Now Under Construction:Intuitionistic Reverse Analysis

Joan Rand Moschovakis

Occidental College (emerita) and MPLA

GothenburgOctober 24, 2014

Each variety of reverse analysis attempts to determine a minimalaxiomatic basis for proving a particular mathematical theorem.Classical reverse analysis asks which set existence axioms areneeded to prove particular theorems of classical second-ordernumber theory. Informal constructive reverse analysis asks whichconstructive principles are needed to prove particular theorems ofBishop’s constructive analysis, and which nonconstructiveprinciples are equivalent over Bishop’s constructive analysis toclassical theorems. Intuitionistic reverse analysis asks whichintuitionistically accepted properties of numbers and functionssuffice to prove particular theorems of intuitionistic analysis usingintuitionistic logic, and may also consider the relative strength ofclassical principles consistent with intuitionistic analysis.

This lecture sketches the current state of intuitionistic reverseanalysis, in relation to its classical counterpart.

In a Nutshell: S. Simpson showed that many theorems of classicalanalysis are exactly provable in one of five subsystems of classicalsecond-order number theory, distinguished by successively strongerset existence axioms. Intermediate systems are considered also.

Intuitionistic analysis depends on function existence principles:countable and dependent choice, fan and bar theorems, continuouschoice. Intuitionistic logic distinguishes classically equivalent formsof countable choice. Many mathematical equivalents of the fantheorem have been identified. Building on a proof by T. Coquand,W. Veldman recently showed that over intuitionistic two-sortedrecursive arithmetic BIM the principle of open induction on Cantorspace is strictly intermediate between the fan and bar theorems,and is equivalent to intuitionistic versions of a number of classicaltheorems. R. Solovay proved that Markov’s Principle is surprisinglystrong in the presence of the bar theorem. This is work in progress.

Primitive recursive arithmetic PRA0 is a quantifier-free systembased on intuitionistic logic with equality, in a language withvariables over numbers, constants for =, 0, ′ and all primitiverecursive functions, and logical symbols & ,∨,¬,→,↔, (, ).¬(0 = x ′) and the definitions of the function constants are axioms.A logical rule allows substitution of terms for variables, andquantifier-free mathematical induction is a rule. x ′ = y ′ → x = yand x = y ∨ ¬(x = y) are provable. x < y abbreviates x ′−y = 0.

There is no difference between classical and intuitionistic PRA0.

If number quantifiers are added, mathematical induction is statedA(0) & ∀x(A(x)→ A(x ′))→ A(x)

(for A(x) quantifier-free) and the mathematical axioms areuniversally quantified, then the substitution rule becomes provable.

By removing the restriction on mathematical induction we get adefinitional extension PRA of intuitionistic first-order arithmetic.

S. Simpson has organized classical reverse mathematics,distinguishing five main subsystems of classical second-orderarithmetic Z2 extending a fragment of Peano arithmetic PA withrestricted induction, in a language with variables and quantifiersover numbers and sets of numbers; individual constants 0, 1;operation constants +, · and predicate constants =, <, ∈.

These subsystems have increasingly strong set existence axioms:

I RCA0 (recursive comprehension and Σ01-induction),

I WKL0 (“weak Konig’s Lemma”),

I ACA0 (arithmetical comprehension),

I ATR0 (arithmetical transfinite recursion),

I Π11-CA0 (Π1

1 comprehension).

(Note that RCA0 + PRA0 is a conservative extension of RCA0.)

More interesting for intuitionism are the versions with unrestrictedmathematical induction: RCA, WKL, ACA, ATR, Π1

1-CA.

An intuitionistic analogue of RCA can be defined in a two-sortedlanguage with constants for a suitable list of primitive recursivefunctions and functionals (including pairing and coding of finitesequences), and variables and quantifiers over numbers andone-place number-theoretic functions. In RCA every set has acharacteristic function so the difference in language is inessentialfor classical reverse mathematics. For intuitionistic reversemathematics it is important because only sets whose membershiprelation is effectively decidable have characteristic functions.

