aa r X i v : . [ m a t h . L O ] J u l THE TAMING OF THE REVERSE MATHEMATICS ZOO
SAM SANDERS
Abstract.
Reverse Mathematics is a program in the foundations of mathe-matics. Its results give rise to an elegant classification of theorems of ordinarymathematics based on computability. In particular, the majority of these the-orems fall into only five categories of which the associated logical systems aredubbed ‘the Big Five’. Recently, a lot of effort has been directed towards find-ing exceptional theorems, i.e. which fall outside the Big Five categories. Theso-called Reverse Mathematics zoo is a collection of such exceptional theorems(and their relations). In this paper, we show that the uniform versions of thezoo-theorems, i.e. where a functional computes the objects stated to exist, allfall in the third Big Five category arithmetical comprehension , inside Kohlen-bach’s higher-order Reverse Mathematics. In other words, the zoo seems todisappear at the uniform level. Our classification applies to all theorems whoseobjects exhibit little structure , a notion we conjecture to be connected to Mon-talb´an’s notion robustness . Surprisingly, our methodology reveals a hithertounknown ‘computational’ aspect of Nonstandard Analysis: We shall formulatean algorithm RS which takes as input the proof of a specific equivalence inNelson’s internal set theory , and outputs the proof of the desired equivalence(not involving Nonstandard Analysis) between the uniform zoo principle andarithmetical comprehension. Moreover, the equivalences thus proved are even explicit , i.e. a term from the language converts the functional from one uniformprinciple into the functional from the other one and vice versa. Introduction: Reverse Mathematics and its zoo
In two words, the subject of this paper is the
Reverse Mathematics classificationin Kohlenbach’s framework ([29]) of uniform versions of principles from the
ReverseMathematics zoo ([14]), namely as equivalent to arithmetical comprehension . Wefirst discuss the italicised notions in more detail.For an introduction to the foundational program Reverse Mathematics (RM forshort), we refer to [45, 46]. One of the main results of RM is that the majority oftheorems from ordinary mathematics , i.e. about countable and separable objects,fall into only five categories of which the associated logical systems are dubbed ‘theBig Five’ (See e.g. [34, p. 432]). In the last decade or so, a huge amount of time andeffort was invested in identifying theorems falling outside of the Big Five categories.All such exceptional theorems (and their relations) falling below the third Big Fivesystem, are collected in the so-called RM zoo (See [14]).In this paper, we shall establish that the exceptional principles inhabiting theRM zoo become non-exceptional at the uniform level , namely that the uniformversions of RM zoo-principles are all equivalent to arithmetical comprehension, the
Department of Mathematics, Ghent University, Belgium & Munich Center for Math-ematical Philosophy, LMU Munich, Germany
E-mail address : [email protected] . TAMING THE REVERSE MATHEMATICS ZOO aforementioned third Big Five system of RM. As a first example of such a ‘uniformversion’, consider the principle
UDNR , to be studied in Section 3.( ∃ Ψ → ) (cid:2) ( ∀ A )( ∀ e )(Ψ( A )( e ) = Φ Ae ( e )) (cid:3) . ( UDNR )Clearly,
UDNR is the uniform version of the zoo principle DNR , defined as:( ∀ A )( ∃ f )( ∀ e ) (cid:2) f ( e ) = Φ Ae ( e ) (cid:3) . ( DNR )The principle
DNR was first formulated in [22] and is even strictly implied by
WWKL (See [1]) where the latter principle sports some
Reverse Mathematics equiv-alences ([34, 48, 49]) but is not a Big Five system. Nonetheless, we shall prove that
UDNR ↔ ( ∃ ), where the second principle is the functional version of arithmeticalcomprehension, the third Big Five system of RM. In other words, the ‘exceptional’status of DNR disappears completely if we consider its uniform version
UDNR . Theproof of the equivalence
UDNR ↔ ( ∃ ) takes place in RCA ω (See Section 2), thebase theory of Kohlenbach’s higher-order Reverse Mathematics .More generally, in Sections 3, 4, and 6, we show that a number of uniform zoo-principles are equivalent to arithmetical comprehension inside RCA ω . In Section 5,we formulate a general template for classifying (past and future) zoo-principles inthe same way. As will become clear, our template provides a uniform and elegant approach to classifying uniform principles originating from the RM zoo; In otherwords, the RM zoo seems to disappear at the uniform level (but see Remark 3.7). Asto a possible explanation for this phenomenon, the axiom of extensionality plays acentral role in our template, as discussed in Remark 4.27. Another key ingredient ofthe template is the presence of ‘little structure’ (which is e.g. typical of statementsfrom combinatorics) on the objects in RM zoo principles, which gives rise to non-robust theorems in the sense of Montalb´an ([34]), as discussed in Section 5.2.To obtain the aforementioned equivalences, Nonstandard Analysis in the formof Nelson’s internal set theory ([36]), is used as a tool in this paper. In particular,these equivalences are formulated as theorems of Kohlenbach’s base theory RCA ω (See [29] and Section 2.2), and are obtained by applying the algorithm RS (SeeSection 5) to associated equivalences in Nonstandard Analysis . Besides providinga streamlined and uniform approach, the use of Nonstandard Analysis via RS alsoresults in explicit equivalences without extra effort . In particular, we shall justprove equivalences inside Nonstandard Analysis without paying any attention toeffective content , and extract the explicit equivalences using the algorithm RS .This hitherto unknown ‘computational aspect’ of Nonstandard Analysis is perhapsthe true surprise of this paper.Finally, as to conceptual considerations, the above-mentioned ‘disappearance’ ofthe RM zoo suggests that Kohlenbach’s higher-order RM ([29]) is not just ‘RMwith higher types’, but a separate field of study giving rise to a completely differentclassification; In particular, the latter comes equipped with its own notion of ex-ceptionality, notably different from the one present in Friedman-Simpson-style RM.In light of the results in Section 6, one could go even as far as saying that, at the We sometimes refer to inhabitants of the RM zoo as ‘theorems’ and sometimes as ‘principles’. For instance, as shown in Section 6, our template is certainly not limited to Π -formulas, andsurprisingly even applies to contrapositions of RM zoo principles, including the Ramsey theorems. An implication ( ∃ Φ) A (Φ) → ( ∃ Ψ) B (Ψ) is explicit if there is a term t in the language suchthat additionally ( ∀ Φ)[ A (Φ) → B ( t (Φ))], i.e. Ψ can be explicitly defined in terms of Φ. AMING THE REVERSE MATHEMATICS ZOO 3 uniform level, weak K¨onig’s lemma is more exceptional than e.g. Ramsey’s theoremfor pairs , as the latter is more robust (at the uniform level) than the former, dueto the behaviour of their contrapositions (at the uniform level). As the saying (sortof) goes, one man’s exception is another’s mainstream.In conclusion, the stark contrast in exceptional behaviour between principlesfrom the RM zoo and their uniform counterparts, speaks in favour of the study ofhigher-order RM. Notwithstanding the foregoing, ‘unconditional’ arguments for thestudy of higher-order RM are also available, as discussed in Section 6.4.2.
About and around internal set theory
In this section, we introduce Nelson’s internal set theory , first introduced in [36],and its fragment P from [4]. We shall also introduce Kohlenbach’s base theory RCA ω from [29], and the system RCA Λ0 , which is based on P .2.1. Introduction: Internal set theory.
In Nelson’s syntactic approach to Non-standard Analysis ([36]), as opposed to Robinson’s semantic one ([38]), a newpredicate ‘st( x )’, read as ‘ x is standard’ is added to the language of ZFC , theusual foundation of mathematics. The notations ( ∀ st x ) and ( ∃ st y ) are short for( ∀ x )(st( x ) → . . . ) and ( ∃ y )(st( y ) ∧ . . . ). A formula is called internal if it doesnot involve ‘st’, and external otherwise. The three external axioms Idealisation , Standard Part , and
Transfer govern the new predicate ‘st’; They are respectivelydefined as: (I) ( ∀ st fin x )( ∃ y )( ∀ z ∈ x ) ϕ ( z, y ) → ( ∃ y )( ∀ st x ) ϕ ( x, y ), for internal ϕ with any(possibly nonstandard) parameters. (S) ( ∀ x )( ∃ st y )( ∀ st z )( z ∈ x ↔ z ∈ y ). (T) ( ∀ st x ) ϕ ( x, t ) → ( ∀ x ) ϕ ( x, t ), where ϕ is internal, t is standard and captures all parameters of ϕ .The system IST is (the internal system)
ZFC extended with the aforementionedexternal axioms; The former is a conservative extension of
ZFC for the internallanguage, as proved in [36].In [4], the authors study G¨odel’s system T extended with special cases of theexternal axioms of IST . In particular, they consider nonstandard extensions of the(internal) systems
E-HA ω and E-PA ω , respectively Heyting and Peano arithmeticin all finite types and the axiom of extensionality . We refer to [4, § x ρ comes equippedwith a superscript denoting its type, which is however often implicit. As to thecoding of multiple variables, the type ρ ∗ is the type of finite sequences of type ρ ,a notational device used in [4] and this paper; Underlined variables x consist ofmultiple variables of (possibly) different type.In the next section, we introduce the system P assuming familiarity with thehigher-type framework of G¨odel’s system T (See e.g. [4, § The superscript ‘fin’ in (I) means that x is finite, i.e. its number of elements are bounded bya natural number. TAMING THE REVERSE MATHEMATICS ZOO
The system P . In this section, we introduce the system P . We first discusssome of the external axioms studied in [4]. First of all, Nelson’s axiom Standardpart is weakened to
HAC int as follows:( ∀ st x ρ )( ∃ st y τ ) ϕ ( x, y ) → ( ∃ st F ρ → τ ∗ )( ∀ st x ρ )( ∃ y τ ∈ F ( x )) ϕ ( x, y ) , ( HAC int )where ϕ is any internal formula. Note that F only provides a finite sequence ofwitnesses to ( ∃ st y ), explaining its name Herbrandized Axiom of Choice . Secondly,Nelson’s axiom idealisation I appears in [4] as follows:( ∀ st x σ ∗ )( ∃ y τ )( ∀ z σ ∈ x ) ϕ ( z, y ) → ( ∃ y τ )( ∀ st x σ ) ϕ ( x, y ) , ( I )where ϕ is again an internal formula. Finally, as in [4, Def. 6.1], we have thefollowing definition. Definition 2.1.
