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This is the published version of an article in the
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Science, 11 (12)
White Rose Research Online URL for this paper:
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Published article:
Rathjen, M (2005)
Constructive set theory and Brouwerian principles.
Journal of
Universal Computer Science, 11 (12). 2008 - 2033. ISSN 0948-695X
Constructive Set Theory and Brouwerian Principles
1Michael Rathjen
(Department of Mathematics, The Ohio State University,2 Email: [email protected])
Abstract: The paper furnishes realizability models of constructive Zermelo-Fraenkel set theory,CZF, which also validate Brouwerian principles such as the axiom of con-tinuous choice (CC), the fan theorem (FT), and monotone bar induction (BIM), and
thereby determines the proof-theoretic strength ofCZFaugmented by these principles. The upshot is thatCZF+CC+FTpossesses the same strength asCZF, or more pre-cisely, thatCZF+CC+FTis conservative overCZFforΠ20statements of arithmetic,
whereas the addition of a restricted version of bar induction toCZF(called decidable bar induction,BID) leads to greater proof-theoretic strength in thatCZF+BIDproves
the consistency ofCZF.
Key Words: Constructive set theory, Brouwerian principles, partial combinatory al-gebra, realizability
Category:
1
Introduction
Constructive Zermelo-Fraenkel Set Theory has emerged as a standard reference theory that relates to constructive predicative mathematics as ZFC relates to classical Cantorian mathematics. The general topic of Constructive Set Theory originated in the seminal 1975 paper of John Myhill (cf. [16]), where a specific ax-iom systemCST was introduced. Myhill developed constructive set theory with the aim of isolating the principles underlying Bishop’s conception of what sets and functions are. Moreover, he wanted “these principles to be such as to make the process of formalization completely trivial, as it is in the classical case” ([16], p. 347). Indeed, while he uses other primitives in his set theoryCST besides the notion of set, it can be viewed as a subsystem ofZF. The advantage of this is that the ideas, conventions and practise of the set theoretical presentation of ordi-nary mathematics can be used in the set theoretical development of constructive mathematics, too. Constructive Set Theory provides a standard set theoretical framework for the development of constructive mathematics in the style of Errett Bishop and Douglas Bridges [6] and is one of several such frameworks for con-structive mathematics that have been considered. Aczel subsequently modified Myhill’s CSTand the resulting theory was calledZermelo-Fraenkel set theory, CZF. A hallmark of this theory is that it possesses a type-theoretic interpre-tation (cf. [1, 2, 3]). Specifically, CZF has a scheme called Subset Collection Axiom (which is a generalization of Myhill’s Exponentiation Axiom) whose for-malization was directly inspired by the type-theoretic interpretation.
1
C. S. Calude, H. Ishihara (eds.). Constructivity, Computability, and Logic. A Collection of Papers in Honour of the 60th Birthday of Douglas Bridges.
2
This paper studies augmentations ofCZFby Brouwerian principles such as the axiom of continuous choice (CC), the fan theorem (FT), and bar induction (BI). The objective is to determine whether these principles increase the proof-theoretic strength of CZF. More precisely, the research is concerned with the question of whether any newΠ0
2 statements of arithmetic (i.e. statements of the
form ∀n∈N∃k∈NP(n, k) withP being primitive recursive) become provable
upon adding CC, FT, and BI(or any combination thereof) to the axioms of CZF.
The first main result obtained here is thatCZF+CC+FTis indeed con-servative overCZFwith respect toΠ0
2 sentences of arithmetic. The first step in
the proof consists of defining a transfinite type structure over a special combi-natory algebra whose domain is the set of all arithmetical functions fromNtoN
with application being continuous function application in the sense of Kleene’s second algebraK2. The transfinite type structure serves the same purpose as a
universe in Martin-L¨of type theory. It gives rise to a realizability interpretation ofCZFwhich also happens to validate the principlesCCandFT. However, to be able to show thatFTis realized we have to employ classical reasoning in the background theory. It turns out that the whole construction can be carried out in a classical set theory known as Kripke-Platek set theory,KP. SinceCZFand KPprove the sameΠ20 statements of arithmetic this establishes the result.
A similar result can be obtained forCZFplus the so-calledRegular Extension Axiom,REA. Here it turns out thatCZF+REA+CC+BIisΠ20conservative
overCZF+REA. This time the choice for the domain of the partial combinatory algebra isNN∩L
ρ, whereρ= supn<ωωnckwithωnckdenoting thenth admissible
ordinal. Application is again continuous function application. The transfinite type structure also needs to be strengthened in that it has to be closed off under
W-types as well. A background theory sufficient for these constructions isKPi, i.e. Kripke-Platek set theory plus an axiom asserting that every set is contained in an admissible set
The question that remains to be addressed is whetherCZF+BIis conser-vative overCZF. This is answered in the negative. It is shown that a restricted form ofBI- calleddecidable bar induction,BID- implies the consistency ofCZF
on the basis ofCZF. The proof makes use of results from ordinal-theoretic proof theory.
2
Constructive Zermelo-Fraenkel Set Theory
Constructive set theory grew out of Myhill’s endeavours (cf. [16]) to discover a simple formalism that relates to Bishop’s constructive mathematics as ZFC relates to classical Cantorian mathematics. Later on Aczel modified Myhill’s set theory to a system which he called Constructive Zermelo-Fraenkel Set Theory, CZF.
Definition: 2.1 (Axioms ofCZF) The language ofCZFis the first order lan-guage of Zermelo-Fraenkel set theory,LST, with the non logical primitive symbol
∈.CZFis based on intuitionistic predicate logic with equality. The set theoretic axioms of axioms ofCZFare the following:
2. Pair∀a∀b∃x∀y(y∈x↔y=a∨y=b).
3. Union∀a∃x∀y(y∈x↔ ∃z∈a y∈z).
4. Restricted Separation scheme∀a∃x∀y(y∈x↔y∈a∧ ϕ(y)),
for everyrestrictedformulaϕ(y), where a formula ϕ(x) is restricted, or∆0,
if all the quantifiers occurring in it are restricted, i.e. of the form∀x∈b or
∃x∈b.
5. Subset Collection scheme
∀a∀b∃c∀u ∀x∈a∃y∈b ϕ(x, y, u) →
∃d∈c(∀x∈a∃y∈d ϕ(x, y, u) ∧ ∀y∈d∃x∈a ϕ(x, y, u)) for every formulaϕ(x, y, u).
6. Strong Collection scheme
∀a ∀x∈a∃y ϕ(x, y) →
∃b(∀x∈a∃y∈b ϕ(x, y)∧ ∀y∈b∃x∈a ϕ(x, y)) for every formulaϕ(x, y).
7. Infinity
∃x∀u
u∈x ↔ 0 =u ∨ ∃v∈x(u=v∪ {v})
wherey+ 1 isy∪ {y}, and 0 is the empty set, defined in the obvious way. 8. Set Induction scheme
(IN D∈) ∀a(∀x∈a ϕ(x) → ϕ(a)) → ∀a ϕ(a),
for every formulaϕ(a).
From Infinity, Set Induction, and Extensionality one can deduce that there exists exactly one setxsuch that∀uu∈x ↔ 0 =u∨ ∃v∈x(u=v∪ {v}); this set will be denoted byω.
2.1 Choice principles
In many a text on constructive mathematics, axioms of countable choice and dependent choices are accepted as constructive principles. This is, for instance, the case in Bishop’s constructive mathematics (cf. [6] as well as Brouwer’s in-tuitionistic analysis (cf. [28], Chap. 4, Sect. 2). Myhill also incorporated these axioms in his constructive set theory [16].
The weakest constructive choice principle we shall consider is the Axiom of Countable Choice, ACω, i.e. whenever F is a function with domain ω such
that ∀i∈ω∃y∈F(i), then there exists a function f with domain ω such that
∀i∈ω f(i)∈F(i).