Troelstra and Veldman treat intuitionistic two-sorted recursivearithmetic as an extension of PRA, but Kleene’s originalaxiomatization of two-sorted intuitionistic arithmetic had onlyfinitely many function constants, with the understanding thatothers could be added as needed. This approach seems more inkeeping with the intuitionistic philosophy and we adopt it here.

Intuitionistic logic also affects the treatment of recursivecomprehension. Classically, every ∆0

1 function is recursive. ForRCA and RCA0 the ∆0

1-comprehension axiom takes the form

∀x(∀yA(x , y)↔ ∃yB(x , y))→ ∃X∀x(x ∈ X ↔ ∃yB(x , y))

for A(x , y), B(x , y) quantifier-free. The intuitionistic analogue

∀x [(∀yα(x , y) = 0↔ ∃yβ(x , y) > 0)

& (∀yβ(x , y) = 0→ ∃yα(x , y) > 0)]

→ ∃γ∀x [γ(x) > 0↔ ∃yα(x , y) > 0]

is more complicated because Markov’s Principle is not acceptedintuitionistically. It is simpler to assume that the functions areclosed under composition and primitive recursion and satisfy aquantifier-free form of the axiom of numerical choice:

qf-AC00: ∀x∃yα(x , y) = 0→ ∃β∀xα(x , β(x)) = 0

which assures the existence of every recursive total function.

Kleene’s fragment IA1 of two-sorted intuitionistic arithmetic hasvariables a, b, . . . , x , y , z over numbers and α, β, . . . over one-placenumber-theoretic functions; finitely many constants0,’,+,·,exp,. . . ,fp for primitive recursive functions and functionalswith their defining equations; full mathematical induction;λ-abstraction and λ-reduction. The equality axioms includex = y → α(x) = α(y), and equality between functions is definedextensionally by α = β ≡ ∀xα(x) = β(x).

Kleene lets 2x · 3y code the pair (x , y). The length lh(x) of x isthe number of nonzero exponents in its prime factorization, and(x)i is the exponent of pi (with p0 = 2). Seq(x) abbreviates∀i < lh(x) ((x)i > 0), indicating that x codes a finite sequence;we write 〈 〉 = 1 and 〈x0, . . . , xn〉 = Πn

i=0 pxi+1i . If Seq(u) then

(u ∗ α)(n) = (u)n−1 if n < lh(u), and (u ∗ α)(lh(u) + n) = α(n).

α(n) = 〈α(0), . . . , α(n−1)〉. If k < lh(u) then u(k) = u ∗ λx .0(k).

Kleene, who was interested in formalizing the full strength ofintuitionistic analysis, originally assumed a strong axiom

AC01: ∀x∃αA(x , α)→ ∃β∀xA(x , λy .β(x , y))

of countable choice. To formalize recursive function theory heweakened AC01 to countable comprehension (“unique choice”)

AC00!: ∀x∃!yA(x , y)→ ∃α∀xA(x , α(x)).

Comparing minimal formal systems for intuitionistic analysis,G. Vafeiadou observed that Kleene’s M1 = IA1 + AC00! proves

CF0: ∀x(A(x) ∨ ¬A(x))→ ∃χ∀x(χ(x) = 0↔ A(x)),

which is as strong, with classical logic, as full comprehension.

Theorem 1. (G. Vafeiadou) Intuitionistic two-sorted recursivearithmetic IRA can be expressed interchangeably by any one of:

I the subsystem IA+1 = IA1 + qf-AC00 of Kleene’s M1 or

I A. S. Troelstra’s Elementary Analysis EL or

I Wim Veldman’s Basic Intuitionistic Mathematics BIM.

The interderivability of CF0 with AC00! over IRA affects thechoice of stronger subsystems of FIM, where “A is a detachablesubset of N” is often expressed by ∀x(A(x) ∨ ¬A(x)).