The set T ∗ is defined as the collection of all the constants in thelanguage of E-PA ω ∗ . The system E-PA ω ∗ st is defined as E-PA ω ∗ + T ∗ st + IA st , where T ∗ st consists of the following axiom schemas.(1) The schema st( x ) ∧ x = y → st( y ),(2) The schema providing for each closed term t ∈ T ∗ the axiom st( t ).(3) The schema st( f ) ∧ st( x ) → st( f ( x )).The external induction axiom IA st is as follows.Φ(0) ∧ ( ∀ st n )(Φ( n ) → Φ( n + 1)) → ( ∀ st n )Φ( n ) . ( IA st )For the full system P ≡ E-PA ω ∗ st + HAC int + I , we have the following theorem.Here, the superscript ‘ S st ’ is the syntactic translation defined in [4, Def. 7.1], andalso listed starting with (2.3) in the proof of Corollary 2.3. Theorem 2.2.
Let Φ( a ) be a formula in the language of E-PA ω ∗ st and suppose Φ( a ) S st ≡ ∀ st x ∃ st y ϕ ( x, y, a ) . If ∆ int is a collection of internal formulas and P + ∆ int ⊢ Φ( a ) , (2.1) then one can extract from the proof a sequence of closed terms t in T ∗ such that E-PA ω ∗ + ∆ int ⊢ ∀ x ∃ y ∈ t ( x ) ϕ ( x, y, a ) . (2.2) Proof.
Immediate by [4, Theorem 7.7]. (cid:3)
It is important to note that the proof of the soundness theorem in [4, §
7] providesa term extraction algorithm A to obtain the term t from the theorem.The following corollary is essential to our results. We shall refer to formulas ofthe form ( ∀ st x )( ∃ st y ) ψ ( x, y, a ) for internal ψ as (being in) the normal form . Corollary 2.3.
If for internal ψ the formula Φ( a ) ≡ ( ∀ st x )( ∃ st y ) ψ ( x, y, a ) satisfies (2.1) , then ( ∀ x )( ∃ y ∈ t ( x )) ψ ( x, y, a ) is proved in the corresponding formula (2.2) .Proof. Clearly, if for ψ and Φ as given we have Φ( a ) S st ≡ Φ( a ), then the corollaryfollows immediately from the theorem. A tedious but straightforward verificationusing the clauses (i)-(v) in [4, Def. 7.1] establishes that indeed Φ( a ) S st ≡ Φ( a ). Forcompleteness, we now list these five inductive clauses and perform this verification. The language of
E-PA ω ∗ st contains a symbol st σ for each finite type σ , but the subscript isalways omitted. Hence T ∗ st is an axiom schema and not an axiom. AMING THE REVERSE MATHEMATICS ZOO 5
Hence, if Φ( a ) and Ψ( b ) in the language of P have the following interpretationsΦ( a ) S st ≡ ( ∀ st x )( ∃ st y ) ϕ ( x, y, a ) and Ψ( b ) S st ≡ ( ∀ st u )( ∃ st v ) ψ ( u, v, b ) , (2.3)then they interact as follows with the logical connectives by [4, Def. 7.1]:(i) ψ S st := ψ for atomic internal ψ .(ii) (cid:0) st( z ) (cid:1) S st := ( ∃ st x )( z = x ).(iii) ( ¬ Φ) S st := ( ∀ st Y )( ∃ st x )( ∀ y ∈ Y [ x ]) ¬ ϕ ( x, y, a ).(iv) (Φ ∨ Ψ) S st := ( ∀ st x, u )( ∃ st y, v )[ ϕ ( x, y, a ) ∨ ψ ( u, v, b )](v) (cid:0) ( ∀ z )Φ (cid:1) S st := ( ∀ st x )( ∃ st y )( ∀ z )( ∃ y ′ ∈ y ) ϕ ( x, y ′ , z )Hence, fix Φ ( a ) ≡ ( ∀ st x )( ∃ st y ) ψ ( x, y, a ) with internal ψ , and note that φ S st ≡ φ for any internal formula. We have [st( y )] S st ≡ ( ∃ st w )( w = y ) and also[ ¬ st( y )] S st ≡ ( ∀ st W )( ∃ st x )( ∀ w ∈ W [ x ]) ¬ ( w = y ) ≡ ( ∀ st w )( w = y ) . Hence, [ ¬ st( y ) ∨ ¬ ψ ( x, y, a )] S st is just ( ∀ st w )[( w = y ) ∨ ¬ ψ ( x, y, a )], and (cid:2) ( ∀ y )[ ¬ st( y ) ∨ ¬ ψ ( x, y, a )] (cid:3) S st ≡ ( ∀ st w )( ∃ st v )( ∀ y )( ∃ v ′ ∈ v )[ w = y ∨ ¬ ψ ( x, y, a )] . which is just ( ∀ st w )( ∀ y )[( w = y ) ∨ ¬ ψ ( x, y, a )]. Furthermore, we have (cid:2) ( ∃ st y ) ψ ( x, y, a ) (cid:3) S st ≡ (cid:2) ¬ ( ∀ y )[ ¬ st( y ) ∨ ¬ ψ ( x, y, a )] (cid:3) S st ≡ ( ∀ st V )( ∃ st w )( ∀ v ∈ V [ w ]) ¬ [( ∀ y )[( w = y ) ∨ ¬ ψ ( x, y, a )]] . ≡ ( ∃ st w )( ∃ y )[( w = y ) ∧ ψ ( x, y, a )]] ≡ ( ∃ st w ) ψ ( x, w, a ) . Hence, we have proved so far that ( ∃ st y ) ψ ( x, y, a ) is invariant under S st . By theprevious, we also obtain: (cid:2) ¬ st( x ) ∨ ( ∃ st y ) ψ ( x, y, a ) (cid:3) S st ≡ ( ∀ st w ′ )( ∃ st w )[( w ′ = x ) ∨ ψ ( x, w, a )] . Our final computation now yields the desired result: (cid:2) ( ∀ st x )( ∃ st y ) ψ ( x, y, a ) (cid:3) S st ≡ (cid:2) ( ∀ x )( ¬ st( x ) ∨ ( ∃ st y ) ψ ( x, y, a )) (cid:3) S st ≡ ( ∀ st w ′ )( ∃ st w )( ∀ x )( ∃ w ′′ ∈ w )[( w ′ = x ) ∨ ψ ( x, w ′′ , a )] . ≡ ( ∀ st w ′ )( ∃ st w )( ∃ w ′′ ∈ w ) ψ ( w ′ , w ′′ , a ) . The last step is obtained by taking x = w ′ . Hence, we may conclude that thenormal form ( ∀ st x )( ∃ st y ) ψ ( x, y, a ) is invariant under S st , and we are done. (cid:3) Finally, the previous theorems do not really depend on the presence of full Peanoarithmetic. Indeed, let
E-PRA ω be the system defined in [29, §
2] and let
E-PRA ω ∗ be its extension with types for finite sequences as in [4, § Corollary 2.4.
The previous theorem and corollary go through for P replaced by P ≡ E-PRA ω ∗ + T ∗ st + HAC int + I .Proof. The proof of [4, Theorem 7.7] goes through for any fragment of
E-PA ω ∗ which includes EFA , sometimes also called I ∆ + EXP . In particular, the exponentialfunction is (all what is) required to ‘easily’ manipulate finite sequences. (cid:3)
Finally, we define
RCA Λ0 as the system P + QF-AC , . Recall that Kohlenbachdefines RCA ω in [29, §
2] as
E-PRA ω + QF-AC , where the latter is the axiom ofchoice limited to formulas ( ∀ f )( ∃ n ) ϕ ( f, n ), ϕ quantifier-free . TAMING THE REVERSE MATHEMATICS ZOO
Notations and remarks.
We introduce some notations regarding
RCA Λ0 .First of all, we shall follow Nelson’s notations as in [5], and given as follows. Remark 2.5 (Standardness) . As suggested above, we write ( ∀ st x τ )Φ( x τ ) and also( ∃ st x σ )Ψ( x σ ) as short for ( ∀ x τ ) (cid:2) st( x τ ) → Φ( x τ ) (cid:3) and ( ∃ x σ ) (cid:2) st( x σ ) ∧ Ψ( x σ ) (cid:3) . Wealso write ( ∀ x ∈ Ω)Φ( x ) and ( ∃ x ∈ Ω)Ψ( x ) as short for ( ∀ x ) (cid:2) ¬ st( x ) → Φ( x ) (cid:3) and ( ∃ x ) (cid:2) ¬ st( x ) ∧ Ψ( x ) (cid:3) . Furthermore, if ¬ st( x ) (resp. st( x )), we also say that x is ‘infinite’ (resp. finite) and write ‘ x ∈ Ω’. Finally, a formula A is ‘internal’ ifit does not involve st, and A st is defined from A by appending ‘st’ to all quantifiers(except bounded number quantifiers).Secondly, we shall use the usual notations for rational and real numbers andfunctions as introduced in [29, p. 288-289] (and [46, I.8.1] for the former). Remark 2.6 (Real number) . A (standard) real number x is a (standard) fast-converging Cauchy sequence q · ) , i.e. ( ∀ n , i )( | q n − q n + i ) | < n ). We freely makeuse of Kohlenbach’s ‘hat function’ from [29, p. 289] to guarantee that every se-quence f can be viewed as a real. Two reals x, y represented by q ( · ) and r ( · ) are equal , denoted x = y , if ( ∀ n )( | q n − r n | ≤ n ). Inequality < is defined similarly.We also write x ≈ y if ( ∀ st n )( | q n − r n | ≤ n ) and x ≫ y if x > y ∧ x y .Functions F mapping reals to reals are represented by functionals Φ → such that( ∀ x, y )( x = y → Φ( x ) = Φ( y )), i.e. equal reals are mapped to equal reals. Finally,sets are denoted X , Y , Z , . . . and are given by their characteristic functions f X ,i.e. ( ∀ x )[ x ∈ X ↔ f X ( x ) = 1], where f X is assumed to be binary.Finally, the notion of equality in RCA Λ0 is important to our enterprise. Remark 2.7 (Equality) . The system
RCA ω includes equality between natural num-bers ‘= ’ as a primitive. Equality ‘= τ ’ for type τ -objects x, y is defined as follows:[ x = τ y ] ≡ ( ∀ z τ . . . z τ k k )[ xz . . . z k = yz . . . z k ] (2.4)if the type τ is composed as τ ≡ ( τ → . . . → τ k → ≈ τ ’ as follows:[ x ≈ τ y ] ≡ ( ∀ st z τ . . . z τ k k )[ xz . . . z k = yz . . . z k ] (2.5)with the type τ as above. Furthermore, the system RCA ω includes the axiom ofextensionality as follows:( ∀ ϕ ρ → τ )( ∀ x ρ , y ρ ) (cid:2) x = ρ y → ϕ ( x ) = τ ϕ ( y ) (cid:3) . ( E )However, as noted in [4, p. 1973], the axiom of standard extensionality ( E ) st cannotbe included in the system P (and hence RCA Λ0 ). Finally, a functional Ξ → is calledan extensionality functional for ϕ → if( ∀ k , f , g ) (cid:2) f Ξ( f, g, k ) = g Ξ( f, g, k ) → ϕ ( f ) k = ϕ ( g ) k (cid:3) . (2.6)In other words, Ξ witnesses ( E ) for Φ. As will become clear in Section 5.1, stan-dard extensionality is translated by our algorithm RS into the existence of anextensionality functional, and the latter amounts to merely an unbounded search. AMING THE REVERSE MATHEMATICS ZOO 7 Classifying
UDNR
In this section, we prove that the principle
UDNR from the introduction is equiv-alent to arithmetical comprehension ( ∃ ) as follows:( ∃ ϕ )( ∀ g ) (cid:2) ( ∃ x ) g ( x ) = 0 ↔ ϕ ( g ) = 0 (cid:3) . ( ∃ )We shall even establish an explicit equivalence between UDNR and a version of ( ∃ ). Definition 3.1. [Explicit implication] An implication ( ∃ Φ) A (Φ) → ( ∃ Ψ) B (Ψ) is explicit if there is a term t in the language such that additionally ( ∀ Φ)[ A (Φ) → B ( t (Φ))], i.e. Ψ can be explicitly defined in terms of Φ.To establish the aforementioned explicit equivalence, we shall obtain a suitable nonstandard equivalence in RCA Λ0 , and apply Corollary 2.3. We first prove thefollowing theorem, where UDNR + is( ∃ st Ψ → ) (cid:2) ( ∀ st A )( ∀ e )(Ψ( A )( e ) = Φ Ae ( e )) ∧ ( ∀ st C , D ) (cid:0) C ≈ D → Ψ( C ) ≈ Ψ( D ) (cid:1)(cid:3) . Note that the second conjunct expresses that Ψ is standard extensional (See Re-mark 2.7). We also need the following restriction of Nelson’s axiom
Transfer :( ∀ st f ) (cid:2) ( ∀ st n ) f ( n ) = 0 → ( ∀ m ) f ( m ) = 0 (cid:3) . (Π - TRANS ) Theorem 3.2. In RCA Λ0 , we have UDNR + ↔ Π - TRANS .Proof.
