, prove the law of the excluded middle for A ( , formulae. Nonetheless, we will again proceed in two steps, and make use of an intermediate theory based on intuitionistic logic. Many of the lemmata below are patterned after similar ones in [7]. In this setting it turns out that the primitive recursive relations of Section 3 are analogous to negative Ao formulae, and the Si formulae of arithmetic are analogous to what I will call "weak Si" formulae in the language of set theory. These are defined to be formulae of the form (7) ]w-i\/x £ nop where ip is negative and Ao, and w does not appear in (p. Being weak Si is more restrictive than being Si: formula (7) does not quite assert that there is an x satisfying -up, but rather that there is a set w of candidates, not all of which satisfy ip.
where ip is negative and Ao. Since the converse direction is intuitionistically valid, (MP r( , s ) implies that the negation of any II i formula is equivalent to something that is weak Si.
In the first step we will use the double-negation translation to interpret KP" 1 ' \ in IKP'"' # \ + {MP rcs ). Since we no longer have the decidability of atomic formulae in the latter, here we must take (y £ x) N to be -i-ij £ x. LEMMA 5.1. Let ip be any formula. Then the following are intuitionistically valid:

{By G xip) N iff -Hy G x^N. Hence, the double-negation translation of any Ao formula is intuitionistically equivalent to a ho formula.
PROOF. The last claim is proved by induction on formulae, using equivalences 2 and 3.
The right-to-left direction of 1 follows from the fact that y G x implies -I->J> G x intuitionistically. Conversely, y G x -> ip N implies -«p N -> ->j G x, and hence ->->j G x -> -I-I</J' V . But since -• -i y? -> ip is classically valid, -i-><p N implies <p N intuitionistically.
Regarding 2, we have which is equivalent to My G x y^ by part 1. Clause 3 is proved by noting that 3y G x<p and ->My G x-np are classically equivalent, and applying 2 with -><p in place of ip. H

LEMMA 5.2. IKP mt# \ + (MP res ) proves the double-negation translation of each axiom of KP in '\.
PROOF. Pair and union imply their double-negation translations, and the doublenegation translation of any instance of A 0 separation is implied by an instance of Ao separation*.
Arguing in IKP' nt# \ + (MP res ), suppose the antecedent is true. By (MP res ) we have Vx G zBs->My G s^ip N (x, y).
By A 0 collection* we have Given such a set w\, let w be the set containing U w\ asserted to exist by the union axiom. Then for every x in z we have weakening the conclusion and taking the contrapositive yields Combining this with (9) yields 3u>Vx G z~Ny G w-up N {x,y), which implies the conclusion of (8).
Finally, since (MP res ) implies that the double-negation of a Si formula is weak Si, the double-negation of an instance of Si foundation is equivalent to an instance of Si foundation*.
-\ Let f r " be the smallest set that is closed under conjunction, implication, and universal quantification, and that contains all weak S] formulae in which every atomic subformula is preceeded by at least one negation. One can show inductively that for each formula tp in f m -IKP m,# \ + (MP rex ) proves that tp is equivalent to <p N . Thus we have THEOREM 5.3. AT 5 "" \ is conservative over IKP'"'* \ + (MP res ) for formulae in W res .
We are now ready to use the forcing framework of Section 2 to reduce IKP"" # \ + (MP res ) to IKP"" \. Here the appropriate conditions are finite sets of Y\\ sentences with parameters. Using the third method described in Section 3, let Tr™ v be a Ao truth predicate for sufficiently many Ao sentences with parameters, and if/? is a set ofFIi formulae {Vxy>i ( x ) , . . . , Vx<^.(x)}. let where Tr™ v is used to express the right-hand side. For any formula tp, define p \-(p = 3u (Tr(/7, u) -> tp).
Intuitively, u provides a "proof" of tp from p by giving a bound on the universal quantifiers that is sufficiently large to witness the fact that tp follows from the formulae in p. In particular, from 2 we have /?, Vxy> Ih ±; that is, and therefore 3w(Tr(/>, M) -> -iVx G wy>). Taking w = u we have 3w/> I-iVx G iu<p. By Lemma 5.6 and the definition of Ih, this is equivalent to

Regarding 4, suppose p Ih Vx G j^. Then
Vx, <j f (# Ih x G j> -> /?, q Ih 77). In particular, if x G y then Ih x G j , and hence p Ih rj.
It should be no surprise that proving 5 requires the use of Ao collection. Suppose p Ih Vx G z3ytp(x, y), where ip is Ao and negative; we need to show p Ih 3wVx G z-iVy G to-iy>(x, >>)• By 4 and the definition of Ih, the assumption implies Vx G z3yp Ih <^(x,_y) which is equivalent to Vx G z3yp h <p(x,y) by Lemma 5.6. This is, by definition, Vx G z3y,u(Tr(p, u) -+ <p(x,y)).