The natural intuitionistic analogue of WKL should be IRA plussome version of the Fan Theorem, and the natural intuitionisticanalogue of ATR should be IRA plus some version of the BarTheorem. But which version? Kleene gave four versions of his “bartheorem” axiom, all equivalent over M1 but not over IRA. Thedifference mattered when Solovay wanted to negatively interpret aclassical system, with arithmetical countable choice and barinduction, in its intuitionistic counterpart.

W. Veldman, who deserves the lion’s share of credit for developingintuitionistic reverse analysis, avoids the issue by working directlywith characteristic functions, replacing ∀x(A(x) ∨ ¬A(x)) by∃ζ∀x(ζ(x) = 0↔ A(x)). In intuitionistic analysis, only detachablesets have characteristic functions.

Brouwer’s binary fan is the tree 2<ω of finite binary sequences,represented by their codes: Bin(u) ≡ ∀i < lh(u)(1 ≤ (u)i ≤ 2).

Proposition. (Veldman) IRA proves detachable fan induction:

∆0-FI: ∀α ∈ 2N∃xβ(α(x)) = 0 & ∀u[Bin(u) &

β(u ∗ 〈0〉) = 0 & β(u ∗ 〈1〉) = 0→ β(u) = 0]→ β(〈 〉) = 0.

The Detachable Fan Theorem for 2N is

∆0-FT: ∀α ∈ 2N∃xβ(α(x)) = 0→ ∃y∀α ∈ 2N∃x ≤ yβ(α(x)) = 0.

The Enumerable Fan Theorem Σ01-FT is similar but with

∃yβ(α(x), y) = 0 in place of β(α(x)) = 0.

Enumerable Bar Induction on 2N, or Σ01-Fan Induction, is

Σ01-FI: ∀α ∈ 2N∃x∃yβ(α(x), y) = 0 & ∀u[Bin(u) &

∃yβ(u ∗ 〈0〉, y) = 0 & ∃yβ(u ∗ 〈1〉, y) = 0→ ∃yβ(u, y) = 0]

→ ∃yβ(〈 〉, y) = 0.

Theorem 2. (H. Ishihara, W. Veldman) Over IRA:

Σ01-FT ⇔ ∆0-FT ⇔ Σ0

1-FI.

Weak Konig’s Lemma for detachable subtrees of 2N is

WKL: ∀y∃α ∈ 2N∀x ≤ yρ(α(x)) = 0→ ∃α ∈ 2N∀xρ(α(x)) = 0.

Adding a strong effective uniqueness hypothesis to WKL gives

WKL!: ∀y∃α ∈ 2N∀x ≤ yρ(α(x)) = 0 &

∀α ∈ 2N∀β ∈ 2N[∃xα(x) 6= β(x)→ ∃x [ρ(α(x)) 6= 0∨ρ(β(x)) 6= 0]]

→ ∃α ∈ 2N∀xρ(α(x)) = 0.

How “at most one” is expressed can be important intuitionistically.If the uniqueness hypothesis in WKL! is weakened to

∀α ∈ 2N∀β ∈ 2N[∀xρ(α(x)) = 0 & ∀xρ(β(x)) = 0→ α = β]

a stronger Weak Konig’s Lemma with uniqueness WKL!! results.

Theorem 3. (JRM)

1. Over IRA: WKL ⇒ WKL!! ⇒ WKL!, and none of thearrows can be reversed.

2. WKL!! is consistent with, but unprovable in, FIM.

The first modern formalization of Brouwer’s results on thestructure of the intuitionistic real numbers was by R. E. Vesley, inKleene-Vesley [1965]. His careful work influenced all that followed.

Theorem 4. The following are equivalent over IRA:

(i) The Detachable Fan Theorem ∆0-FT for 2N.

(ii) Each pointwise continuous function on [0, 1] with a modulusof continuity is uniformly continuous.