To prove Π - TRANS → UDNR + , define:Θ( A, M )( e ) := ( Φ Ae,M ( e ) + 1 ( ∃ y, s ≤ M )(Φ Ae,s ( e ) = y )0 otherwise . (3.1)Assuming Π - TRANS , the functional from (3.1) clearly satisfies:( ∀ st e , A )( ∀ M, N ∈ Ω) (cid:2) Θ( A, M )( e ) = Θ( A, N )( e ) (cid:3) . (3.2)The formula (3.2) clearly implies( ∀ st e , A )( ∃ k )( ∀ M, N ≥ k ) (cid:2) Θ( A, M )( e ) = Θ( A, N )( e ) (cid:3) . (3.3)Since RCA Λ0 proves minimisation for Π -formulas, there is a least k as in (3.3),which must be finite by (3.2). Hence, we obtain:( ∀ st e , A )( ∃ st k )( ∀ M, N ≥ k ) (cid:2) Θ( A, M )( e ) = Θ( A, N )( e ) (cid:3) . (3.4)Applying HAC int , there is a standard functional Ψ such that( ∀ st e , A )( ∃ k ∈ Ψ( A, e ))( ∀ M, N ≥ k ) (cid:2) Θ( A, M )( e ) = Θ( A, N )( e ) (cid:3) . (3.5)Now define Ξ( A )( e ) as Θ( A, ζ ( A, e ))( e ), where ζ ( A, e ) is the maximum of Ψ(
A, e )( i )for i < | Ψ( A, e ) | . We then have that:( ∀ st e , A )( ∀ M ∈ Ω) (cid:2) Θ( A, M )( e ) = Ξ( A )( e ) (cid:3) . (3.6)By definition of Θ in (3.1), Ξ is standard extensional (which follows from applyingΠ - TRANS to the associated axiom of extensionality) and satisfies, for standard A , the formula ( ∀ st e ) (cid:2) Ξ( A )( e ) = Φ Ae ( e ) (cid:3) , where the ‘st’ predicates in the latterformula may be dropped by Π - TRANS . Hence, Ξ is as required for
UDNR + .We now prove UDNR + → Π - TRANS . To this end, assume the former andsuppose the latter is false, i.e. there is standard h such that ( ∀ st n ) h ( n ) = 0 and( ∃ m ) h ( m ) = 0. Next, fix a standard pairing function π and its inverse ξ . Nowlet the standard number e be the code of the following program: On input n , set TAMING THE REVERSE MATHEMATICS ZOO k = n and check if k ∈ A and if so, return the second component of ξ ( k ); If k A ,repeat for k + 1. Intuitively speaking, e is such that Φ Ae ( n ) outputs m if startingat k = n , we eventually find π (( l, m )) ∈ A , and undefined otherwise. Furthermore,define C = ∅ (which is the sequence 00 . . . ) and D = { π ( e, Ψ( C )( e )) : h ( e ) = 0 ∧ ( ∀ i < e ) h ( i ) = 0 } , where h is the exception to Π - TRANS . Note that C ≈ D by definition, implyingthat Ψ satisfies Ψ( C ) ≈ Ψ( D ) due to its standard extensionality. However, thelatter combined with UDNR gives us:Ψ( C )( e ) = Ψ( D )( e ) = Φ De ,m ( e ) = Ψ( C )( e ) , (3.7)for large enough (infinite) m . This contradiction yields the theorem. (cid:3) For the following theorem, we require Feferman’s mu-operator:( ∃ µ ) (cid:2) ( ∀ f )(( ∃ n ) f ( n ) = 0 → f ( µ ( f )) = 0) (cid:3) , ( µ )which is equivalent to ( ∃ ) over RCA ω by [29, Prop. 3.9]. As to notation, denote by MU ( µ ) the formula in square brackets in ( µ ) and denote by UDNR (Ψ) the formulain square brackets in
UDNR . We have the following theorem.
Theorem 3.3.
From the proof of
UDNR + ↔ Π - TRANS in RCA Λ0 , two terms s, u can be extracted such that RCA ω proves: ( ∀ µ ) (cid:2) MU ( µ ) → UDNR ( s ( µ )) (cid:3) ∧ ( ∀ Ψ → ) (cid:2) UDNR (Ψ) → MU ( u (Ψ , Φ)) (cid:3) , (3.8) where Φ is an extensionality functional for Ψ Proof.
We prove the second conjunct in (3.8); The first conjunct is analogous. Wefirst show that
UDNR + → Π - TRANS can be brought in the normal form fromCorollary 2.3. First of all, note that Π - TRANS is easily brought into the form:( ∀ st f )( ∃ st y ) (cid:2) ( ∃ x ) f ( x ) = 0 → ( ∃ z ≤ y ) f ( z ) = 0 (cid:3) . (3.9)In UDNR + , resolve the predicates ‘ ≈ ’ in the second conjunct to obtain:( ∀ st X , Y , k )( ∃ st N )( XN = Y N → Ψ( X ) k = Ψ( Y ) k ) . Apply
HAC int to obtain standard Φ such that ( ∃ N ∈ Φ( X, Y, k )). Define Ξ(
X, Y, k )as max i< | Φ( X,Y,k ) | Φ( X, Y, k )( i ) to obtain( ∃ st Ξ)( ∀ st X , Y , k ) (cid:2) X Ξ( X, Y, k ) = Y Ξ( X, Y, k ) → Ψ( X ) k = Ψ( Y ) k (cid:3) . Let B (Ξ , X, Y, k ) be the formula in square brackets in the previous formula and let C ( f, y ) be the formula in square brackets in (3.9). So far, we have derived (cid:2) ( ∃ st Ψ)( ∀ st Z ) A ( Z, Ψ) ∧ ( ∃ st Ξ)( ∀ st X , Y , k ) B (Ξ , X, Y, k ) (cid:3) → ( ∀ st f )( ∃ st y ) C ( f, y ) , from UDNR + → Π - TRANS in RCA Λ0 , where A ( Z, Ψ) ≡ ( ∀ e )(Ψ( Z )( e ) = Φ Ze ( e )).By bringing outside all the standard quantifiers, we obtain( ∀ st f, Ψ , Ξ)( ∃ st y , X , Y , Z , k ) (cid:2) [ A ( Z, Ψ) ∧ B (Ξ , X, Y, k )] → C ( f, y ) (cid:3) , (3.10)where the formula in square brackets is internal. Thanks to Corollary 2.3, the termextraction algorithm A applied to ‘ RCA Λ0 ⊢ (3.10)’, provides a term t such that( ∀ f, Ψ , Ξ)( ∃ ( y , X , Y , Z , k ) ∈ t ( f, Ψ , Ξ)) (cid:2) [ A ( Z, Ψ) ∧ B (Ξ , X, Y, k )] → C ( f, y ) (cid:3) AMING THE REVERSE MATHEMATICS ZOO 9 is provable in
RCA ω . Now let s be the term t with all entries not pertaining to y omitted; We have( ∀ f, Ψ , Ξ)( ∃ k , X , Y , Z )( ∃ y ∈ s ( f, Ψ , Ξ)) (cid:2) [ A ( Z, Ψ) ∧ B (Ξ , X, Y, k )] → C ( f, y ) (cid:3) . Now define u ( f, Ψ , Ξ) as the maximum of all entries of s ( f, Ψ , Ξ); We have( ∀ f, Ψ , Ξ)( ∃ k , X , Y , Z ) (cid:2) [ A ( Z, Ψ) ∧ B (Ξ , X, Y, k )] → C ( f, u ( f, Ψ , Ξ)) (cid:3) . Bringing all quantifiers inside again as far as possible, we obtain( ∀ Ψ , Ξ) (cid:2) [( ∀ Z ) A ( Z, Ψ) ∧ ( ∀ k , X , Y ) B (Ξ , X, Y, k )] → ( ∀ f ) C ( f, u ( f, Ψ , Ξ)) (cid:3) . Hence, if Ψ is as in UDNR and Ξ witnesses the extensionality of Ψ , then u ( · , Ψ , Ξ ) is Feferman’s my-operator, and we are done. (cid:3) Corollary 3.4. In RCA ω , we have the explicit equivalence UDNR ↔ ( µ ) . Clearly, there is a general strategy to obtain the normal form as in (3.10) forprinciples similar to
UDNR + , as discussed in the following remark. Remark 3.5 (Algorithm B ) . Let T ≡ ( ∀ X )( ∃ Y ) ϕ ( X, Y ) be an internal formulaand define the ‘strong’ uniform version
U T + as( ∃ st Φ → ) (cid:2) ( ∀ st X ) ϕ ( X, Φ( X )) ∧ Φ is standard extensional (cid:3) . The proof of Theorem 3.3 provides a normal form algorithm B to convert the im-plication U T + → Π - TRANS into the normal form ( ∀ st x )( ∃ st y ) ϕ ( x, y ) as in (3.10).In general, if T → DNR and the proof of the implication is sufficiently uniform,then
U T → ( ∃ ) and this implication is explicit. We now list some examples. Remark 3.6 (Immediate consequences) . First of all, let DNR k be DNR where thefunction f satisfies f ≤ k , and let UDNR k be UDNR with the same restrictionon Ψ( A ). Clearly, for any k ≥
1, we have the explicit equivalence
UDNR k ↔ ( ∃ ).Secondly, let RKL be the ‘Ramsey type’ version of
WKL from [19] and let
URKL be its obvious uniform version. In
RCA ω , we have the explicit equivalence URKL ↔ ( ∃ ), as it seems the proof of RKL → DNR from [19, Theorem 8] can be uniformized.Indeed, in this proof,
RKL is applied to a specific tree T from [19, Lemma 7] toobtain a certain set H . Then the function g is defined such that W g ( e ) is the least e + 3 elements of H . This function g is then shown to be fixed-point free, whichmeans it gives rise to a DNR -function by [47, V.5.8, p. 90]. Noting that the tree T has positive measure, we even obtain WRKL → DNR (and the associated uniformequivalence to ( ∃ )), where the tree has positive measure in the latter (See [7]).Thirdly, let SEM be the stable
Erd¨os-Moser theorem from [32]. In [37, Theo-rem 3.11], the implication
SEM → DNR is proved, and the proof is clearly uniform.Hence, for
USEM the uniform version of
SEM , we have (explicitly) that
USEM ↔ ( ∃ ). The same obviously holds for EM , the version of SEM without stability.In the next section, we shall study principles from the zoo for which a ‘uniformis-ing’ proof as in the previous remark is not immediately available. We finish thissection with a remark on the Reverse Mathematics zoo. Since the system