By Lemma 5.6 and the definition of Ih, this is equivalent to
p Ih 3u>Vx G z-A/j G w~^ip(x, y), as desired. Finally, regarding 6, note that 4 takes care of the forwards direction. For the converse direction, suppose where <p is Ao and negative. Then by the definition of Ih and Lemma 5.6 we have Vx G y3zp I-iVi; G zip.
Since p I-<\/v G z<^ is Si, we can pair existential quantifiers and use Ao collection to obtain 3*iVx G y3z G s\p I-iVv G zy>.
If s is IJ «i, then z G JI implies z C s and so ->\/v G z</? implies -iVv G sip. As a result, the last formula implies 3Wx G yp I-'Vv G sip.
By Lemma 5.5.6, this is equivalent to 3sp h Vx G y~Nv G jy> and hence p Ih 3Wx G y-Mv G sy.
Lemma 5.7.3 implies that (MP res ) is forced, and Lemma 5.7.5 takes care of A 0 collection**. Finally, to handle Si foundation*, suppose where y/ is weak Si. By definition this means Vx,q(q Ih Vy G xy/(y) -^ p,q Ih y(x)).
In particular, taking q to be p and applying Lemma 5.7.6, we have that for every x Vy G xp Ih \//(y) implies p Ih y/(x). What about adding the full foundation schema to both sides? Interpreting the Ei foundation axiom of IKP m '*\ made use of Lemma 5.7.6, which asserts that p\\-\fx e yn is equivalent to Vx € yp Ih n when n is weak £]. To interpret KP'"', it suffices to allow foundation for negative formulae in KP'"'*; and to interpret that, we need to know that the equivalence given by Lemma 5.7.6 holds for arbitrary negative formulae. This fact is supplied by the following two lemmata. where v is any new variable.
PROOF. Arguing in IKP'"'\, suppose tp -> 3uy/. We can use pairing and Ao separation to prove the existence of {0}, and then the existence of a set z such that \/w{w e z <-* w e {0} A tp).
If there is any w in z, then tp holds, and hence so does 3uy/. In other words, we have Vu> s z3u\fj. PROOF. The second claim follows easily from the first, and proving the right-toleft direction of the first claim is straightforward, as in the proof of Lemma 5.7.4. The left-to-right direction of the first claim is proved by induction on the complexity of tp.
By the previous lemma, there is a v\ such that x e y -f 3u e v\ (Tr(p, u) -> 0).
Letting v be [j v\, we have x e j^ ( T r ( / M ; ) -0 ) .
Rearranging the antecedents in this last formula yields which is the same as p\-(xe y ->6).

• (»/ -• (<p -• v ) ) <-»• {(v ->• <p) ->• in ->• v ) ) -
• (?7 -> Vx<^) <-> Vx(?7 -> y?), if x is not free in rj. This completes the proof. H Since we can now interpret foundation for negative formulae as in the proof of Lemma 5.8, we have THEOREM 5.14. KP is interpretable in IKP"". Here and for the rest of this section, I will take the wording of Theorem 5.14 as an abbreviation for the assertion that Theorem 5.11 still holds when one replaces KP\ and IKP"" \ by the theories mentioned.
Since (10) is in 9J res , IKP mt \ proves that it is forced if and only if it is true. Furthermore, since (10) is implied by (infinity), we have THEOREM 5.15. KPco\ is interpretable in IKPa>""\, and KPco is interpretable in IKPa>"".
Here the symbol a> in a theory's name indicates that infinity is to be included among the axioms.
Suppose that instead of adding an axiom of infinity, we take KPu to be the theory of [16], with a set N of natural numbers as urelements. Then, according to the discussion at the end of Section 4, KPu is interpreted in IKPu"". As a result, if KPu proves Vx e N3j 6 N^4(x,y) for some primitive recursive predicate A, then IKPu" 1 ' proves VJC G N-A/J e N-i^4(x, y). Since the latter theory is closed under the Friedman-Dragalin translation (see [14]), we have THEOREM 5.17. KPu is conservative over IKPu"" for TI2 sentences of arithmetic. In [16], Jager also considers theories in which one drops the foundation axiom and replaces it with various forms of induction over the natural numbers. The methods discussed in this section apply to these theories as well: the analogues of Theorem 5.10 hold for the intensional versions of KPu 0 + (£/ induction) and KPu" + (induction). However, the corresponding version of Theorem 5.11 does not follow, since the interpretation of extensionality in Section 4 requires £1 foundation. §6. Subsystems of second-order arithmetic. The theory KPu° + (induction), discussed at the end of the previous section, has the same strength as the subsystem of second-order arithmetic, IL'j-AC (see [16]). In this section I will show that the methods we have been using can be applied to ~L\-AC directly, reducing it, as well, to its intuitionistic counterpart. 4 Here the task is somewhat easier than that of interpreting KP, for two reasons: we do not have to worry about extensionality, and induction is easier to interpret than foundation.
The language of second-order arithmetic is two-sorted, extending the language of first-order arithmetic with variables X, Y,Z,... ranging over sets of natural numbers, and a relation € between terms of the two sorts. Equality between second-order objects is taken to be defined in terms of first-order equality, so that X = Y is given by Mz(z £ X <-> z € Y). A formula is said to be arithmetic if it contains no second-order quantifiers, though it may contain second-order variables; it is said to be S] if it is either arithmetic or of the form 3 Yip, where <p is arithmetic.
If we let (•, •} denote a primitive recursive pairing function on the natural numbers and read / e Y x as (t, x) e Y, we can think of the set Y as coding a countable sequence of sets indexed by x. We can also interpret a single set Y as coding a countable collection of sets, and introduce bounded second-order quantification by While this device is suggestive, one should keep in mind that in this context that apparent second-order quantifiers are, in reality, first-order.
The axioms ofL'j-AC are as follows: 1. Quantifier-free defining equations for the first-order symbols of arithmetic.