(iii) The Heine-Borel Theorem: Each enumerable open covering of[0, 1] by intervals with rational endpoints has a finite subcover.

(iv) Brouwer’s Approximate Fixed-Point Theorem for enumerablecontinuous functions on U (= [0, 1]× [0, 1]).

(v) WKL!.

Proofs in recent literature are by J. Berger ((i) ⇔ (ii)); H. Ishihara,I. Loeb, W. Veldman (independently) ((i) ⇔ (iii)); W. Veldman((i) ⇔ (iv)); H. Ishihara and H. Schwichtenberg ((i) ⇔ (v)).

Theorem 5. (essentially Kleene) Over IRA the ∆0-FT for 2N isequivalent to the version for fans with bounded branching:

∆0-FT: ∀α[∀xα(x) ≤ β(α(x))→ ∃xρ(α(x)) > 0]

→ ∃y∀α[∀xα(x) ≤ β(α(x))→ ∃x ≤ yρ(α(x)) > 0].

Simpson observed that classical WKL proves the correspondinggeneralization of Weak Konig’s Lemma.

Since ∆0-FT is interderivable with WKL over the classical versionIRAc = IRA + (A ∨ ¬A) of IRA, it seems reasonable to conclude

I WFT = IRA + ∆0-FT is an intuitionistic analogue of WKL.

The next question is whether or not there is a natural intuitionisticanalogue of ACA. Since Kleene showed that FIM proves

¬∀α(∀xα(x) = 0 ∨ ¬∀xα(x) = 0)

the answer will require some thought.

Brouwer proved the Full Fan Theorem using a stronger principle ofbar induction on the universal spread NN.

A spread is (determined by) a binary function σ satisfying

1. σ(〈 〉) = 0,

2. σ(u) = 0↔ Seq(u) & ∃xσ(u ∗ 〈x〉) = 0.

The elements of a spread are its infinite branches:

α ∈ σ ≡ ∀xσ(α(x)) = 0.

A fan is a finitely branching spread, satisfying also

3. ∀u[σ(u) = 0→ ∃y∀x(σ(u ∗ 〈x〉) = 0→ x ≤ y)].

Let Spr(σ) ≡ (1) & (2) and Fan(σ) ≡ (1) & (2) & (3).

While the bounded fan theorem is equivalent to ∆0-FT over IRA,the literal statement of the detachable fan theorem is stronger:

∆0-FTσ: Fan(σ)→[∀α ∈ σ∃xρ(α(x)) > 0→ ∃y∀α ∈ σ∃x ≤ yρ(α(x)) > 0].

A formula is arithmetical if it contains no function quantifiers.Parameters of both sorts are allowed.

Arithmetical countable comprehension (“unique choice”) is

AC−00!: ∀x∃!yA(x , y)→ ∃!α∀xA(x , α(x))

for arithmetical formulas A(x , y).

Proposition. (Vafeiadou) Over IRA the following are equivalent:

(i) Arithmetical countable comprehension AC−00!.

(ii) Every arithmetical relation A(x) satisfying the law of excludedmiddle has a characteristic function:

CF−0 : ∀x(A(x) ∨ ¬A(x))→ ∃ζ∀x(ζ(x) = 0↔ A(x)).

With classical logic, IRA + CF−0 gives full arithmeticalcomprehension. It seems reasonable to conclude that

I IAC ≡ WFT + AC−00! is an intuitionistic analogue of ACA.

To prove ∆0-FT ⇒ ∆0-FTσ one needs e.g. Veldman’s axiom

W-Π01-AC00: ∀x∃y∀z ≥ y α(x , z) = 0→ ∃β∀x α(x , β(x)) = 0.

This is an instance of AC−00! because ∃y∀z ≥ yα(x , z) = 0↔∃!y [∀z ≥ yα(x , z) = 0 & (y > 0→ α(x , y−1) > 0)].