RCA ω includes the axiom of extensionality ( E ) and QF-AC , , and in lightof the elementary nature (an unbounded search) of an extensionality functional Ξ, we will stillcall t (Ξ , · ) ‘explicit’ if t is a term from the language. Remark 3.7 (A higher-order zoo) . Since
DNR is rather ‘low’ in the zoo, it is to beexpected that uniform versions of ‘most’ of the zoo’s principles will behave as
UDNR ,i.e. turn out equivalent to ( ∃ ) (as we will establish below). In particular, sinceFriedman-Simpson style Reverse Mathematics is limited to second-order arithmetic,the proof of Theorem 3.2 will go through for principles other than UDNR as theassociated functionals can only have type 1 → higher-type principles, to whichthe proof of Theorem 3.2 does not apply, will populate a ‘higher-order’ RM zoo.4. Classifying the Reverse Mathematics zoo
In this section, we classify uniform versions of a number principles from the RMzoo, based on the results in the previous section. After these case studies, we shallformulate in Section 5.1 a template which seems sufficiently general to apply tovirtually any (past or future) principle from the RM zoo.4.1.
Ascending and descending sequences.
In this section, we study the uni-form version of the ascending-descending principle
ADS (See e.g. [24, Def. 9.1]).
Definition 4.1.
For a linear order (cid:22) , a sequence x n is ascending if x ≺ x ≺ . . . and descending if x ≻ x ≻ . . . . Definition 4.2 ( ADS ) . Every infinite linear ordering has an ascending or a de-scending sequence.
Recall that LO( X ) is short for ‘ X is a linear order’; We append ‘ ∞ ’ to ‘LO’to stress that X is an infinite linear order, meaning that its field is not boundedby any number (See [46, V.1.1]). With this in place, uniform ADS is as follows:
Definition 4.3 ( UADS ) . ( ∃ Ψ → )( ∀ X ) (cid:2) LO ∞ ( X ) → ( ∀ n )Ψ( X )( n ) < X Ψ( X )( n + 1) ∨ ( ∀ m )Ψ( X )( m ) > X Ψ( X )( m + 1) (cid:3) . (4.1)Note that we can decide which case of the disjunction of UADS holds by testingΨ( X )(0) < X Ψ( X )(1). We have the following theorem. Theorem 4.4. In RCA ω , we have the explicit equivalence UADS ↔ ( µ ) .Proof. Since all notions involved are arithmetical, the explicit implication ( µ ) → UADS is straightforward. For the remaining explicit implication, we will prove
UADS + → Π - TRANS , where the former is( ∃ st Ψ → ) (cid:2) ( ∀ st X ) A ( X, Ψ) ∧ ( ∀ st X , Y ) (cid:0) X ≈ Y → Ψ( X ) ≈ Ψ( Y ) (cid:1)(cid:3) , where A ( X, Ψ) is the formula in square brackets in (4.1). It is then easy to bringthe implication
UADS + → Π - TRANS in the normal form as in (3.10) using thealgorithm B from Remark 3.5. Applying the term extraction algorithm A usingCorollary 2.3 then establishes the explicit implication UADS → ( µ ).Thus, assume UADS + and suppose Π - TRANS is false, i.e. there is a standard h such that ( ∀ st n ) h ( n ) = 0 and ( ∃ m ) h ( m ) = 0. Now let m be the least numbersuch that h ( m ) = 0 and define the ordering ‘ ≺ ’ as follows: · · · ≺ m + 2 ≺ m + 1 ≺ ≺ ≺ ≺ · · · ≺ m . (4.2) Here, ‘infinite’ should not be confused with the notation ‘ M is infinite’ for ¬ st( M ); Note thetype mismatch between numbers and orders. AMING THE REVERSE MATHEMATICS ZOO 11
It is straightforward to define the standard ordering ≺ using the function h . Nowconsider the usual strict ordering < and note that ( ≺ ) ≈ ( < ) (with someabuse of notation in light of [46, V.1.1]). By the standardness of ≺ and standardextensionality for the standard Ψ functional from UADS + , we have Ψ( ≺ ) ≈ Ψ( < )(again with some abuse of notation). However, this leads to a contradiction as < only has ascending infinite sequences, while ≺ only has descending infinitesequences. Indeed, while only the first case in (4.1) can hold for Ψ( < ), onlythe second case can hold for Ψ( ≺ ). But then Ψ( ≺ ) ≈ Ψ( < ) is impossible. Thiscontradiction guarantees that UADS + → Π - TRANS , and we are done. (cid:3)
In [23, Prop. 3.7], it is proved that
ADS is equivalent to the principle
CCAC .In light of the uniformity of the associated proof, the uniform version of the latteris also equivalent to ( ∃ ). Furthermore, the equivalence in the previous theoremtranslates into a result in constructive Reverse Mathematics (See [26]) as follows.
Remark 4.5 (Constructive Reverse Mathematics) . The ordering ≺ defined in (4.2)yields a proof that ADS → Π -LEM over the (constructive) base theory from [26].Indeed, for a function h , define the ordering ≺ h from Footnote 8. By ADS , thereis a sequence x n which is either ascending or descending in ≺ h . It is now easy tocheck that if x ≺ h x , then ( ∀ n ) h ( n ) = 0, and if x ≻ h x then ¬ [( ∀ n ) h ( n ) = 0].Hence, ADS provides a way to decide whether a Π -formula holds or not, i.e. thelaw of excluded middle limited to Π -formulas.Next, we consider a special case of ADS . The notion of discrete and stable linearorders from [24, Def. 9.15] is defined as follows.
Definition 4.6. [Discrete and stable orders] A linear order is discrete if everyelement has an immediate predecessor, except for the first element of the order ifthere is one, and every element has an immediate successor, except for the lastelement of the order if there is one. A linear order is stable if it is discrete and hasmore than one element, and every element has either finitely many predecessors orfinitely many successors. (Note that a stable order must be infinite.)Again, to be absolutely clear, the notion of ‘finite’ and ‘infinite’ in the previousdefinition constitutes the ‘usual’ internal definitions of infinite orders in
RCA ω andhave nothing to do with our notation ‘ M is infinite’ for ¬ st( M ). In particular,note the type mismatch between orders and numbers.Now denote by SADS the principle
ADS limited to stable linear orderings, andlet
USADS be its uniform version.
Corollary 4.7. In RCA ω , we have the explicit equivalence USADS ↔ ( µ ) ,Proof. Note that both the orderings < and ≺ defined in the proof of the theoremare stable and this proof thus also yields USADS + → Π - TRANS . (cid:3) Let
SRT be Ramsey’s theorem for pairs limited to stable colourings (See e.g.[24, Def. 6.28]), and let USRT be its uniform version where a functional Ψ → takesas input a stable 2-colouring of pairs of natural numbers and outputs an infinitehomogeneous set. Corollary 4.8. In RCA ω , we have the explicit equivalence USRT ↔ ( µ ) . The order ≺ from (4.2) can be defined as: i ≺ j holds if i < j ∧ ( ∀ k ≤ j − h ( k ) = 0 or i > j ∧ ( ∃ k ≤ j − h ( k ) = 0 or i > j ∧ ( ∃ k ≤ j − h ( k ) = 0 ∧ ( ∀ k ≤ i − h ( k ) = 0. Proof.
By [23, Prop. 2.8], we have
SRT → SADS . The proof of the latter is clearlyuniform, yielding the forward implication by Corollary 4.7. By [39, Theorem 4.2],the reverse implication follows. (cid:3)
We can prove similar results for
SRAM and related principles from [15], but donot go into details. Our next corollary deals with the chain-antichain principle . Definition 4.9. [ CAC ] Every infinite partial order ( P, ≤ P ) has an infinite subset S that is either a chain , i.e. ( ∀ x , y ∈ S )( x ≤ P ∨ y ≤ P x ), or an antichain , i.e.( ∀ x , y ∈ S )( x = y → x P ∨ y P x ).Let UCAC be the principle
CAC with the addition of a functional Ψ → such thatΨ( P, ≤ P ) is the infinite subset which is either a chain or antichain. Let USCAC be UCAC limited to stable partial orders (See [23, Def. 3.2]).