LetxRy stand for hx, yi ∈ R. A mathematically very useful axiom to have in set theory is theDependent Choices Axiom,DC, i.e., for all setsaand (set) relationsR⊆a×a, whenever
(∀x∈a) (∃y∈a)xRy
andb0∈a, then there exists a functionf :ω→asuch thatf(0) =b0and
Even more useful in constructive set theory is the Relativized Dependent Choices Axiom,RDC. It asserts that for arbitrary formulaeφandψ, whenever
∀xφ(x) → ∃y φ(y)∧ ψ(x, y)
andφ(b0), then there exists a functionf with domainωsuch thatf(0) =b0and
(∀n∈ω)φ(f(n)) ∧ψ(f(n), f(n+ 1)).
One easily sees thatRDC impliesDCandDCimpliesACω.
2.2 The strength of CZF
In what follows we shall use the notions of proof-theoretic equivalence of theories and proof-theoretic strength of a theory whose precise definitions one can find in [18]. For our purposes here we take proof-theoretic equivalence of set theoriesT1
andT2to mean that these theories prove the sameΠ20 statements of arithmetic
and that this insight can be obtained on the basis of a weak theory such as primitive recursive arithmetic,PRA.
Theorem: 2.2 Let KP be Kripke-Platek Set Theory (with the Infinity Axiom) (see [4]).
(i) CZFandCZF−are of the same proof-theoretic strength asKPand the clas-sical theoryID1 of non-iterated positive arithmetical inductive definitions.
These systems prove the sameΠ0
2 statements of arithmetic.
(ii) The system CZF augmented by the Power Set axiom is proof-theoretically stronger than classical Zermelo Set theory, Z(in that it proves the consis-tency ofZ).
(iii) CZFdoes not prove the Power Set axiom.
Proof: Let Pow denote the Power Set axiom. (i) follows from [17] Theorem 4.14. Also (iii) follows from [17] Theorem 4.14 as one easily sees that 2-order Heyting arithmetic has a model in CZF+Pow. Since second-order Heyting arithmetic is of the same strength as classical second-order arithmetic it follows that CZF+Pow is stronger than classical second-order arithmetic (which is much stronger thanKP). But more than that can be shown. Working inCZF+ Pow one can iterate the power set operation ω+ω times to arrive at the set
Vω+ω which is readily seen to be a model of intuitionistic Zermelo Set Theory,
Zi. AsZcan be interpreted in Zi by means of a double negation translation as
was shown in [13] Theorem 2.3.2, we obtain (ii). ut
The first large set axiom proposed in the context of constructive set theory was theRegular Extension Axiom,REA, which was introduced to accommodate inductive definitions inCZF(cf. [?], [2]).
Definition: 2.3 A setc is said to beregular if it is transitive, inhabited (i.e.
∃u u∈c) and for anyu∈c and setR⊆u×cif∀x∈u∃y hx, yi ∈Rthen there is a setv∈csuch that
∀x∈u∃y∈vhx, yi ∈R ∧ ∀y∈v∃x∈uhx, yi ∈R.
We writeReg(a) for ‘ais regular’. REAis the principle
Theorem: 2.4 Let KPi be Kripke-Platek Set Theory plus an axiom asserting that every set is contained in an admissible set (see [4]).
(i) CZF+REAis of the same proof-theoretic strength asKPiand the subsys-tem of second-order arithmetic with∆12-comprehension and Bar Induction.
(ii) CZF+REAdoes not prove the Power Set axiom.
Proof:(i) follows from [17] Theorem 5.13. (ii) is a consequence of (i) and
The-orem 2.2. ut
3
Russian constructivism
We give a brief review of Russian constructivism which is intended to serve the purpose of enhancing our account of Brouwerian intuitionism by contrast. The concept of algorithm or recursive function is fundamental to theRussian schools of Markov and Shanin. Contrary to Brouwer, this school takes the viewpoint that mathematical objects must be concrete, or at least have a constructive description, as a word in an alphabet, or equivalently, as an integer, for only on such objects do recursive functions operate. Furthermore, Markov adopts what he calls Church’s thesis, CT, which asserts that whenever we see a quantifier combination ∀n∈N∃m∈NA(n, m), we can find a recursive function f which
produces m from n, i.e. ∀n ∈ NA(n, f(n)). On the other hand, as far as pure
logic is concerned he augments Brouwer’s intuitionistic logic by what is known asMarkov’s principle,MP, which may be expressed as
∀n∈N[A(n)∨ ¬A(n)] ∧ ¬∀n∈N¬A(n) → ∃n∈NA(n),
withAcontaining natural number parameters only. The rationale for accepting
M P may be phrased as follows. Suppose A is a predicate of natural numbers which can be decided for each number; and we also know by indirect arguments that there should be an n such that A(n). Then a computer with unbounded memory could be programmed to search through N for a number n such that A(n) and we should be convinced that it will eventually find one. As an example for an application ofMPto the reals one obtains∀x∈R(¬x≤0 → x >0).
In the next section we shall recall that Church’s thesis is incompatible with Brouwer’s principlesCCandFTD.
CT is also incompatible with the axiom of choice AC1,0 to be defined in
Definition 4.1(2).
Lemma: 3.1 AC1,0 refutesCT.
Proof:See [27] or [5], Theorem 19.1. ut
One of the pathologies of CTis that it refutesUC.
Theorem: 3.2 CTimplies that there exists a continuous functionf : [0,1]→R
which is unbounded and hence not uniformly continuous.
Proof:[28], 6.4.4. ut
However, in Russian constructivism one can also prove that all functions from
Definition: 3.3 Extended Church’s Thesis,ECT, asserts that
∀n∈Nψ(n)→ ∃m∈Nϕ(n, m) implies
∃e∈N∀n∈Nψ(n)→ ∃m, p∈N[T(e, n, p)∧ U(p, m)∧ ϕ(n, m)]
wheneverψ(n) is an almost negative arithmetic formula and ϕ(u, v) is any for-mula. A formula θ of the language of CZF with quantifiers ranging over N is
said to bealmost negative arithmeticif∨does not appear in it and instances of
∃m∈Nappear only as prefixed to primitive recursive subformulae ofθ.
Note thatECTimpliesCT, takingψ(n) to ben=n.
Theorem: 3.4 Under the assumptionsECTandMP, all functionsf :R→R
are continuous.
Proof:[28], 6.4.12. ut
4
Brouwer’s world
This section expounds on principles specific to Brouwer’s intuitionism and de-scribes their mathematical consequences.
Intuitionistic mathematics diverges from other types of constructive math-ematics in its interpretation of the term ‘sequence’. This led to the following principle of continuous choice, abbreviatedCC, which we divide into a con-tinuity part and a choice part:
Definition: 4.1 CCis the conjunction of (1) and (2):
1. Any function fromNNto Nis continuous.
2. IfP ⊆NN×N, and for eachα∈NNthere existsn∈Nsuch that (α, n)∈P,
then there is a functionf :NN→
Nsuch that (α, f(α))∈P for allα∈NN.
The first part ofCC will also be denoted byCont(NN,N). The second part of
CCis often denoted byAC1,0.
The justification forCCsprings from Brouwer’s ideas about choice sequences. Let P ⊆ NN×N, and suppose that for each α ∈ NN there exists n ∈ N such
that (α, n)∈P. According to Brouwer, the construction of an element ofNN is
forever incomplete. A generic sequenceαis purely extensional, in the sense that at any given moment we can know nothing aboutαother than a finite number of its terms. It follows that for a given sequence α, our procedure for finding an n ∈ N such that (α, n) ∈ P must be able to calculate n from some finite initial sequence ¯α(m).3 If β is another such sequence, and ¯α(m) = ¯β(m), then our procedure must return the samenforβ as it does forα. So this procedure defines a continuous functionf :NN→Nsuch that (α, f(α))∈P for allα∈NN,
where we assume the topology onNNto be the product topology.