Theorem 6. (essentially Kleene-Vesley) In IAC one can prove

1. Every finitely branching spread σ has bounded branching.

2. The literal version ∆0-FTσ of the detachable fan theorem.

3. The fan theorem with a thin arithmetical bar A:

Fan(σ)→ [∀α ∈ σ∃!xA(α(x))→ ∃y∀α ∈ σ∃!x ≤ yA(α(x))].

4. Full bar induction on a fan with a thin arithmetical bar A:Fan(σ)→[∀α ∈ σ∃!xA(α(x)) & ∀u[σ(u) = 0 & A(u)→ C (u)] &

∀u[∀y(σ(u ∗ 〈y〉) = 0→ C (u ∗ 〈y〉))→ C (u)]→ C (〈 〉)].

Each of the last three is equivalent to ∆0-FT over IRA + AC−00!.

Proof of Theorem 6.3 in IAC:

Fan(σ)→ [∀α ∈ σ∃!xA(α(x))→ ∃y∀α ∈ σ∃x ≤ yA(α(x))].

Assume Fan(σ) and ∀α ∈ σ∃!xA(α(x)). Then ∀w [σ(w) = 0→A(w) ∨ ¬A(w)], so ∃β∀w [σ(w) = 0→ (β(w) = 0↔ A(w))].For such a β we have ∀α ∈ σ∃!xβ(α(x)) = 0 so by ∆0-FT:∃y∀α ∈ σ∃x ≤ yβ(α(x)) = 0, so ∃y∀α ∈ σ∃!x ≤ yA(α(x)).

Proof of Theorem 6.4 in IAC:

Fan(σ)→ [∀α ∈ σ∃!xA(α(x)) & ∀u[σ(u) = 0 & A(u)→ C (u)] &

∀u[∀y(σ(u ∗ 〈y〉) = 0→ C (u ∗ 〈y〉))→ C (u)]→ C (〈 〉)].

Proof. Assume the hypotheses, so by Theorem 6.3 there is an nsuch that ∀α ∈ σ∃!x ≤ nA(α(x)). If σ(u) = 0 and lh(u) ≤ n thereare two cases: (i) ∃x ≤ lh(u)A(u(x)) or (ii) otherwise. We want toshow that in each case ∃x ≤ lh(u)C (u(x)). Case (i) is no problem.If σ(u) = 0 & ∀x ≤ lh(u)¬A(u(x)) then lh(u) < n, and we maychoose u so that n − lh(u) is as small as possible for case (ii).Then for all y with σ(u ∗ 〈y〉) = 0: A(u ∗ 〈y〉), so C (u ∗ 〈y〉), soC (u) after all. In particular, C (〈 〉).

Three parameters determine the strength of a fan theorem and itscorresponding fan induction principle:

(a) How is “finitely branching” expressed?

(b) What are the restrictions on the inductive predicate?

(c) What are the restrictions on the predicate defining the bar?

Intuitionistically there is more than one notion of finiteness.

Veldman calls a detachable set A ⊆ N finite if∃y∀x(A(x)→ x ≤ y); or bounded-in-number if there is a y suchthat A has no more than y elements. Formally, “A has no morethan y elements” can be expressed in Π0

1 form by

∀u[lh(u) = y + 1 & Inc(u)→ ∃i ≤ y¬A((u)i )],

where Inc(u) ≡ Seq(u) & ∀i(i + 1 < lh(u)→ (u)i < (u)i+1).

A set A ⊆ N is almost finite if ∀α[Inc(α)→ ∃x¬A(α(x))], whereInc(α) ≡ ∀n(α(n) < α(n + 1)).

An approximate fan is a spread σ such that if σ(u) = 0 then theset of all x such that σ(u ∗ 〈x〉) = 0 is bounded-in-number.