Corollary 4.10. In RCA ω , we have the explicit equivalences UCAC ↔ ( µ ) ↔ USCAC .Proof.
In [23, Prop. 3.1],
CAC → ADS is proved. The proof is clearly uniform,implying the explicit implication
UCAC → UADS . By Theorem 4.4, we obtain thefirst forward implication in the theorem. The first reverse implication is proved asin the final part of the proof of Theorem 4.4. For the final reverse implication, theimplication
SCAC → SADS is proved in [23, Prop. 3.3]. Since the latter proof isclearly uniform, we have (explicitly) that
USCAC → ( ∃ ) by Corollary 4.7. Theremaining implication is immediate. (cid:3) Finally, we point out one important feature of the above proofs.
Remark 4.11 (Discontinuities) . We show that the construction (4.2) which givesrise to
UADS + → Π - TRANS , also implies the existence of a discontinuity (in thesense of Nonstandard Analysis). Indeed, for infinite M , define g ≡ . . . . . . where g ( M ) = 1. Let the (nonstandard) order ⊳ be the order ≺ as in (4.2) butwith g instead of h . Then clearly ( < ) ≈ ( ⊳ ) ∧ Ψ( < ) Ψ( ⊳ ) for Ψ as in UADS , i.e. this functional is not nonstandard continuity ‘around’ < . A similarconstruction involving g gives rise to a discontinuity around any standard input. Inconclusion, the functional Ψ from UADS + is ‘everywhere discontinuous’ in the senseof Nonstandard Analysis. This observation applies to all the RM zoo principlesdiscussed in this section. Thus, principles of the form ( ∀ X )( ∃ Y ) ϕ ( X, Y ) fromthe RM zoo can be said to be ‘not continuous in their input parameter X ’.4.2. Thin and free sets.
In this section, we study the so-called thin- and free settheorems from [11]. In the latter, the thin set theorem TS is defined as follows; TS ( k ) is TS limited to some fixed k ≥ Principle 4.12 ( TS ) . ( ∀ k )( ∀ f : [ N ] k → N )( ∃ A )( A is infinite ∧ f ([ A ] k ) = N ) . We define
UTS (2) as follows:( ∃ Ψ → )( ∀ f : [ N ] → N ) (cid:2) Ψ( f ) is infinite ∧ ( ∃ n ) (cid:2) n f (cid:0) [Ψ( f )] (cid:1)(cid:3)(cid:3) . ( UTS (2))We did not use ‘ N ’ to avoid confusion. Recall that ‘Ψ( f ) is infinite’ has nothing todo with infinite numbers M ∈ Ω; Note in particular the type mismatch.
Theorem 4.13. In RCA ω , we have the explicit equivalence ( µ ) ↔ UTS (2) . AMING THE REVERSE MATHEMATICS ZOO 13
Proof.
The forward (explicit) implication is immediate from the results in [11, § RCA Λ0 and applythe algorithms B and A using Corollary 2.3. Hence, let Ψ be as in UTS (2) andapply
QF-AC , to ( ∀ f : [ N ] → N )( ∃ n ) (cid:2) n f (cid:0) [Ψ( f )] (cid:1)(cid:3) to obtain Ξ witnessing n . In this way, UTS (2) becomes( ∃ Φ → (1 × )( ∀ f : [ N ] → N ) (cid:2) Φ( f )(1) is infinite ∧ Φ( f )(2) f (cid:0) [Φ( f )(1)] (cid:1)(cid:3) . Let A (Φ , f ) be the formula in square brackets and define UTS (2) + as( ∃ st Φ → (1 × )( st ∀ f : [ N ] → N ) (cid:2) A (Φ , f ) ∧ Φ is standard extensional (cid:3) . We now prove that
UTS (2) + → Π - TRANS ; To this end, assume the latter andsuppose h is a counterexample to Π - TRANS , i.e. ( ∀ st n ) h ( n ) = 0 ∧ ( ∃ m ) h ( m ) = 0.Fix standard f : [ N ] → N and define g : [ N ] → N as: g ( k, l ) := ( f ( k, l ) ( ∀ i ≤ max( k, l )) h ( i ) = 0Φ( f )(2) otherwise . (4.3)By assumption, f ≈ g , and we obtain Φ( f ) ≈ (1 × Φ( g ) by the standard exten-sionality of Φ. Note that in particular Φ( f )(2) = Φ( g )(2), and since Φ( g )(1) isinfinite, there are some k ′ > k > m such that k , k ′ ∈ Φ( g )(1) where m is suchthat h ( m ) = 0. However, by the definition of g , we obtain Φ( f )(2) ∈ g ([Φ( g )(1)] ),as we are in the second case of (4.3) for g ( k , k ′ ). Since Φ( f )(2) = Φ( g )(2), theprevious yields the contradiction Φ( g )(2) ∈ g ([Φ( g )(1)] ), and hence Π - TRANS must hold. Now bring
UTS (2) + → Π - TRANS in the normal form using B andapply term extraction via A , using Corollary 2.3. (cid:3) Clearly, the previous proof also goes through for the uniform version of
STS (2),which is TS (2) limited to stable functions, i.e. for functions f : [ N ] → N such that( ∀ x )( ∃ y )( ∀ z ≥ y )( f ( x, y ) = f ( x, z )).Next, we consider the following corollary regarding the free set theorem, where UTS ( k ) and UFS ( k ) have obvious definitions in light of the notations in [11]. Corollary 4.14. In RCA ω , we have ( explicitly ) that ( µ ) ↔ UTS ( k ) ↔ UFS ( k ) ,where k ≥ .Proof. The case k ≥ ACA proves FS ([11]). To obtain the set B in the proof of the former theorem, apply QF-AC , to the fact that the free set isinfinite. For the case k = 1, proceed as in the theorem. (cid:3) As noted by Kohlenbach in [29, § FS (1) in [11, Theorem 2.2] uses this law,explaining the equivalence to ( ∃ ) of the associated uniform version.4.3. Cohesive sets.
In this section, we study principles based on cohesiveness (Seee.g. [24, Def. 6.30]). We start with the principle
COH . Definition 4.15.
A set C is cohesive for a collection of sets R , R , . . . if it isinfinite and for each i , either C ⊆ ∗ R i or C ⊆ ∗ R i . Here, A is the complement of A and A ⊆ ∗ B means that A \ B is finite. Definition 4.16. [ COH ] Every countable collection of sets has a cohesive set.
It is important to note that
COH involves multiple significant existential quan-tifiers: The ‘( ∃ C )’ quantifier, but also the existential type 0-quantifiers in C ⊆ ∗ R i ∨ C ⊆ ∗ R i . As we will see, it is important that the functional from the uniformversion of COH outputs both the set C and an upper bound to C \ R i or C \ R i . Itwould be interesting, but beyond the scope of this paper, to study a weak versionof UCOH only outputting C . Definition 4.17. [ UCOH ] There is Φ (0 → → (1 × such that for all R → ( ∀ k )( ∃ l > k )[ l ∈ Φ( R )(1)] ∧ ( ∀ i ) h(cid:0) ∀ n ∈ Φ( R )(1) (cid:1) ( n ≥ Φ( R )(2)( i ) → n ∈ R ( i )) ∨ (cid:0) ∀ m ∈ Φ( R )(1) (cid:1) ( m ≥ Φ( R )(2)( i ) → m ∈ R ( i )) i . (4.4)Note that we may treat the collection R → as a type 1-object, namely as adouble sequence (See for instance [46, p. 13]), and the same holds for Φ( R ). Theorem 4.18. In RCA ω , we have the explicit equivalence UCOH ↔ ( µ ) .Proof. For the reverse implication, since cohesiveness is an arithmetical property,it is easy to build the functional Φ from
UCOH assuming ( ∃ ).For the forward implication, consider UCOH and apply
QF-AC , to the first con-junct of (4.4) to obtain Ξ such that ( ∀ R → , k )[Ξ( R, k ) > k ∧ Ξ( R, k ) ∈ Φ( R )(1)].Define UCOH + as the resulting formula but starting with ( ∃ st Φ , Ξ)( ∀ st R → ) andthe addition that Φ and Ξ are standard extensional. Note that we can decidewhich disjunct holds (for given i ) in the second conjunct of (4.4) by checking ifΞ( R, Φ( R )(2)( i )) ∈ R ( i ). For standard R, i , the latter only involves standard ob-jects. We now prove
UCOH + → Π - TRANS , from which the theorem follows byapplying the algorithms B and A using Corollary 2.3.Now assume UCOH + and suppose there is standard h such that ( ∀ st n ) h ( n ) =0 ∧ ( ∃ m ) h ( m ) = 0. Suppose for some fixed standard R , there is standard i such thatthe first disjunct holds in the second conjunct of (4.4). Now define R ′ as follows: k ∈ R ′ ( j ) ↔ [ k ∈ R ( j ) ∧ ( ∀ n ≤ max( j, k )) h ( n ) = 0]. Clearly, R ′ is standard andwe have R ≈ → R ′ , implying Φ( R ) ≈ × Φ( R ′ ). In particular, Φ( R )(2)( i ) = Φ( R ′ )(2)( i ), and Φ( R )(1) ≈ Φ( R ′ )(1). However, then the first disjunct holdsin the second conjunct of (4.4) for R ′ , i too, since Ξ( R ′ , Φ( R ′ )(2)( i )) ∈ R ′ ( i )is equivalent to Ξ( R, Φ( R )(2)( i )) ∈ R ( i ). However, now let m be such that h ( m ) = 0 and take m < l ∈ Φ( R ′ )(1). Clearly, l > Φ( R ′ )(2)( i ) as the firstnumber is infinite and the second finite. But then l ∈ R ′ ( i ) by UCOH + , which isimpossible by the definition of R ′ . A similar procedure leads to a contradiction incase the second disjunct holds in the second conjunct of (4.4) for some standard i .In light of these contradictions, the implication UCOH + → Π - TRANS follows. (cid:3)
While Ramsey’s theorem for pairs RT does not imply WKL (See e.g. [24, 33]),the uniform versions are equivalent.