It is plain thatCont(NN,N) is incompatible with classical logic.
The other principle central to Brouwerian mathematics is the so-called Fan Theoremwhich is also classically valid and equivalent to K¨onig’s lemma,KL.
3
¯
Definition: 4.2 Let 2N be the set of all binary sequencesα:
N→ {0,1} and
let 2∗ be the set of finite sequences of 0s and 1s. For s, t∈2∗ we write s⊆t to mean thatsis an initial segment of t. Abar of2∗ is subset Rof 2∗ such that the following property holds:
∀α∈2N∃nα¯(n)∈R. The barR isdecidableif it also satisfies
∀s∈2∗(s∈R ∨ s /∈R).
FTD is the statement that every decidable bar R of 2∗ is uniform, i.e., there
exists a natural number msuch that
∀α∈2N∃k≤mα¯(k)∈R.
TheFan TheoremorGeneral Fan Theorem,FT, is the statement that every bar
R of 2∗ is uniform.
Lemma: 4.3 FTD refutes CT.
Proof:ApplyFTD to Kleene’s singular tree. For details see [28] 4.7.6. ut
4.1 Decidable Bar induction
Brouwer justified FTD by appealing to a principle known as decidable Bar
Induction,BID.
Definition: 4.4 Let N∗ be the set of all finite sequences of natural numbers.
If s ∈ N∗, m ∈ N and s = hs0, . . . ski then s ∗ hmi denotes the sequence
hs0, . . . , sk, mi. Abar ofN∗ is defined in the same vein as a bar of 2∗.
BID is asserts that for every decidable bar RofN∗ and arbitrary classQ,
∀s∈N∗(s∈R→s∈Q) ∧
∀s∈N∗[(∀k∈Ns∗ hki ∈Q)→s∈Q] →
hi ∈Q.
Monotone Bar Induction,BIM, asserts that for every barRofN∗and arbitrary
classQ,
∀s, t∈N∗(s∈R→s∗t∈R) ∧
∀s∈N∗(s∈R→s∈Q) ∧
∀s∈N∗[(∀k∈Ns∗ hki ∈Q)→s∈Q) →
hi ∈Q.
It is easy to see thatBIMentailsBID (cf. [10], Theorem 3.7).
Corollary: 4.5 BID impliesFTD andBIM impliesFT.
4.2 Local continuity
In connection with Brouwer’s intuitionism, one often works with the local con-tinuity, LCP, rather thanCC.LCPentailsCont(NN,N) but notAC1,0.
Definition: 4.6 Forα, β∈NNwe writeβ ∈α¯(n) to mean ¯β(n) = ¯α(n).
TheLocal Continuity Principle,LCP, states that
∀α∈NN∃n∈NA(α, n)→
∀α∈NN∃n, m∈N∀β ∈NN[β∈α¯(m)→A(β, n)].
LCPis also known as theWeak Continuity Principle(WC-N) (see [28], p. 371) orBrouwer’s Principle for Numbers(BP0).
Some obvious deductive relationships between some of the principles are recorded in the next lemma.
Lemma: 4.7 (i) LCP⇒Cont(NN,N).
(ii) CC⇔LCP ∧ AC1,0
Proof:Obvious. ut
While CCentails the choice principle AC1,0, it is not compatible with choice
for higher type objects. In point of fact, the incompatibility already occurs in connection with a consequence ofCC.
Lemma: 4.8 Let AC2,0 be the following principle:
If P is a subset of (NN → N)×N such that for every f : NN → N
there exists n ∈ N such that (f, n) ∈ P, then there exists a function F: (NN→
N)→Nsuch that for all f :NN→N,(f, F(f))∈P.
AC2,0 refutesCont(NN,N).
Proof:See [27] or [5], Theorem 19.1. ut
LCP and a fortiori CC refute principles of omniscience. For α ∈ 2N let
αn:=α(n).
Definition: 4.9 Limited Principle of Omniscience (LPO):
∀α∈2N[∃n αn= 1 ∨ ∀n αn= 0]. Lesser Limited Principle of Omniscience (LLPO):
∀α∈2N ∀n, m[α
n=αm= 1→n=m] → [∀n α2n= 0 ∨ ∀n α2n+1= 0]
.
Lemma: 4.10 (i) LCP⇒ ¬LPO. (ii) CC⇒ ¬LLPO.
Proof:(i) follows from [28] 4.6.4. (ii) is proved in [7], 5.2.1. ut
LCPis also incompatible with Church’s thesis.
Proof:[28], 4.6.7. ApplyLCP toCT. ut
With the help ofLCP one deduces Brouwer’s famous theorem.
Theorem: 4.12 If LCP holds, then every mapf :R→Ris pointwise
contin-uous.
Proof:[28] Theorem 6.3.4. ut
Brouwer needed the fan theorem to derive the classically validUniform Con-tinuity Principle:
UC Every pointwise continuous function from 2Nto N is uniformly continuous.
Lemma: 4.13 FT impliesUC.
Proof:[28] Theorem 6.3.6. ut
UnderLCP we can drop the condition of continuity.
Corollary: 4.14 If LCP and FThold, then every f : [a, b]→R is uniformly
continuous and has a supremum.
Proof:[28] Theorem 6.3.8. ut
Theorem: 4.15 Under the hypothesis CC, the statements UC and FTD are
equivalent.
Proof. See [7] Theorem 5.3.2 and Corollary 5.3.4. ut
Corollary: 4.16 If CCandFTD hold, then every mapping of a nonvoid
com-pact metric space into a metric space is uniformly continuous.
Proof:[7], Theorem 5.3.6. ut
5
More Continuity Principle
Another continuity principle one finds in the literature isStrong Continuity for Numbers:
C-N ∀α∈NN∃x∈
NA(α, n) → ∃γ∈K∗∀α∈NNA(α, γ(α)), (1)
whereK∗ is the class ofneighbourhood functions, i.e.γ∈K∗ iffγ:
N∗→Nand γ(hi) = 0∧ ∀s, t∈N∗[γ(s)6= 0→γ(s) =γ(s∗t)]∧ ∀α∈NN∃n∈Nγ( ¯α(n))6= 0
and
γ(α) =niff∃m∈N[γ( ¯α(m)) =n+ 1].
Dummett in [10], p. 60 refers toC-N as ’the Continuity Principle’ and assigns it the acronym CP∃n.
Definition: 5.1 We introduce abbreviations for partial continuous application:
α(β) =x:=∃yα( ¯β(y)) =x+ 1 ∧ ∀y0< y(α( ¯β(y0)) = 0, α|β=γ:=∀x[λn.α(hxi ∗n)(β) =γ(x)] ∧ α(0) = 0.
A yet stronger continuity principle is functional continuous choice F-CC or CONT1(cf. [28],7.6.15) (also denoted by C-C in [28],7.6.15 and by CP∃β in [10],
p. 60):
F-CC ∀α∈NN∃β∈NNA(α, β)→ ∃γ∈K∗∀α∈NNA(α, γ|α).
F-CCis not considered to be part of Brouwer’s realm as it is actually inconsistent with some of Brouwer’s later, though controversial, proposals about the “creative subject” (see [10] 6.3).
In point of fact,F-CCis equivalent to the schema
F-CC0 ∀α∈NN∃β∈NNA(α, β)→ ∃γ∈NN∀α∈NNA(α, γ|α)
because ifγ|αis defined for allα, one can defineγ∗∈K∗such thatγ∗|α=γ|α
for allα∈NN whereγ∗is recursive inγ: Putγ∗(hi) = 0 and
γ∗(s∗ hni) =
(γ∗(s) ifγ∗(s)>0
γ(s) ifγ∗(s) = 0 andγ(s)>0 0 ifγ∗(s) =γ(s) = 0.
Lemma: 5.2 (i) C-N ⇔ CC. (ii) F-CCimpliesCC.