Veldman’s Approximate Fan Theorem is

AFT: EAF (σ) & ∀α[α ∈ σ → ∃!xρ(α(x)) > 0]

→ ∀β[Inc(β) & ∀xσ(β(x)) = 0→ ∃xρ(β(x)) = 0],

where EAF (σ) expresses: σ is an approximate fan, and there is a βsuch that for each x the set of all finite sequence codes u withlh(u) = x & σ(u) = 0 has no more than β(x) elements.

Theorem 7. (Veldman) Over IRA: AFT ⇒ ∆0-FTσ ⇒ ∆0-FT.The first arrow cannot be reversed even over IRA + AC01.

Veldman proved the last statement by deriving in AFT anintuitionistic version of the Ramsey Theorem, which entails theParis-Harrington formula, which is not provable in IRA + AC01 bya result of Goodman. And Troelstra proved that ∆0-FTσ isconservative over IRA + AC01 for arithmetical formulas.

Brouwer’s Bar Theorem (really an axiom, as Kleene showed) isclassically equivalent to transfinite recursion up to any countableordinal. The principle of Monotone Σ0

1-Bar Induction on NN is

Σ01-BI

m: ∀α∃x∃yβ(α(x), y) = 0

& ∀u[Seq(u) & ∀x∃yβ(u ∗ 〈x〉, y) = 0↔ ∃yβ(u, y) = 0]

→ ∃yβ(〈 〉, y) = 0.

The Principle of Open Induction on NN or 2N is

OI: ∀β[∀α(α < β → ∃yρ(α(y)) > 0)→ ∃yρ(β(y)) > 0]

→ ∀α∃yρ(α(y) > 0where < is the lexicographic ordering on sequences.

Theorem 8. (T. Coquand) In IRA one can prove:

Σ01-BIm ⇒ OI on 2N (and [0, 1]) ⇒ Heine-Borel for [0, 1].

The proofs of OI actually used bar induction on a subspread of NN

with at most binary branching, an approximate fan!

Theorem 9. (Veldman) Over IRA:

OI on 2N ⇔ OI on [0, 1] ⇔ AFT

Corollary. Open Induction on 2N (and [0, 1]) is unprovable in IACbut can be proved in IAC + Σ0

1-BIm.

Proof: By the proof of Theorem 7 with Troelstra’s conservativityresult (which holds for IAC); and by Theorems 8 and 9.

Veldman found other equivalents over IRA of Open Induction,including contrapositive versions of the Bolzano-Weierstrass andMonotone Convergence Theorems, and the following statement(whose converse is provable in IRA):

If ∀α∃xβ(α(x)) > 0 then {u|Seq(u) & ∀x < lh(u)β(u(x)) = 0} iswell-founded under the Kleene-Brouwer ordering.

This suggests that perhaps

I ABI = IAC + Σ01-BIm is an intuitionistic analogue of ATR.

A Short Story. A few years ago R. Solovay wanted to prove that aclassical system S with arithmetical comprehension and barinduction could be negatively interpreted in Kleene’s neutralsubsystem B of FIM. I pointed out that the negative interpretationof arithmetical comprehension is not accepted intuitionistically.Clearly Markov’s Principle in the form

MP1: ¬¬∃x α(x) = 0→ ∃x α(x) = 0

together with Kleene’s bar induction schema in the form x26.3b ofKleene-Vesley:

BI1: ∀α∃xρ(α(x)) = 0 & ∀u[Seq(u) & ρ(u) = 0→ A(u)]

& ∀u[Seq(u) & ∀nA(u ∗ 〈n〉)→ A(u)]→ A(〈 〉)proves the negative interpretation of BI1. (This was not obviousfor the other forms x26.3a,c,d.)

Finally Solovay finessed the issue of arithmetical comprehension,and completed the proof of his theorem, by proving

Solovay’s Lemma. In IRA + BI1 + MP1 one can prove

∀ᬬ∃ζ∀x [ζ(x) = 0↔ ∃yT (x , x , α(y))]

with Kleene’s primitive recursive T -predicate.