Corollary 4.19. In RCA ω , we have the explicit equivalence URT ↔ UWKL .Proof.
The implication RT → COH is proved in [24, 6.32]. This proof is clearlyuniform (as also noted at the end of [24, p. 85]), yielding
URT → UCOH , andthe theorem implies the forward implication, since ( ∃ ) ↔ UWKL ([29]). By [39,Theorem 4.2], the reverse implication follows. (cid:3)
AMING THE REVERSE MATHEMATICS ZOO 15
Next, we study the cohesive version of
ADS . Recall the definition of a stableorder from Definition 4.6. Denote by
CADS the statement that every infinite linearorder has a stable suborder. The connection between
CADS and cohesiveness isdiscussed between [24, 9.17-9.18]. Now let
UCADS be the ‘fully’ uniform version of
CADS as follows.
Definition 4.20. [ UCADS ] There is Φ → (1 × such that for infinite linear orders X , Y ≡ Φ( X )(1) is a stable suborder of X and Φ( X )(2) witnesses this, i.e. for y ∈ Y :( ∀ w )( y ≤ Y w → w ≤ Y Φ( X )(2)( y )) ∨ ( ∀ v )( y ≥ Y v → v ≥ Y Φ( X )(2)( y )) . (4.5) Theorem 4.21. In RCA ω , we have the explicit equivalence UCADS ↔ ( µ ) .Proof. The reverse implication is immediate in light of Theorem 4.18 and the uni-formity of the proofs of [23, Prop. 1.4 and 2.9]. For the forward implication,we proceed as in the proof of Theorem 4.18: Consider
UCADS and apply
QF-AC , to the formula expressing that Φ( X )(1) is infinite to obtain Ξ such that( ∀ X , k )[Ξ( X, k ) > k ∧ Ξ( X, k ) ∈ Φ( X )(1)]. Define UCADS + as the resulting for-mula but starting with ( ∃ st Φ , Ξ)( ∀ st X ) and the addition that Φ and Ξ are standardextensional.Now assume UCADS + and suppose h is a counterexample to Π - TRANS . Con-sider again the orders < and ≺ from the proof of Theorem 4.4. Since < ≈ ≺ (again with some abuse of notation), we have Φ( < ) ≈ × Φ( ≺ ). Now take standard n ∈ Φ( < )(1) (which exist by the standardness of Ξ and also satisfies n ∈ Φ( ≺ )(1)by standard extensionality) and consider the standard number Φ( < )(2)( n ) = Φ( ≺ )(2)( n ), the latter equality again by standard extensionality. However, by theinfinitude of Φ( < )(1) (resp. of Φ( ≺ )(1)) only the second (resp. first) disjunct of(4.5) can hold for < (resp. for ≺ ). Then, the second (resp. first) disjunct of (4.5)for < (resp. ≺ ) implies n ≥ Φ( < )(2)( n ) (resp. n (cid:22) Φ( ≺ )(2)( n )). Since allobjects are standard, we obtain n = Φ( < )(2)( n ) = Φ( ≺ )(2)( n ). However,then Φ( < )(1) ≈ Φ( ≺ )(1) is impossible as the ‘overlap’ between the latter twoorders is a singleton, namely { n } . (cid:3) In [23, Prop. 2.9], a uniform proof of
CADS from
CRT , a cohesive version of RT , is presented. Hence, it follows that the (fully) uniform version of CRT is alsoequivalent to ( ∃ ). Finally, we discuss a connection between our results and [5]. Remark 4.22 (Alternative approach) . The above non-explicit results can also beobtained in a different way: It is established in [5, Cor. 12] that ( ∃ ) ↔ Π - TRANS over a suitable (nonstandard) base theory. The essential ingredient in the latter sys-tem is parameter-free Transfer
PF-TP ∀ , i.e. Nelson’s axiom Transfer (See Section 2)where the formulas ϕ have no parameters. In contrast to Transfer , parameter-freeTransfer does not carry any logical strength. However, the principle UDNR isparameter-free, implying that the functional Ψ from the former is standard , assum-ing
PF-TP ∀ . Similarly, the principle ‘There is Ψ , Ξ such that
UDNR (Ψ) and Ξ is anextensionality functional for Ψ’ does not have any parameters, and
PF-TP ∀ yieldsthat Ψ and Ξ are standard. However, the standardness of Ξ also implies that Ψ isstandard extensional . Hence, Theorem 3.2 and [5, Cor. 12] immediately yield that UDNR ↔ Π - TRANS ↔ ( ∃ ), assuming PF-TP ∀ . These equivalences were provedin a conservative extension of RCA ω , implying that the latter proves ( ∃ ) ↔ UDNR . Classifying the strong Tietze extension theorem.
In this section, westudy a uniform version of the Tietze (extension) theorem. Non-uniform versionsof the Tietze theorem are studied in [46, II.7] and [22]. We are interested in the‘strong’ Tietze theorem [22, 6.15.(5)] since it implies
DNR and is implied by
WKL (See [22, § µ ) (and hence UWKL by [29, § f ∈ C rm ( X ) mean that f is continuousin the sense of Reverse Mathematics on X , i.e. as in [46, II.6.1] or [22, Def. 2.7]. Fur-thermore, let C ( X ) be the Banach space used in the Tietze theorem [22, 6.15.(5)] asdefined in [22, p. 1454]. Finally, we use the same definition for closed and separablyclosed sets as in [22]. Principle 4.23 ( UTIE ) . There is a functional Ψ (1 × → such that for closed andseparably closed sets A ⊆ [0 , and for f ∈ C rm ( A ) with modulus of uniform conti-nuity g , we have Ψ( f, g, A ) ∈ C ([0 , and f equals Ψ( f, g, A ) on A . We also study the following uniform version of Weierstraß’ (polynomial) approx-imation theorem. The non-uniform version is equivalent to
WKL by [46, IV.2.5]
Principle 4.24 ( UWA ) . There is Ψ → such that ( ∀ f ∈ C rm [0 , ∀ x ∈ [0 , , n ) (cid:2) Ψ( f )( n ) ∈ POLY ∧ | f ( x ) − Ψ( f )( n )( x ) | < n (cid:3) . Theorem 4.25. In RCA ω , we have the explicit equivalences UWA ↔ UTIE ↔ ( µ ) .Proof. As in the proof of [29, Prop. 3.14], it is straightforward to obtain
UWA using ( ∃ ) from the associated non-uniform proof, even when f is a type 1 → ε - δ -continuous. Indeed, it is well-known thatlim n →∞ B n ( f )( x ) = f ( x ) uniformly for x ∈ [0 , f is continuous on [0 ,
1] and B n ( f ) are the associated Bernstein polynomials ([35, p. 6]). Using ( ∃ ) it is theneasy to define Ψ( f )( n ) as the least N such sup x ∈ [0 , | B N ( f )( x ) − f ( x ) | ≤ n +2 .For the explicit implication UTIE → ( ∃ ), we will use of Corollary 3.4 and [22, § DNR . In this proof, a function f defined on a set C is constructed in RCA (Seethe proof of [22, Lemma 6.16]). This function satisfies all conditions of the strongTietze theorem; In particular, it has a modulus of uniform continuity of f . Applying[22, 6.15.(5)], one obtains F ∈ C [0 , f to [0 , C ( X ) from [22, p. 1454], F is coded by a sequenceof polynomials p n such that k p n − F k < n +2 , and we can define h ( n ) := ♯ ( p n ). Thelatter is then such that ( ∀ e )( h ( e ) = Φ e ( e )). The case of DNR where A = ∅ is thenstraightforward. Indeed, the initial function f (from the proof of [22, Lemma 6.16])is defined using a recursive counterexample to the Heine-Borel lemma. Such acounterexample can be found in [46, I.8.6] and clearly relativizes (uniformly) toany set A . Let us use f A to denote the function f obtained from the previous AMING THE REVERSE MATHEMATICS ZOO 17 construction relative to the set A , and let C A and g A be the relativized domain andmodulus. Now let Ψ be the functional from UTEI and define Ξ → byΞ( A ) := ♯ (cid:0) Ψ( f A , g A , C A ) (cid:1) , where f A , g A , and C A are as in the previous paragraph of this proof. In the sameway as in the proof of [22, Lemma 6.17], one proves that for any A , we have( ∀ e )(Ξ( A )( e ) = Φ Ae ( e )). However, this yields the explicit implication UTIE → UDNR and the latter explicitly implies ( µ ) by Corollary 3.4.Next, to prove the explicit implication UWA → UTIE , note that Simpson provesan effective version of the Tietze theorem in [46, II.7.5]. Following the proof ofthe latter, it is clear that there is a functional Φ in
RCA ω such that for closedand separably closed A and f ∈ C rm ( A ), the image Φ( f, g, A ) ∈ C rm [0 ,
1] is theextension of f to [0 ,
1] provided by [46, II.7.5]. For Ψ as in
UWA , the functionalΨ(Φ( f, g, A )) is as required by
UTIE . (cid:3) Let
UTIE ′ and UWA ′ be the versions of UTIE and
UWA with the usual epsilon-delta definition of continuity instead of the Reverse Mathematics definition of con-tinuity. The following corollary is immediate from the proof of the theorem.
Corollary 4.26. In RCA ω , we have UWA ′ ↔ UTIE ′ ↔ ( ∃ ) . In the proof of the theorem, we established
UTIE → ( ∃ ) by showing that UTIE → UDNR , and then applying Corollary 3.4. The latter implication goes throughbecause of the uniformity of the proof of
DNR from the strong Tietze theorem (See[22, Lemma 6.17]).
Remark 4.27 (The role of extensionality) . At the risk of stating the obvious, theaxiom of extensionality is central in proving all above equivalences; In particular,half of the explicit implications obtained above all have an extensionality functional‘buit-in’. Hence, an approach to uniform computability not involving the axiom ofextensionality will yield different results. It is a matter of opinion whether in thelatter such ‘non-extensional framework’, the glass is half-full (finer distinctions)or half-empty (more complicated picture). In our opinion, it is remarkable howuniform our uniform classification has turned out.5.
Taming the future Reverse Mathematics zoo
In this secton, we formulate a general template for obtaining (explicit) equiva-lences between ( µ ) and uniform versions of principles from the RM zoo.5.1. General template.
Our template is defined as follows.
Template.