Arithmetic comprehension
where <p is arithmetic and Y does not appear in ip.

Arithmetic choice
where <p is arithmetic. 4. Induction for arbitrary formulae in the language. By coding pairs of sets as a single set, one can easily extend the choice principle to E j formulae, which explains the name. Z' r ACo denotes the theory in which induction is restricted to sets of natural numbers: in the presence of arithmetic comprehension, this set induction axiom implies the schema of induction for arbitrary arithmetic formulae. "L\-AC l and 2,',-AC'Q denote the corresponding intuitionistic theories.
The following lemma makes the analogy to KP more salient. LEMMA  Let X'j-AC* + (2/ induction**) denote the theory in which induction is restricted to formulae of the form 3 Yip. where <p is arithmetic and negative. As in Section 5, I will say that a formula is weak S] if it is of the form 3 W~NX e Wip, where ip is arithmetic and negative and W is not free in <p. Thanks to the fact that induction is easier to interpret than foundation, note that here we do not have to restrict Ej induction* to weak Sj formulae.
Finally, let us define Let ffarith be the smallest class of formulae that is closed under conjunction, implication, and universal quantification, and that contains all the weak Sj formulae in which every subformula of the form / e X is preceeded by at least one negation. Then we have for formulae in % ri , s . and similarly for l,' r AC andH^-AC** + {MP arit h).
To interpret the intermediate theories, let us take our forcing conditions to be finite sets P = {VXtp x {X),...,VXip k (X)} of n j sentences with first-and second-order parameters. In order to code these parameters, conditions must be represented by second-order objects. Let Tr™ /;/ , be a truth predicate for a sufficiently large subset of the arithmetic sentences, and use this to define k Tr(/> U) = f\VX e U<pi(X).

Then define
p\\-e = p\-e for atomic formulae 6, and extend the forcing definition to arbitrary formulae in the usual way. Since most of the proofs from Section 5 now carry over, mutatis mutandis, I will only sketch the details below. PROOF. Lemma 6.5.1 takes care of the quantifier-free axioms and {ACA # ), and we can verify that induction is forced just as we did for first-order arithmetic, in the proof of Lemma 3.5. Lemma 6.5.3 shows that (MP ari ,i,) is forced as well.

P\\-\/x3Y<p(x, Y),
where ip is negative and arithmetic. This implies Since P h ip(x, Y) is 1\, the desired conclusion follows from an application of (I. 1 !-AC). H If ^aruh is the smallest set containing the weak 2 J formulae and closed under conjunction, disjunction, and universal and existential quantification, we have Combining Theorems 6.3 and 6.7 we have THEOREM 6.8. IfL^-ACo + (Ej induction) proves\/X3Y(p(X, Y), where ip is arithmetic, then X'j-AQ + (?,', induction) proves 3WVX3Y e Wip N (X, Y). The corresponding assertion also holds for ~L\-AC andY.\-AC l .
Both classical theories are stable under the Friedman-Dragalin translation (see [14,Section 3]). As a result, here too we can recapture the theorems that are arithmetic FT2- §7. Questions. Given a proof of -*->3xA(x) in Heyting arithmetic, where A is primitive recursive, one can use either the Friedman-Dragalin translation or the interpretation described above to extract a proof of 3xA (x). What can one can say about the relationship between the two methods?
In a sense, Buchholz' interpretations in [7] are more general than the ones described here, since they allow one to interpret iterations of the basic theory. An "iterated" version of I. 1 ,-AC yields the theory ATRn, whose main axiom is equivalent to the assertion that every set X is contained in a coded model of Y.\-AC (see [4] or [21]); and the theories KPI and KPi can be seen as "iterated" versions of KPco, since they axiomatize segments of the constructible set hierarchy that correspond to limits (resp. admissible limits) of admissible ordinals. Can the methods described here be used to provide direct interpretations of ATRQ, KPI, and KPi, in intuitionistic versions thereof? Such an interpretation of KPi would be particularly interesting, because at present the only means of reducing it to its intuitionistic counterpart involves an ordinal analysis.