Proof:For (i) see [7], p. 119. (ii) is to be found in [28], 7.6.15. ut
In the presence ofCC, one can actually dispense with the decidability of the bar inFTD andBID.
Lemma: 5.3 Assuming CC,FTD impliesFTandBID implies BIM.
Proof: This follows fromC-N ⇔ CCby [10], Theorem 3.8 and the proof of
the general fan theorem of [10], page 64. ut
6
Elementary analysis augmented by Brouwerian principles
Realizability interpretations of Brouwerian principles have been given for elemen-tary analysis. It should be instructive to review these results before venturing to the more complex realizability models required for set theory.
Elementary analysis, EL, is a two-sorted intuitionistic formal system with variablesx, y, z, . . .intended to range over natural numbers and variablesα, β, γ, . . .
intended to range over one-place total functions fromN toN. The language of ELis an extension of the language of of Heyting Arithmetic. In particular,πis a symbol for a primitive-recursive pairing function onN×NandSis the symbol
for the successor function.
for explicit definitions of functions, and a recursion-operator Rec such that (ta numerical term, φa function term;φ(t, t0) :=φ(π(t, t0)))
Rec(t, φ)(0) =t, Rec(t, φ)(Sx) =φ(x,Rec(t, φ)(x)).
Induction is extended to all formulas in the new language. The functions ofEL are assumed to be closed under “recursive in”, which is expressed by including a weak choice axiom for quantifier-freeA:
QF-AC ∀x∃y A(x, y)→ ∃α∀x A(x, α(x)).
We may introduce the operations|,·(·) of Definition 5.1 as primitive operators in a conservative extension EL∗ based on the logic of partial terms.
It was shown by Kleene in [15],Ch.II thatELaugmented by the principlesBIM
and C-N is consistent. For this purpose he used a realizability interpretation based on continuous function application, i.e. the second Kleene algebra K2.
As this paper will follow a similar strategy for gauging the strength of CZF extended by Brouwerian principles it is instructive to review Kleene’s results.
Definition: 6.1 (Function realizability) InELequality of functions α=β
is not a prime formula and defined by∀x α(x) =β(x).
α|β ↓stands for∃γ α|β =γ.
The function realizability interpretation is an inductively defined translation of a formulaA ofELinto a formula αrfAofEL, whereαis a fresh variable not occurring inA:
αrf⊥ iff ⊥
αrft=s iff t=s
αrfA∧B iff π0αrfA ∧ π1αrfB
αrfA∨B iff [α(0) = 0→α+rfA] ∧ [α(0)6= 0→α+rfB]
αrfA→B iff ∀β [β rfA→α|β↓ ∧α|β rfB]
αrf∀xA(x) iff ∀x[α|λn.x↓ ∧α|λn.xrfA(x)]
αrf∃xA(x) iff α+rfA(α(0))
αrf∀βA(β) iff ∀β[α|β ↓ ∧α|β rfA(β)]
αrf∃βA(β) iff π1αrfA(π0α)
with π0, π1 being the projection functions with respect to some fixed primitive
recursive pairing function π : N×N → N and α+ being the tail of α (i.e. α=hα0i ∗α+).
Lemma: 6.2 Let A be a closed formula of EL. If EL+C-N `A, then EL` ∃αrfA.
Proof:See [26], theorem 3.3.11. ut
Kleene shows in [15], Ch.11, Lemma 9.10 that the arithmetical functions constitute the least class of functionsC ⊆NNclosed under general recursiveness
and the jump operation ’ (look there for precise definitions).
An inhabited setC ⊆NNgives rise to a structureNC for the language ofEL
as follows: The domain ofNC isN ∪ C, the number variables range overN; the
Lemma: 6.3 (KP) For any set of functionsC ⊆NN closed under general
re-cursiveness and the jump operation ’: the fan theorem FTholds in the classical model NC of EL, when the representing function of the barR belongs toC.
Proof:[15], Lemma 9.12. ut
Theorem: 6.4 (KP) For any set of functions C ⊆ NN closed under general
recursiveness and the jump operation ’ (e.g. the arithmetical functions): IfEL+ C-N+FT`A, thenNC |=∃α αrfA.
Proof:[15], Ch.11 Theorem 9.13. ut
Kleene’s proof of Theorem 6.4 can actually be extended to includeBIM.
Definition: 6.5 An inhabited set of functionsC ⊆NNis said to be aβ-model
ifCis closed under general recursiveness and the jump operation ’ and whenever
≺ is a binary relation on N whose representing function is in C and NC |=
∀α∃n¬α(n+ 1)≺α(n), then≺is well-founded.
Lemma: 6.6 (KPi)IfC ⊆NNis aβ-model then monotone bar induction holds
in the classical model NC of EL, when the representing function of the bar R
belongs toC.
Proof:Similar to [15], Ch.11 Lemma 9.12. ut
Theorem: 6.7 (KPi)For anyβ-modelC ⊆NN (e.g. the functions inNN∩Lρ,
where ρ = supn<ωωck
n with ωnck being the nth admissible ordinal; cf. [4]): If
EL+C-N+BIM`A, then NC |=∃α αrfA.
Proof:SinceBIMfollows fromBIDon the basis ofEL+C-Nit suffices to find
realizers for instances ofBID.
So assume that
β rf ∀n[R(n)∨ ¬R(n)], (2)
γ rf ∀α∃n R( ¯α(n)), (3)
δ rf ∀s[R(s)→Q(s)], (4)
η rf ∀s[∀x Q(s∗ hxi) → Q(s)]. (5) Setβn :=β|λx.n. (2) implies that
βn(0)→βn+rfR(n) andβn(0)6= 0→βn+rf¬R(n) (6) while (3) yields that
∀α π1(γ|α)rfR( ¯α((π0(γ|α))(0))),
so that
∀α∃m βα¯(m)(0) = 0. (7)
Now define aonNbyts:=βs(0)6= 0 ∧ ∃u t=s∗ hui. On account of (7)
Define a functionψ:N→ C by transfinite recursion onas follows:
ψ(s) =
δ|βs+ ifβs(0) = 0 (η|λu.s)|`(ψ, s) ifβs(0)6= 0,
where`is aC-valued operation to the effect that`(α, s)|λu.k=α(s∗ hki). Note that formal terms denoting ψ and ` in the the model NC (uniformly in the parameters β, γ, δ, η) can be defined in the system EL∗ (based on the logic of partial terms) using the recursion theorem and other gadgets.
By transfinite induction onwe shall prove that for all s∈N,
ψ(s)rfQ(s) (8)
Case 1: Suppose that βs = 0. Using (6) we get β+
s rfR(s), and henceδ|βs+ ↓
∧δ|β+
s rfQ(s) by (4); thusψ(s)rfQ(s).
Case 2: Now suppose thatβs6= 0. Thens∗hkisfor allk∈N, and the inductive
hypothesis yieldsψ(s∗ hki)rfQ(s∗ hki) for allk; thence`(ψ, s)rf∀x Q(s∗ hxi). By (5) we haveη|λu.srf∀x Q(s∗ hxi) → Q(s), so that
(η|λu.s)`(ψ, s)rfQ(s).
In sum, we haveψ(s)rfQ(s), confirming (8).
In view of the above we conclude the realizability ofBID in the modelNC.
u t
7
Combinatory Algebras
The notion of function is crucial to any concrete BHK-interpretation in that it will determine the set theoretic and mathematical principles validated by it. The most important semantics for intuitionistic theories, known asrealizability interpretations, also require that we have a set of (partial) functions on hand that serve as realizers for the formulae of the theory. An abstract and therefore “cleaner” approach to this semantics considers realizability over general domains of computations allowing for recursion and self-application. These structures have been variably calledpartial combinatory algebras,applicative structures, or Sch¨onfinkel algebras. They are closely related to models of theλ-calculus.