Let T ≡ ( ∀ X )( ∃ Y ) ϕ ( X, Y ) be a RM zoo principle and let
U T be( ∃ Φ → )( ∀ X ) ϕ ( X, Φ( X )). To prove the explicit implication U T → ( µ ), executethe following steps:(i) Let U T + be ( ∃ st Φ → )( ∀ st X ) ϕ ( X, Φ( X )) where the functional Φ is addition-ally standard extensional. We work in RCA Λ0 + U T + .(ii) Suppose the standard function h is such that ( ∀ st n ) h ( n ) = 0 and ( ∃ m ) h ( m ) =0, i.e. h is a counterexample to Π - TRANS .(iii) For standard V , use h to define standard W ≈ V such that Φ( W ) Φ( V ),i.e. W is V with the nonstandard elements changed sufficiently to yield adifferent image under Φ. (iv) The previous contradiction implies that RCA Λ0 proves U T + → Π - TRANS .(v) Bring the implication from the previous step into the normal form( ∀ st x )( ∃ st y ) ψ ( x, y ) ( ψ internal) using the algorithm B from Remark 3.5.(vi) Apply the term extraction algorithm A using Corollary 2.3. The resultingterm yields the explicit implication U T → ( µ ).The explicit implication ( µ ) → U T is usually straightforward; Alternatively, es-tablish Π - TRANS → U T + in RCA Λ0 and apply steps (v) and (vi).The algorithm RS is defined as the steps (v) and (vi) in the template, i.e. theapplication of the algorithms B and A to suitable implications. In Section 5.2,we speculate why uniform principles U T originating from RM zoo-principles areequivalent to ( ∃ ) en masse . We conjecture a connection to Montalb´an’s notion of robustness from [34].Finally, the above template treats zoo-principles in a kind of ‘Π -normal form’,for the simple reason that most zoo-principles are formulated in such a way. Nonethe-less, it is a natural question, discussed in Section 6, whether principles not formu-lated in this normal form gives rise to uniform principles not equivalent to ( ∃ ).Surprisingly, the answer to this question turns out to be negative.5.2. Robustness and structure.
In this section, we try to explain why our tem-plate works so well for RM zoo principles. We conjecture a connection to Mon-talb´an’s notion of robustness from [34].First of all, standard computable functions are determined by their behaviouron the standard numbers (by the
Use principle from [47, p. 50]), while e.g. astandard Turing machine may well halt at some infinite number (given e.g. thefan functional from [29] or ¬ Π - TRANS ), i.e. non-computable problems, like theHalting problem for standard Turing machines, are not necessarily determined bythe standard numbers.Now in step (iii), the assumption ¬ Π - TRANS allows us to change the nonstan-dard part of a standard set V , resulting in standard W ≈ V . Since Φ( V ) (resp.Φ( W )) is not computable from V (resp. W ), the former depends on the nonstan-dard numbers in V (resp. W ). However, making the nonstandard parts of V and W different enough, we can guarantee Φ( W ) Φ( V ), and obtain a contradictionwith standard extensionality. Hence, Π - TRANS follows and so does
U T → ( ∃ ).Alternatively, as noted in Remark 4.11, we can define standard V and nonstandard W such that V ≈ W ∧ Φ( V ) Φ( W ) without assuming ¬ Π - TRANS . Hence Φ isnot nonstandard continuous and Kohlenbach has pointed out that a discontinuousfunction can be used to define ( ∃ ) using Grilliot’s trick (See [29, Prop. 3.7]).Secondly, note that step (iii) crucially depends on the fact that we can modify thenonstandard numbers in the set V without changing the standard numbers , i.e. whileguaranteeing V ≈ W . Such a modification is only possible for structures which are not closed downwards: For instance, our template will fail for the fan theorem (SeeSection 6), as the latter deals with (finite) binary trees, which are closed downwards.Of course, many of the zoo-principles have a distinct combinatorial flavour, whichimplies that the objects at hand exhibit little structure. Furthermore, as notedin Remark 4.11, this absence of structure directly translates into a (nonstandard)discontinuity in the input parameter X in ( ∀ X )( ∃ Y ) ϕ ( X, Y ). AMING THE REVERSE MATHEMATICS ZOO 19
Thirdly, in light of this absence of structure in principles of the RM zoo, weconjecture that robust theorems (in the sense of [34, p. 432]) are (exactly) thosewhich deal with mathematical objects with lots of structure like trees, continuousfunctions, metric spaces, et cetera. These theorems are also (exactly) those whichare continuous in their input parameter(s). In particular, the presence of thisstructure ‘almost guarantees’ a place in one of the Big Five categories. The non-robust theorems, by contrast, deal with objects which exhibit little structure (andhence can be discontinuous in their input parameters), and for this reason have thepotential to fall outside the Big Five and in the RM zoo. However, as we observedin the previous paragraph, the absence of structure in RM zoo principles, is exactlywhat makes our template from Section 5.1 work.In conclusion, what makes the principles in the RM zoo exceptional (namely thepresence of little structure on the objects at hand) guarantees that the uniformversions of the RM zoo principles are non-exceptional (due to the fact that theabove template works form them).6. Converse Mathematics
In this section, we classify the uniform versions of the contrapositions of zoo-principles. This study is motivated by the question whether the template fromSection 5.1 ‘always’ works, i.e. perhaps we can find counterexamples to this templateby studying contrapositions of zoo-principles, as these do not necessarily have a Π -structure? We first discuss this motivation in detail.First of all, the weak K¨onig’s lemma ( WKL ) is rejected in all varieties of con-structive mathematics, while the (classical logic) contraposition of
WKL , called the fan theorem is accepted in Brouwer’s intuitionistic mathematics (See e.g. [10, § WKL itself. (See [29, 42]). Hence, we observe that,from the constructive and uniform point of view, a principle can behave ratherdifferently compared to its contraposition.Secondly, the template from Section 5.1 would seem to work for any Π -zooprinciple T ≡ ( ∀ X )( ∃ Y ) ϕ ( X, Y ) and the associated ‘obvious’ uniform version
U T ≡ ( ∃ Φ → )( ∀ X ) ϕ ( X, Φ( X )). Nonetheless, while U T is the most natural uni-form version of T (in our opinion), there sometimes exists an alternative uniformversion of T , similar to the uniform version of the fan theorem. With regard to ex-amples, the principle ADS from Section 4.1 is perhaps the most obvious candidate,while various Ramsey theorems can also be recognised as suitable candidates.In conclusion, it seems worthwhile investigating the uniform versions of contra-posed zoo-principles, inspired by the difference in behaviour of the fan theoremand weak K¨onig’s lemma. However, somewhat surprisingly, we shall only obtainprinciples equivalent to arithmetical comprehension, i.e. our study will not yieldexceptions to our observation that the RM zoo disappears at the uniform level.6.1.
The contraposition of
ADS . In this section, we study the uniform versionof the contraposition of ADS . Recall that
ADS states that every infinite linear ordereither has an ascending or a descending chain. Hence, the contraposition of
ADS is the statement that if a linear order has no ascending and descending sequences,then it must be finite, as follows:( ∀ X ) (cid:2) LO( X ) ∧ ( ∀ x · ) ∈ Seq( X ))( ∃ n , k )( x n ≤ X x n +1 ∧ x k ≥ X x k +1 ) → ( ∃ l , k ∈ field( X ))( ∀ m ∈ field( X ))( k ≤ X m ≤ X l ) (cid:3) . (6.1)By removing all existential quantifiers, we obtain the following alternative uniformversion of ADS . Principle 6.1 ( UADS ) . There is Φ such that for all linear orders X and g ( ∀ x · ) ∈ Seq( X ))( ∃ n , k ≤ g ( x ( · ) ))( x n ≤ X x n +1 ∧ x k ≥ X x k +1 ) → ( ∀ m ∈ field( X ))(Φ( X, g ))(1) ≤ X m ≤ X Φ( X, g )(2)) . (6.2) Theorem 6.2. In RCA ω , we have the explicit equivalence UADS ↔ ( µ ) .Proof. The reverse direction is immediate since ( µ ) implies ADS and the upper andlower bounds to ≤ X in the consequent of (6.1) can be found using the same searchoperator. For the forward direction, we shall apply the template from Section 5.1.Hence, let UADS +2 be as in the template and fix standard X = ∅ and g suchthat the antecedent of UADS +2 holds. Then Φ( X , g ) is standard and considerthe standard function h which is constant and always outputs Φ( X , g )(1) +Φ( X , g )(2) + 4. Clearly, we have:( ∀ x · ) ∈ Seq( X ))( ∃ n , k ≤ h ( x ( · ) ))( x n ≤ X x n +1 ∧ x k ≥ X x k +1 ) , (6.3)as there are less than Φ( X , g )(1) + Φ( X , g )(2) + 2 distinct elements in the finitelinear order induced by X , by the consequent of UADS +2 . Indeed, by the definitionof linear order ([46, V.1.1]), if x ≤ X y ∧ x ≥ X y , then x = y , i.e. equality in thesense of X is equality on the natural numbers. By (6.3), the associated consequentof UADS +2 also follows for Φ( X , h ).Now suppose that Π - TRANS is false, i.e. there is standard function h such that( ∀ st n )( h ( n ) = 0) and ( ∃ n ) h ( n ) = 0. Following [46, V.1.1], define the standardset Y by adding to X the pairs ( x, m ) for x ∈ field( X ) and where m is suchthat ( ∀ i < m ) h ( i ) = 0 ∧ h ( m ) = 0. Intuitively speaking, the standard set Y represents the linear order X with a ‘point at infinity’ m added (in a standardway, thanks to h ). Since the order induced by Y is only a one-element extensionof the order induced by X , we also have( ∀ x · ) ∈ Seq( Y ))( ∃ n , k ≤ h ( x ( · ) ))( x n ≤ Y x n +1 ∧ x k ≥ Y x k +1 ) , i.e. the antecedent of UADS +2 holds for Y and h . Hence, the order induced by Y is bounded by Φ( Y , h ) as in the consequent of UADS +2 . However, by definition,we have X ≈ Y , implying Φ( X , h ) = ∗ Φ( Y , h ). By the latter, we cannothave m ≤ Y Φ( Y , h ) for the unique (and necessarily infinite) element m ∈ field( Y ) \ field( X ), i.e. a contradiction. Hence, we obtain UADS +2 → Π - TRANS and applying RS finishes the proof. (cid:3) In the previous proof, we added the ‘point at infinity’ m to the finite linearorder induced by X ; Such a modification is only possible for structures which are not closed downwards. In particular, the above approach clearly does not workfor theorems concerned with trees, like e.g. the fan theorem. On the other hand,we can easily obtain a version of the previous theorem for e.g. the chain-antichainprinciple CAC , and of course for stable versions of the latter and of
ADS . AMING THE REVERSE MATHEMATICS ZOO 21
The contraposition of Ramsey theorems.