Theorem 10. (Solovay) IRA + BI1 + MP1 proves:

1. ∀ᬬ∃ζ∀x [ζ(x) = 0↔ ∃yα(x , y) = 0].

2. ∀ᬬ∃ζ∀x [ζ(x) = 0↔ A(x , α)] for A(x , α) arithmetical.

3. Kuroda’s Principle (arithmetical double negation elimination):

DNS−0 : ∀x¬¬A(x)→ ¬¬∀xA(x) (A(x) arithmetical).

Theorem 11. (JRM) IRA + BI1 + MP1 proves

1. The constructive arithmetical hierarchy is proper.

2. An intuitionistic version of ∆11 comprehension:

∀x [¬¬∃α∀zβ(x , α(z)) = 0↔ ∀β∃zγ(x , α(z)) = 0]

→ ¬¬∃δ∀x [δ(x) = 0↔ ∀β∃zγ(x , α(z)) = 0].

Veldman’s EnDec?! is a logical principle equivalent to AFT overIRA, hence derivable in IRA + Σ0

1-BIm. Working from Solovay’sresult and my observation that EnDec?! follows in IRA + MP1

from ∀ᬬ∃ζ∀x [ζ(x) = 0↔ ∃yα(x , y) = 0], Veldman proved

Theorem 12. (Veldman) Over IRA + MP1 :

EnDec?! ⇔ ∀ᬬ∃ζ∀x [ζ(x) = 0↔ ∃yα(x , y) = 0] ⇔ Σ01-BIm.

So with classical logic, Σ01-BIm would not add strength to IAC.

Markov’s Principle MP1, although not accepted by Brouwer, isconsistent with FIM relative to B + MP1, as Kleene showed usingfunction-realizability. F. Waaldijk and others have shown that MP1

settles many questions in intuitionistic analysis and topology.

So the proposal for ABI = IAC + Σ01-BIm as an intuitionistic

analogue of ATR must be considered tentative.

An intuitionistic analogue of Π11-CA is needed to complete this

proposed correspondence with Simpson’s five subsystems.

So far we have not considered the axioms of countable choice,which were accepted by Brouwer and Kleene. Even to prove thatevery Cauchy sequence of rationals has a modulus of convergence,one seems to need full Π0

1 countable choice:

Π01-AC00: ∀x∃y∀zα(x , y , z) = 0→ ∃β∀x∀zα(x , β(x), y) = 0,

which does not follow from AC00!, as S. Weinstein showed in hisPhD dissertation.

Proposition. Over IRA the following are equivalent:

1. Veldman’s axiom W-Π01-AC00

2. ∆0-ACm00: ∀x∃my α(x , y) = 0→ ∃β∀x α(x , β(x)) = 0

where ∃myB(y) ≡ ∃yB(y) & ∀y∀z(B(y) & y ≤ z → B(z))

3. Π01-AC00!.

Note that ∃myB(y) is in general stronger than ∃y∀x ≥ yB(y).This will be important in the sequel.

Kleene implicitly incorporated a classically correct choice principle

(*): ∀α∃yR(α(y))→∃σ∀α[∃!xσ(α(x)) > 0 & ∀x∀y(σ(α(x)) = y + 1→ R(α(y))],

into his version CC10 of Brouwer’s classically false axiom ofcontinuous choice. Troelstra’s “neighborhood function principle” isanother version of (*). Since classical bar induction conflicts withcontinuous choice, the conclusion cannot be sharpened to∃σ∀α[∃!xσ(α(x)) > 0 & ∀x(σ(α(x)) > 0→ R(α(x)))].

However, monotone bar induction holds intuitionistically:

BImon: ∀α∃myR(α(y)) & ∀u[Seq(u) & R(u)→ A(u)]

& ∀u[Seq(u) & ∀nA(u ∗ 〈n〉)→ A(u)]→ A(〈 〉).