Let (M,·) be a structure equipped with a partial operation, that is,·is a binary function with domain a subset ofM×M and co-domainM. We often omit the sign “·” and adopt the convention of “association to the left”. Thusexymeans (e·x)·y. We also sometimes writee·xin functional notation ase(x). Extending this notion to several variables, we write e(x, y) forexyetc.
Definition: 7.1 AP CAis a structure (M,·), where·is a partial binary opera-tion onM, such thatM has at least two elements and there are elementskand sinM such that kxyandsxyare always defined, and
(i) kxy=x
where'means that the left hand side is defined iff the right hand side is defined, and if one side is defined then both sides yield the same result.
(M,·) is atotalPCA ifa·bis defined for alla, b∈M.
Definition: 7.2 Partial combinatory algebras are best described as the models of a formal theoryPCA. The language ofPCAhas two distinguished constants kands. To accommodate the partial operation in a standard first order language, the language ofPCAhas a trinary relation symbolAp. ThetermsofPCAare just the variables and constants.Apwill almost never appear in what follows as we prefer to writet1t2't3forAp(t1, t2, t3). In order to facilitate the formulation
of the axioms, the language ofPCAis expanded definitionally with the symbol
'and the auxiliary notion of anapplication termor partial termis introduced. The set of application terms is given by two clauses:
1. All terms of PCAare application terms; and
2. Ifsand tare application terms, then (st) is an application term.
Forsandtapplication terms, we have auxiliary, defined formulae of the form:
s't := ∀y(s'y↔t'y),
iftis not a variable. Heres'a(foraa free variable) is inductively defined by:
s'a is
s=a, ifsis a term ofPCA,
∃x, y[s1'x∧ s2'y∧ Ap(x, y, a)] ifsis of the form (s1s2). Some abbreviations aret1t2. . . tn for ((...(t1t2)...)tn);t↓for∃y(t'y) andφ(t)
for∃y(t'y∧φ(y)).
In this paper, the logic of PCA is assumed to be that of intuitionistic predicate logic with identity.PCA’snon-logical axiomsare the following:
Axioms of PCA
1. ab'c1 ∧ ab'c2 → c1=c2.
2. (kab)↓ ∧kab'a. 3. (sab)↓ ∧sabc'ac(bc).
The following shows howλ-terms can be constructed in PCA.
Lemma: 7.3 For each application term t and variable x, one can construct a term λx.t, whose free variables are those of t, excluding x, such that PCA `
λx.t ↓ and PCA ` (λx.t)u' t[x/u] for all application terms u, where t[x/u] results fromt by replacingxint throughout by u.
Proof:We proceed by induction on the buildup oft. (i)λx.xisskk; (ii)λx.tiskt
forta constant ofPCAor variable other thanx; (iii)λx.t1t2iss(λx.t1)(λx.t2).
u t
Lemma: 7.4 (Recursion Theorem) There is an application term r such that PCA proves:
rx↓ ∧rxy'x(rx)y.
Proof:Takerto beλx.ggwhereg:=λzy.x(zz)y. Thenrx'gg'(λzy.x(zz)y)g'
λy.x(gg)y, so thatrx↓by Lemma 7.3. Moreover,rxy'x(gg)y'x(rx)y. ut
Corollary: 7.5 PCA ` ∀f∃g∀x1. . .∀xng(x1, . . . , xn)'f(g, x1, . . . , xn).
It often convenient to equip a PCA with additional structure such as pairing, natural numbers, and some form of definition by cases. In fact, these gadgets can be constructed in any PCA, as Curry showed. Nonetheless, it is desirable to consider richer structures as the natural models for PCAs we are going to study come already furnished with a “natural” copy of the natural numbers, natural pairing functions, etc., which are different from the constructions of combinatory logic.
Definition: 7.6 The language of PCA+ is that of PCA, with a unary rela-tion symbol N (for a copy of the natural numbers) and additional constants 0,sN,pN,d,p,p0,p1 for, respectively, zero, successor onN, predecessor onN,
definition by cases on N, pairing, and the corresponding two projections. TheaxiomsofPCA+are those of PCA, augmented by the following:
1. (pa0a1)↓ ∧(p0a)↓ ∧(p1a)↓ ∧pi(pa0a1)'aifori= 0,1.
2. N(c1) ∧ N(c2) ∧ c1=c2→dabc1c2↓ ∧dabc1c2'a.
3. N(c1) ∧ N(c2) ∧ c16=c2→dabc1c2↓ ∧dabc1c2'b.
4. ∀x N(x)→
sNx↓ ∧sNx6=0 ∧ N(sNx)
. 5. N(0) ∧ ∀x N(x) ∧ x6=0→
pNx↓ ∧sN(pNx) =x
. 6. ∀x
N(x)→pN(sNx) =x.
The extension ofPCA+ by the schema of induction for all formulae,
ϕ(0) ∧ ∀x
N(x) ∧ ϕ(x)→ϕ(sNx)
→ ∀x
N(x)→ϕ(x)
is is known by the acronymEON(elementary theory ofoperations andnumbers) or APP (applicative theory). For full details about PCA, PCA+, andEON see [11, 12, 5, 28].
Let1:=sN0. The applicative axioms entail that1is an application term that
evaluates to an object falling under N but distinct from0, i.e., 1↓,N(1) and 0 6= 1. More generally, we define the standard integers of a P CA to be the interpretations of the numerals, i.e. the terms ¯ndefined by ¯0 =0and n+ 1 = sNn¯ forn∈N. Note thatPCA+`¯n↓.
AP CA+(M,·, . . .) whose integers are standard, meaning that{x∈M|M |=
N(x)} is the set consisting of the interpretations of the numerals inM, will be called anω-P CA+. Note that anω-P CA+ is also a model ofAPP.
Some further conventions are useful. Systematic notation forn-tuplesis intro-duced as follows: (t) ist, (s, t) ispst, and (t1, . . . , tn) is defined by ((t1, . . . , tn−1), tn).
Lemma: 7.7 PCA+ is conservative overPCA.
7.1 Kleene’s Examples of Combinatory Algebras
The primordial PCA is furnished by Turing machine application on the integers. There are many other interesting PCAs that provide us with a laboratory for the study of computability theory. As the various definitions are lifted to more general domains and notions of application other than Turing machine applica-tions some of the familiar results break down. By studying the noapplica-tions in the general setting one sees with a clearer eye the truths behind the results on the integers.
7.1.1 Kleene’s first model
The “standard” applicative structure is Kleene’s first model, calledK1, in which
the universe|K1| isNandApK1(x, y, z) is Turing machine application:
ApK1(x, y, z) iff {x}(y)'z.
The primitive constants of PCA+ are interpreted over N in the obvious way,
andN is interpreted asN.
Kleene’s second model
The universe of “Kleene’s second model” ofAPP,K2, isNN. The most
interest-ing feature of K2 is that in the type structure overK2, every type-2 functional
is continuous.
We shall use α, β, γ, . . .as variables ranging over functions fromNto N. In
order to describe this PCA, it will be necessary to review some terminology.
Definition: 7.8 We assume that every integer codes a finite sequence of inte-gers. For finite sequences σ and τ, σ ⊂ τ means thatσ is an initial segment ofτ;σ∗τ is concatenation of sequences;hiis the empty sequence;hn0, . . . , nki
displays the elements of a sequence; if this sequence isτthenlh(τ) =k+ 1 (read “length ofτ”); ¯α(m) =hα(0), . . . , α(m−1)iifm >0; ¯α(0) =hi. A functionα
and an integer n produce a new functionhni ∗α which is the function β with
β(0) =nandβ(k+ 1) =α(k).
Application requires the following operations onN
N:
αβ = m iff ∃n[α( ¯βn) =m+ 1 ∧ ∀i < n α( ¯βi) = 0] (α|β)(n) = α(hni ∗β)
We would like to define application onNNbyα|β, but this is in general only a partial function, therefore we set:
α·β =γ iff ∀n(α|β)(n) =γ(n). (9) Theorem: 7.9 K2 is a model ofAPP.