In this section, we study thewell-known
Ramsey’s theorem for pairs RT . The latter is the statement that everycolouring with two colours of all two-element sets of natural numbers must have aninfinite homogenous subset, i.e. of the same colour. Now, RT has an equivalentversion (See [24, § Principle 6.3 (Contraposition of RT ) . ( ∀ X , c : [ X ] → h ( ∀ H ⊆ X )( ∀ i < (cid:2) ( ∀ s ∈ [ H ] )( c ( s ) = i ) (6.4) → H is finite (cid:3) → X is finite i . Here, ‘ Z is finite’ is short for ( ∃ n )( ∀ σ ∗ ) (cid:2) ( ∀ i < | σ | )( σ ( i ) ∈ Z ) → | σ | ≤ n (cid:3) . Wealso abbreviate the previous formula by ( ∃ n )( | Z | ≤ n ), where obviously | Z | ≤ n is a Π -formula. Note that we used the usual notation [ H ] n for the set of n -elementsubsets of H , which of course has nothing to do with the typing of variables.Based on the previous principle, define URTP as the following principle. Principle 6.4.
There is Φ such that for all g , X , c : [ X ] → , we have ( ∀ H ⊆ X )( ∀ i < (cid:2) ( ∀ s ∈ [ H ] ) c ( s ) = i → | H | ≤ g ( H ) (cid:3) → | X | ≤ Φ( X, g, c ) . (6.5)Note that g does not depend on i , as the quantifier ( ∀ i <
2) can be broughtinside the square brackets to obtain ( ∀ s ∈ [ H ] )( c ( s ) = 0) ∨ ( ∀ t ∈ [ H ] )( c ( t ) = 1). Theorem 6.5. In RCA ω , we have the explicit equivalence ( µ ) ↔ URTP .Proof. The forward direction is immediate as ( µ ) implies RT and the upper boundto | X | in the former’s contraposition can be found using this search operator. Forthe reverse direction, we work following the template from Section 5.1. Hence,consider URTP +2 and let g , X , c : [ X ] → | X | ≤ Φ( X , g , c ), where X = ∅ . Nowdefine h to be the functional which is constantly Φ( X , g , c ) + 1, and note that:( ∀ H ⊆ X )( ∀ i < (cid:2) ( ∀ s ∈ [ H ] )( c ( s ) = i ) → | H | ≤ h ( H ) (cid:3) , as H ⊆ X implies that | H | ≤ | X | . By URTP , we also have | X | ≤ Φ( X , h , c ).Now suppose Π - TRANS is false, i.e. there is some standard h such that ( ∀ st n ) h ( n ) =0 and ( ∃ m ) h ( m ), and define the standard set Y as X ∪ { m , m + 1 , . . . , m +Φ( X , h , c ) } , where m is the least number k such that h ( k ) = 0. Now define thestandard colouring d as follows: d ( s ) is 0 if both elements of s are at least m , 1if one element of s is at least m and the other one is not, and c ( s ) otherwise. Bythe definition of Y and d , we have( ∀ H ⊆ Y )( ∀ i < (cid:2) ( ∀ s ∈ [ H ] )( d ( s ) = i ) → | H | ≤ h ( H ) (cid:3) , (6.6)as for H ⊆ Y with more than Φ( X , g , c ) + 1 elements, the set H is not ho-mogenous for d . By URTP , we obtain | Y | ≤ Φ( Y , h , d ), but we also haveΦ( Y , h , d ) = Φ( X , h , c ) by standard extensionality since X ≈ Y and c ≈ d . However, Y by definition has more elements than Φ( X , h , c ), a contradiction.Hence, we have URTP +2 → Π - TRANS and applying RS finishes the proof. (cid:3) Contraposition of thin and free set theorems.
In this section, we againstudy the thin- and free set theorems from [11]. These results are similar to thosein the previous two sections, hence our treatment will be brief. Notations are as in[11], except that we write f : [ X ] k → N instead of f : [ X ] k → N .Recall the equivalent version of Ramsey’s theorem from [24, §
6] in Principle 6.3.Because of the extra set parameter X in the latter, (6.4) is amenable to ourtreatment as in Theorem 6.5. As it turns out, the free and this set theorems alsohave such equivalent versions by [11, Lemma 2.4 and Corollary 3.6].For instance, by the aforementiond lemma, FS ( k ), the free set theorem for index k , is equivalent to the statement that for every infinite set X and f : [ X ] k → N ,there is infinite A ⊂ X which is free for f . The contraposition of the latter is:( ∀ X , f : [ X ] k → N ) h ( ∀ A ⊆ X ) (cid:2) ( ∀ s ∈ [ A ] k )( f ( s ) A ∨ f ( s ) ∈ s ) (6.7) → H is finite (cid:3) → X is finite i . which is neigh identical to Principle 6.3 for k = 2. Now let UFSP k be the uniformversion of (6.7) similar to URTP . Similar to Theorem 6.5, one proves the following. Theorem 6.6. In RCA ω , we have the explicit equivalence ( µ ) ↔ UFSP . The version of the thin set theorem from [11, Corollary 3.6] is not so elegant,hence we do not consider it. We finish this section with some concluding remarks
Remark 6.7.
First of all, Kohlenbach claims in [29, §
1] that ( ∃ ) sports a rich andvery robust class of equivalent principles, which seems to be ‘more than’ confirmedby the above results, especially those in this section.Secondly, if one were to categorise principles according to robustness at theuniform level , ADS and other principles studied in this section would rank veryhigh, as even their contrapositions give rise to uniform principles equivalent to ( ∃ ).By contrast, WKL would rank lower, as the uniform version of the fan theorem, theclassical contraposition of
WKL , is not stronger than
WKL , as discussed in the firstpart of this section. In other words,
ADS is exceptional in Friedman-Simpson-styleRM, while it is not in the aforementioned ‘uniform’ categorisation.6.4.
Motivation for higher-order Reverse Mathematics.
The reader unac-customed to higher-order arithmetic may deem higher-order principles like
UDNR unnatural, compared to e.g. second-order arithmetic. We now argue that, at leastfrom the point of view of second-order RM, higher-order RM is also natural. Itshould also be mentioned that Montalb´an includes higher-order RM among the‘new avenues for RM’ in [34].First of all, Fujiwara and Kohlenbach have established the connection (and evenequivalence in some cases) between (classical) uniform existence as in
U T andintuitionistic provability ([20, 21]). Hence, the investigation of uniform principleslike
UDNR may be viewed as the (second-order) study of intuitionistic provability.Secondly, the author shows in in [43] that higher-order statements are implicitin (second-order) RM-theorems concerning continuity, due to the special nature ofthe RM-definition of continuity. In particular, consider the statementAll continuous functions on Canter space are uniformly continuous.
AMING THE REVERSE MATHEMATICS ZOO 23
Let (H) be the previous statement with continuity as in the RM -definition . Onecan then prove (H) ↔ (UH), where:There is a functional which witnesses the uniform RM-continuityon Cantor space of any RM-continuous function. (UH)From the treatment in [43], it is clear that the functional in (UH) can only beobtained because the RM-definition of continuity greatly reduces quantifier com-plexity. In conclusion, higher-order RM is already implicit in second-order RM dueto the RM-definition of continuity involving codes . Similar results are in [41, 42].Thirdly, RM can be viewed as a classification based on computability : Theoremsprovable in RCA are part of ‘computable mathematics’; An equivalence between atheorem and a Big Five system classifies the computational strength of the theorem,as the Big Five have natural formulations in terms of computability. Furthermore,as noted by Simpson in [46, I.8.9 and IV.2.8], theorems are analysed in RM ‘as theystand’, in contrast to constructive mathematics, where extra conditions are addedto enforce a constructive solution. In other words, the goal of RM is not to enforcecomputability onto theorems, but to classify how ‘non-computable’ the latter are.In light of the previous, it is a natural question whether there are other naturalways of classifying theorems of ordinary mathematics. As noted in [41, 42], thestudy of uniform versions of theorems constitutes a classification based on thecentral tenet of Feferman’s Explicit Mathematics (See [16–18]), which is:
A proof of existence of an object yields a procedure to compute said object .Indeed, in the same way as the RM-classification is based on the question whichaxioms (and hence ‘how much’ non-computability) are necessary to prove a theorem,the study of uniform versions of theorems is motivated by the following question:
For a given theorem T , what extra axioms are needed to compute the objectsclaimed to exist by T ? Similar to RM, we do not enforce the central tenet of Explicit Mathematics inhigher-order RM: We measure ‘how much extra’ is needed to obtain
U T , the uniformversion of T where a functional witnesses the existential quantifiers.7. Conclusion
In conclusion, by establishing the template and associated algorithm RS inSection 5.1, we have exhibited a hitherto unknown ‘computational aspect’ of Non-standard Analysis. In particular, we have shown that for a theorem T from theRM zoo, to obtain the explicit equivalence U T ↔ ( µ ) for the associated uniformversion U T , we can just apply RS to the proof of the nonstandard equivalence U T + ↔ Π - TRANS . This conclusion suggests the following observations.(1) The Reverse Mathematics of Nonstandard Analysis gives rise to explicit equivalences in classical Reverse mathematics without the need to actuallyconstruct the terms in the explicit equivalence .(2) Nonstandard Analysis carries plenty of computational content, in directcontrast to the claims made by e.g. Bishop (See [9, p. 513] and [8], whichis the review of [28]) and Connes (See [13, p. 6207] and [12, p. 26]) . The proof takes place in
RCA ω + QF-AC , , a conservative extension of RCA by [25, § (3) To extract more computational information from Nonstandard Analysis, weshould study which notions (like continuity, Riemann integration, compact-ness, et cetera) can be brought into the normal form from Corollary 2.3.As will be shown in [44], this turns out to be a very large class. Acknowledgement 7.1.
This research was supported by the following fundingbodies: FWO Flanders, the John Templeton Foundation, the Alexander von Hum-boldt Foundation, and the Japan Society for the Promotion of Science. The authorexpresses his gratitude towards these institutions. The author would like to thankUlrich Kohlenbach, Benno van den Berg, Steffen Lempp, Paulo Oliva, Paul Shafer,Mariya Soskova, Vasco Brattka, and Denis Hirschfeldt for their valuable advice.
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