Vafeiadou and I have suggested a monotone choice principle

ACm1/2,0: ∀α∃myR(α(y))→

∃σ∀α[∃!xσ(α(x)) = 0 & ∀x(σ(α(x)) = 0→ R(α(x))].

Proposition.

1. BImon is interderivable with BI1 over IAC + ACm1/2,0.

2. Σ01-BIm is interderivable with Σ0

1-BI1 in IAC + Σ01-ACm

1/2,0.

3. In IAC + Π01-ACm

00 one can prove that every Cauchy sequenceof rationals has a modulus of convergence.

4. In IAC + Π11-ACm

1/2,0 one can prove that every continuous

function from NN to N has a modulus of convergence andevery Cauchy sequence of reals has a Cauchy modulus.

I Is MBI = ABI + Π11-ACm

1/2,0 + Π11-BI1 a reasonable

intuitionistic analogue of Π11-CA?

The subsystem IRA + AC01 + BI1 + ACm1/2,0 of Kleene’s FIM

properly extends his neutral basic system B, and proves BImon, thefull fan theorem FT and full fan induction FI.

I Does IRA + AC01 + BI1 + ACm1/2,0 correspond to Z2?

Some References:

1. S. C. Kleene and R. Vesley, Foundations of IntuitionisticMathematics, North-Holland (1965).

2. S. C. Kleene, Formalized recursive functionals and formalizedrealizability, AMS Memoirs (1969).

3. A. S. Troelstra, “Note on the Fan Theorem,” JSL 39 (1974)584-596.

4. A. S. Troelstra and D. van Dalen, Constructivism inMathematics I and II, North-Holland Elsevier (1988).

5. I. Loeb, “Equivalents of the (Weak) Fan Theorem,” APAL132 (2005) 51-66.

6. J. Berger and H. Ishihara, “Brouwer’s fan theorem and uniqueexistence in constructive analysis,” Math. Log. Quart. (2005)360-364.

7. JRM, “Classical and constructive hierarchies in extendedintuitionistic analysis,” JSL 69 (2003), 1015-1043.

8. JRM, “The effect of Markov’s Principle on the intuitionisticcontinuum,” Oberwolfach Proceedings (2005).

9. JRM, “Another weak Konig’s Lemma WKL!!,” in U. Berger,H. Diener, P. Schuster and M. Seisenberger (eds.), Logic,Construction, Computation, Ontos (2012).

10. JRM and G. Vafeiadou, “Some axioms for constructiveanalysis,” Arch. Math. Logic (2012) 443-459.

11. G. Vafeiadou, “Formalizing constructive analysis: acomparison of minimal systems and a study of uniquenessprinciples,” PhD dissertation, Graduate Program in Logic andAlgorithms (MPLA), University of Athens (2012).

12. F. Waaldijk, “On the foundations of constructive mathematics– especially in relation to the theory of continuous functions,”Found. of Sci. Vol. 3 Issue 10 (2005) 249-324.

13. H. Ishihara, “Constructive reverse mathematics: compactnessproperties,” Chapter 16 of L. Crosilla and P. Schuster (Eds.),From Sets and Types to Topology and Analysis, Oxford(2005).

14. S. Simpson, Subsystems of Second Order Arithmetic, 2nd Ed.,Perspectives in Logic, Springer (2009).

15. W. Veldman, “Brouwer’s approximate fixed-point theorem isequivalent to Brouwer’s Fan Theorem,” in S. Lindstrom et al(eds.), Logicism, Intuitionism and Formalism, SyntheseLibrary 341, Springer (2009) 277-299.

16. W. Veldman, “Brouwer’s Fan Theorem as an axiom and as acontrast to Kleene’s alternative,” to appear in Arch. Math.Logic.

17. W. Veldman, “The Principle of Open Induction on CantorSpace and the Approximate Fan Theorem,” preprint.

and many more by W. Veldman, most available from arXiv.