Proof:For the natural numbers ofK2takeN :={nˆ|n∈N}, where ˆndenotes the
constant function onNwith valuen. For pairing define the functionP :N
N→NN
by P(α, β)(n) = α(n/2) if n is even andP(α, β)(n) =β(n−21) if n is odd. We then have to find a specific π ∈NNsuch that (π|α)|β =P(α, β) for all α and
β. Details on how to define all the constants of APP in K2 can be found in
Substructures of Kleene’s second model
Inspection of the definition of application in K2 shows that subcollections of
NNclosed under “recursive in” give rise to substructures ofK
2that are models
of APP as well. Specifically, the set of unary recursive functions forms a sub-structure ofK2 as does the set of arithmetical functions fromNto N, i.e., the
functions definable in the standard model of Peano Arithmetic, furnish a model ofAPP when equipped with the application of (9).
8
Type Structures over Combinatory Algebras
We shall define an “internal” version of a transfinite type structure with depen-dent products and dependepen-dent sums over any applicative structure.
Definition: 8.1 Let P = (P,·, . . .) be an ω-P CA+. The types of P and their
elements are defined inductively. The set of elements of a type A is called its extensionand denoted by ˆA. The type structure will be denoted by TP.
1. NP is a type with extension the set of integers ofP, i.e.,{x∈P|P|=N(x)}. 2. For each integern,N
P
n is a type with extension{k¯
P
|k= 0, . . . n−1}ifn >0 andN
P
0=∅.
3. UP is a type with extension P.
4. IfA andB are types, thenA+PB is a type with extension
{(0, x)|x∈Aˆ} ∪ {(1, x)|x∈Bˆ}.
5. If A is a type and for each x∈Aˆ, F(x) is a type, where F ∈P and F(x) meansF·x, then
P Y
x:A F(x)
is a type with extension{f ∈P| ∀x∈A fˆ ·x∈F[(x)}.
6. IfA is a type and for eachx∈Aˆ,F(x) is a type, whereF ∈P, then P
X
x:A F(x)
is a type with extension{(x, u)|x∈Aˆ ∧ u∈F[(x)}.
The obvious question to ask is: Why should we distinguish between a type A
and its extension ˆA. Well, the reason is that we want to apply the application operation ofPto types. For this to be possible, types have to be elements ofP. Thus types aren’t sets. Alternatively, however, we could identify types with sets and require that they be representable inPin some way. This can be arranged by associating G¨odel numbers inPwith types and operations on types. This is
easily achieved by employing the coding facilities of theP CA+
P. For instance,
A+B has G¨odel number (1,pAq,pBq), and ifC is a type with G¨odel number
pCq, F ∈P, and for all x∈Cˆ, F(x) is the G¨odel number of a type Bx, then (2,pCq, F) is the G¨odel number of the dependent type QP
x:CBx, etc. In what
follows we will just identify types with their extensions (or their codes) as such ontological distinctions are always retrievable from the context.
Remark: 8.2 The ordinary product and arrow types can be defined with the aid of dependent products and sums, respectively. LetA, B be types andF ∈P
be a function such that F(x) =B for allx∈P.
A×B := P X
x:A
F(x) A→B := P Y
x:A F(x).
.
Definition: 8.3 (The set-theoretic universe VP) Starting from the inter-nal type structure over an ω-P CA+
P, we are going to construct a universe
of sets for intuitionistic set theory. The rough idea is that a set X is given by a type A together with a set-valued function f defined on A (or rather the extension of A) such that X ={f(x)|x∈ Aˆ}. Again, the objects of this uni-verse will be coded as elements ofP. The above set will be coded as sup(A, f), where sup(A, f) = (8,(A, f)) or whatever. We sometimes write {f(x)|x∈A}
for sup(A, f).
Frequently we shall writex∈Arather thanx∈Aˆ. The universe of sets over the type structure ofP,V
P
, is defined inductively by a single rule:
ifA is a type overP,f ∈P, and∀x∈A fˆ ·x∈V
P
, thensup(A, f)∈VP. We shall use variablesx,y,z, . . .to range over elements of VP. Each x∈VP is of the form sup(A, f). Define ¯x:=A and ˜x:=f.
An essential characteristic of set theory is that sets having the same elements are to be identified. So if{f(x)|x∈A}and{g(y)|y∈B}are inVPand for every
x∈ A there exists y ∈ B such that f(x) andg(y) represent the same set and conversely for everyy∈B there existsx∈Asuch thatf(x) andg(y) represent the same set, then{f(x)|x∈A}and{g(y)|y∈B}should be identified as sets. This idea gives rise to an equivalence relation onVP.
Definition: 8.4 (Kleene realizability over VP) We will introduce a seman-tics for sentences of set theory with parameters fromVP. Bounded set quantifiers will be treated as quantifiers in their own right, i.e., bounded and unbounded quantifiers are treated as syntactically different kinds of quantifiers. Letx,y∈VP ande, f ∈P. We writeei,j for ((e)i)j.
e
Px∈yiff (e)0∈¯y ∧ (e)1Px= ˜y(e)0
e
Px=yiff∀i∈¯x[e0,0i∈¯y ∧ e0,1iP˜xi= ˜y(e0,0i)] ∧
∀i∈¯y[e1,0i∈¯x ∧ e1,1iP˜yi= ˜x(e1,0i)]
e
Pφ∨ψiff
(e)0=0∧ (e)1Pφ
∨
(e)0=1 ∧ (e)1Pψ
e
P ¬φ iff∀f ∈P ¬f Pφ
eP φ→ψiff∀f ∈Pf Pφ → ef Pψ eP∀x∈xφ(x) iff∀i∈¯xeiPφ(˜xi)
eP∃x∈xφ(x) iff (e)0∈¯x ∧ (e)1Pφ(˜x((e)0))
eP ∀xφ(x) iff∀x∈VPexPφ(x)
eP ∃xφ(x) iff (e)0∈V
P
∧ (e)1P φ((e)0). The definitions ofe
Px∈yandePx=yfall under the scope of definitions by transfinite recursion, i.e. by recursion on the inductive definition of VP.
Theorem: 8.5 Let Pbe anω-P CA+. Letϕ(v1, . . . , vr)be a formula of set
the-ory with at most the free variables exhibited. If
CZF+RDC`ϕ(v1, . . . , vr)
then there exists a closed application termtϕ ofPCA +
such that for allx1, . . . ,xr∈
VP,
P|=tϕx1. . .xr↓
and
tϕx1. . .xr P ϕ(x1, . . . ,xr).
The termtϕcan be effectively constructed from theCZF-deduction ofϕ(v1, . . . , vr).
Remark: 8.6 A background theory sufficient for carrying out the definition of VP and establishing Theorem 8.5 is KP. More precisely, if KP proves that P
is an ω-P CA+ and CZF+RDC ` ϕ(v1, . . . , vr), then there exists a closed
application term tϕ of PCA+ such that KP proves for all x1, . . . ,xr ∈ V
P ,
P|=tϕx1. . .xr↓ andtϕx1. . .xr P ϕ(x1, . . . ,xr).
To obtain a similar result for CZFplus the regular extension axiom we need a stronger type structure.
Definition: 8.7 LetP= (P,·, . . .) be anω-P CA+.
The type structure TP
W is defined by adding one more inductive clause to
Definition 8.1.
(7) IfA is a type and for each x∈Aˆ, F(x) is a type, where F ∈P and F(x) meansF·x, then
WxP:AF(x)
is a type with extension S, where S is the set inductively defined by the following clause:
Ifa∈Aˆ,f ∈P, and∀x∈F[(a)f·x∈S, thenp(a, f)∈S. The set-theoretic universe VP
W is defined in the same vein asV
P
except that it is built over the type structureTP
W.
Realizability overVP
W is defined similarly as in Definition 8.4 with V
P
W
Theorem: 8.8 Let Pbe anω-P CA+. Letϕ(v
1, . . . , vr)be a formula of set
the-ory with at most the free variables exhibited. If
CZF+REA+RDC`ϕ(v1, . . . , vr)
then there exists a closed application termtϕ ofPCA+such that for allx1, . . . ,xr∈
VWP,
P|=tϕx1. . .xr↓
and
tϕx1. . .xr P ϕ(x1, . . . ,xr).
The term tϕ can be effectively constructed from the CZF+REA-deduction of
ϕ(v1, . . . , vr).
Remark: 8.9 A background theory sufficient for carrying out the definition of VWP and establishing Theorem 8.8 isKPi.
9
The set-theoretic universe over Kleene’s second model
Henceforth let U be a subset of NN closed under ‘recursive in’ and the jump
operator. LetAbe Kleene’s second model based onU, i.e. the applicative struc-ture with domain U and application being continuous function application, |. The interpretation of the natural numbers inAthat is the interpretationNA of
the predicate symbol N in A is the set of all constant functions. We use ˆn to
denote the constant function with valuen. In particular there are the interpreta-tionskA,sA,0A,sA
N,pAN,dA,pA,pA0,pA1 of the constants ofAPP inU. We shall,
however, mostly drop the superscriptA.
Our goal is to show that in addition to the axioms ofCZF,VA also realizes CCandFT. The first step is to single out the element ofVA that plays the role ofω and the Baire space ωω, respectively. We use variables α, β, γ, . . .to range
overU. LetV:=V
A
. Define
∅:= sup(N0A, λα.α) x0:= sup¯x+
AN A
1, λβ.d(˜x(p1β),x,p0β,0)
and∆∈ U by
∆·η=
∅ ifη(0) = 0
(∆·(η−1))0ifη(0)6= 0,
where η−1 is the function γ withγ(n) = η(n)−1 if η(n) >0 and γ(n) = 0 otherwise. The definition of∆ appeals to the recursion theorem forA.4Finally, ω is defined by
ω := sup(NA, ∆).
By induction onnone shows that∆·ˆn↓and∆·ˆn∈V, thusω∈V.
The representationω of the set of von Neumann integers has an important property.
4
Definition: 9.1 We use
A Ato convey thatηA Afor some η∈ U.
x∈Visinjectively presentedif for all α, β∈¯x, whenever
A˜x(α) = ˜x(β)
thenα=β.
Lemma: 9.2 ω is injectively presented.
Proof:We must show thatA ∆·nˆ=∆·mˆ impliesn=m. This can be verified by a routine double induction, first on nand within that onm. For details see
[1] Lemma 5.5 or [23] Theorem 4.24. ut
Corollary: 9.3 The Axiom of Countable Choice,ACω, and the Dependent Choices Axiom, RDC, are validated inV.
Proof:This is an immediate consequence of the injective presentation ofω. The details are similar to the proof of [1] Theorem 5.7 or [23] Theorem 4.26. ut
Next we aim at finding an injective presentation of Baire space in V. We will
need internal versions of unordered and ordered pairs inV.
Definition: 9.4 There is a closed application term OP ofAPPsuch thatA|=
OP↓ and
A|= OP(α, β,ˆ0) =α ∧ OP(α, β,ˆ1) =β
for allα, β∈ U. Now let
{x,y}
V = sup N A
2, λα.OP(x,y, α)
; hx,yi
V = {{x,x}V,{x,y}V}V for x,y ∈ V. The internal versions of α ∈ U, denoted αV, and of Baire space, denoted BV, are the following:
αV := supNA, λγ.h∆·γd(0), ∆·α\(γ(0))i V
B
V := sup(U A, λα.α
V).
Corollary: 9.5 For all x,y ∈ V, {x,y}V,hx,yiV ∈ V. For all α ∈ U, αV ∈ V. Moreover,BV ∈V.
Proof:These claims are obviously true. ut
Corollary: 9.6 BV is injectively presented.
Proof:Let∆∗(n) :=∆·nˆ. As a first step, one must show that
A h∆
∗(n
1), ∆∗(m1)iV =h∆
∗(n
2), ∆∗(m2)iV impliesn1=m1andn2=m2. This follows from the fact that
A “hx,yiV is the ordered pair ofxandy”
holds and thatω is injectively presented according to Corollary 9.2. ut
Lemma: 9.7 There is a closed application term tsuch that A|=t↓ and
αtA ”BV is the set of all functions fromω toω” wheret evaluates toαtin A.
Proof:Suppose
β
A “f is function fromω toω. Then from β one can distill β∗ such that β∗
A ∀n ∈ ω∃k ∈ ω hn, ki ∈ f. Thus β∗·nˆ A ∃k ∈ω h∆·n, kˆ i ∈ f, so that (β∗·nˆ)0 ∈N˚A and (β∗·nˆ)1 A
h∆·ˆn, ∆·(β∗·nˆ)i ∈f. Now defineβ# byβ#(n) = (β∗·nˆ)
0(0). Then one can
effectively construct β, β[fromβ such thatβA f =β#
V andβ
[
A f ∈ BV. Conversely, if γ A f ∈ BV, one can construct γ† from γ such that γ† A
“f is a function fromω toω.”.
As all of the above transformations can be effected by application terms, the
desired assertion follows. ut
Theorem: 9.8 The principles CCandAC2 are valid in V.
Proof:Suppose
η
A ∀f ∈ BV∃n∈ω A(f, n). (10) Then, for allα∈ U,η·αA ∃n∈ω A(αV, n), so that
(η·α)0∈NA ∧ (η·α)1A A(αV, ∆·(η·α)0). (11) Define
η∗:= sup UA, λα.hα
V, ∆·(η·α)0iV
.
Obviously we can construct a closed termt#such that
A|=t#↓and withϑ∈ U
such thatA|=t#'ϑwe obtain
ϑ·ηA η∗:BV →ω ∧ ∀f ∈ BVA(f, η∗(f)). (12) We can thus cook up another closed application term t+ which evaluates to a
functionΞ inAsuch that
Ξ·η=p(η∗, ϑ·η).
In view of (12) we arrive at
Ξ
A ∀f ∈ BV∃n∈ω A(f, n) → ∃F [F :BV →ω ∧ ∀f ∈ BVA(f, F(f))]. One can also show that the functionη∗in (12) constructed fromηis a continuous function in the realizability model V. By the previous Lemma 9.7, BV is also realizably the Baire space. So the upshot is thatCCis realized.
Moreover, for the above proof the restriction of the existential quantifier to
ω in (10) is immaterial. As a result, the above proof establishes realizability of AC2 in V as well, wherebyAC2 stands for the following statement: If F is
a function with domain NN such that ∀α ∈ NN∃x ∈ F(α) then there exists a
functionf with domainNNsuch that∀α∈NNf(α)∈F(α).
Theorem: 9.9 Letϕ(v1, . . . , vr)be a formula of set theory with at most the free
variables exhibited.
(i) If
CZF+CC+FT+AC2+RDC`ϕ(v1, . . . , vr)
then there exists a closed application term tϕ of PCA+ such that for all
x1, . . . ,xr∈V
A ,
A|=tϕx1. . .xr↓
and
tϕx1. . .xr A ϕ(x1, . . . ,xr).
The termtϕcan be effectively constructed from the deduction ofϕ(v1, . . . , vr).
(ii) Suppose that the domain ofAis aβ-model and that
CZF+REA+CC+BIM+AC2+RDC`ϕ(v1, . . . , vr).
Then there exists a closed application term sϕ of PCA +
such that for all x1, . . . ,xr∈V
A
W,
A|=sϕx1. . .xr↓
and
sϕx1. . .xr A ϕ(x1, . . . ,xr).
The termsϕcan be effectively constructed from the deduction ofϕ(v1, . . . , vr).
Proof: (i): In view of Theorem 8.5 and Theorem 9.8, it suffices to show real-izability of FT. This is basically the same proof as for Theorem 6.4 only in a more involved context. So we omit the details.
(ii): By Theorem 8.8 and Theorem 9.8, it remains to verify realizability of BIM. This is similar to the proof of Theorem 6.7. ut
Theorem: 9.10 (i) CZFandCZF+CC+FT+AC2+RDChave the same
proof-theoretic strength and prove the sameΠ0
2 sentences of arithmetic.
(ii) CZF+REAandCZF+REA+CC+BIM+AC2+RDChave the same
proof-theoretic strength and prove the sameΠ0
2 sentences of arithmetic.
(i) follows from Theorem 9.9(i), the fact that the proof of 9.9(i) can be carried out in the background theory KP, and thatCZFand KPprove the same Π20
sentences.
(ii) follows from Theorem 9.9(ii) together with the insight that the existence of NN ∩ Lρ (whereρ = supn<ωωnck) can be shown inKPi and that KPi is a
background theory sufficient for the construction of VWP. Moreover, KPi is of the same strength asCZF+REAand the theories prove the sameΠ0
2 sentences
10
BI
Dstrengthens CZF
The last section will show that the addition ofBIDtoCZFis not a conservative
extension of CZF with respect to Π0
1 sentences in that we shall prove that
CZF+BID proves the 1-consistency ofCZF.
To set the stage for the proof we will be needing some definitions.
Definition: 10.1 LetCZFbe the acronym for Constructive Zermelo-Fraenkel set theory (see [?, 3]). LetCZF0be the systemCZFwithout Subset Collection. CZFR0 results from CZF by deleting Subset Collection and replacing Strong Collection with Replacement. LetCZFR,E be obtained fromCZFby replacing
Strong Collection with Replacement and Subset Collection with Exponentiation, respectively.
Note that Strong Collection implies Replacement and that Subset Collection implies Exponentiation. Thus, bothCZFR0andCZFR,Eare subtheories ofCZF.
Another important axiom for constructive set theory is the so-called regular extension axiom,REA(see [2, 19]). A weaker version ofREAis the functional regular extension axiom, fREA, which is formulated in terms of functions in place of multi-valued functions.
Definition: 10.2 If R is a binary relation, that is a set of ordered pairs, we writexRy forhx, yi ∈R.
Apartial orderingis a pairhA, Risuch thatRpartially ordersA- that is,A
is a set,Ris a binary relation,R istransitive onA:∀x, y, z∈A[xRy ∧ yRz→
xRz],andR isirreflexive onA:∀x∈A¬xRx.
Note that we do not assumeR ⊆A×A. So if hA, Riis a partial ordering, then for anyB ⊆A,hB, Riis a partial ordering.
hA, Ri is a total ordering if it is a partial ordering and trichotomy holds:
∀x, y∈A[xRy ∨ x=y ∨ yRx].
Theorem: 10.3 (CZFR0)IfhA, Riis a partial ordering then there is a smallest classX such that for all a∈A, whenever ∀u∈A(uRa→u∈X)then a∈X. This class will be denoted byWF(A, R)(the well-founded part of hA, Ri).
Proof:[22] Theorem 2.6. ut
Definition: 10.4 SupposehA, Riis a partial ordering andφ(u) is a set-theoretic formula. Let
ProghA,Ri(φ) iff (∀u∈A)
(∀v∈A) (vRu→φ(v))→φ(u)
.
IfX is a class{u|φ(u)}we also writeProghA,Ri(X) instead ofProghA,Ri(φ).
Corollary: 10.5 Let hA, Ri be a partial ordering and φ(u) be a set-theoretic formula.
(i) (∀x∈A)
x∈WF(A, R) ↔ (∀u∈A)(uRx→u∈WF(A, R))
. (ii) ProghA,Ri(φ) → (∀x∈WF(A, R))φ(x).
Theorem: 10.6 (CZFR,E +BID) Let hA,≺i be a partial ordering such that
A⊆N,≺ ⊆A×A, andA and≺ are decidable, i.e.,∀n∈N(n∈A ∨ n /∈A)
and∀n, m∈N(n≺m ∨ n⊀m). ThenWF(A,≺)is a set. Proof:Set
X ={x∈A| ∀f ∈N
N[f(0)≺x→ ∃n∈Nf(n+ 1)⊀f(n)]}. (13)
Note thatX is a set owing to Exponentiation and Bounded Separation. We claim that
WF(A,≺)⊆ X. (14)
According to Corollary 10.5 (ii) it suffices to show ProghA,Ri(X). So suppose
a∈A and for allu≺a,u∈ X. To showa∈ X suppose f ∈N
N andf(0)≺a.
We have to find ak∈Nsuch thatf(k+ 1)⊀f(k). Owing to the decidability of
≺we either havef(1)≺f(0) orf(1)⊀f(0). In the latter case choosek= 0. In the former case definef∗:N→Nbyf∗(n) =f(n+ 1). Asf∗(0) =f(1)≺f(0)
and f(0) ∈ X there exists m ∈ N such that f∗(m+ 1) ⊀ f∗(m). Thus with
k=m+ 1 it holds thatf(k+ 1)⊀f(k).
As a result of the foregoing we havea∈ X. Next we show
X ⊆WF(A,≺). (15) This part will requireBID. Fixa∈ X. For a finite sequenceσof natural numbers
we useSa(σ) to convey thatσ=hior σ=hσ0, . . . , σriwithr≥0,σ0≺a, and
σi+1≺σi for all 0≤i < r. DefineRa(σ) by¬Sa(σ). Note thatRa is decidable
because≺is decidable.
Letf :N→N.f(0)⊀ayieldsRa( ¯f(1)). Iff(0)≺athen there existsm∈N
suchf(m+ 1)⊀f(m) becausea∈ X; and hence Ra( ¯f(m+ 2)). Therefore we
can conclude that
∀f ∈N
N∃k∈NRa( ¯f(k)). (16)
We say that W(τ) holds if τ is of the form hτ0, . . . , τri with r ≥ 0 and τr ∈
WF(A,≺). Further define
Qa(σ) ↔ Ra(σ) ∨ W(hai ∗σ).
Clearly,
Ra(σ)→Qa(σ). (17)
Suppose that ∀m ∈ NQa(σ∗ hmi). Ra(σ) implies Qa(σ). So assume¬Ra(σ).
Then hai ∗σ is of the form hτ0, . . . , τri with r≥0,τ0 =a, andτ0 . . . τr.
For all u ≺ τr we then also have ¬Ra(σ∗ hui) by definition of Ra. Thus, as
Qa(σ∗ hmi) holds for allm by supposition, we conclude thatW(hai ∗σ∗ hui)
holds for all u ≺ τr. This entails that u ∈ WF(A,≺) for all u ≺τr, yielding
τr∈WF(A,≺); and consequently,W(hai ∗σ) andQa(σ). As a result, we have shown that
In view of (16), (17), and (18), BID allows us to conclude that Qa(hi), i.e. a∈WF(A,≺). This completes the proof of (15).
From (14) and (15) we get X = WF(A,≺), showing that WF(A,≺) is a
set. ut
Definition: 10.7 IfT is a theory, the 1-consistency of T is the schema
∀u∈N[P rT(pF( ˙u)q)→F(u)]
forΣ10formulae F(u) with one free variableu.
Corollary: 10.8 CZFR,E+BID proves the 1-consistency ofCZFandKP.
Proof:From Theorem 10.6 it follows that the classAccof [22] Definition 4.2 is a set. The order-type of this set is the Bachmann-Howard ordinal which is the proof-theoretic ordinal ofCZFandKP(see [19] Theorem 2.1 (i)). As a result, the ordinal analyses ofCZFandKPcan be carried inCZFR,E+BID, yielding
the provability of the 1-consistency of these theories inCZFR,E+BID. For more
details see [22] remark 4.14. ut
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