aa r X i v : . [ m a t h . L O ] N ov A NOTE ON NON-CLASSICAL NONSTANDARD ARITHMETIC
SAM SANDERS
Abstract.
Recently, a number of formal systems for Nonstandard Analysisrestricted to the language of finite types, i.e. nonstandard arithmetic , have beenproposed. We single out one particular system by Dinis-Gaspar, which is cat-egorised by the authors as being part of intuitionistic nonstandard arithmetic .Their system is indeed in consistent with the Transfer axiom of NonstandardAnalysis, and the latter axiom is classical in nature as it implies (higher-order)comprehension. Inspired by this observation, the main aim of this paper is toprovide answers to the following questions:(Q1) In the spirit of
Reverse Mathematics , what is the minimal fragment of
Transfer that is inconsistent with the Dinis-Gaspar system?(Q2) What other axioms are inconsistent with the Dinis-Gaspar system?Our answer to the first question suggests that the aforementioned inconsis-tency actually derives from the axiom of extensionality relative to the standardworld, and that other (much stronger) consequences of
Transfer are actuallyharmless. Perhaps surprisingly, our answer to the second question shows thatthe Dinis-Gaspar system is inconsistent with a number of (non-classical) conti-nuity theorems which one would -in our opinion- categorise as intuitionistic inthe sense of Brouwer. Finally, we show that the Dinis-Gaspar system involvesa standard part map , suggesting this system also pushes the boundary of whatstill counts as ‘Nonstandard Analysis’ or ‘internal set theory’. Introduction
Aim and motivation.
In the last decade, a number of versions of Heytingand Peano arithmetic in all finite types have been introduced ([2, 5, 9, 10]) which arebased on (fragments of) Nelson’s internal set theory ([18]). Such systems allow forthe extraction of the (copious) computational content of Nonstandard Analysis, asdiscussed at length in [22]. In this paper, we study the system DG by Dinis-Gasparto be found in [5] and Section 2; DG has been described as follows:We present a bounded modified realisability and a bounded func-tional interpretation of intuitionistic nonstandard arithmetic withnonstandard principles. ([5, Abstract], emphasis added)Similar claims may be found in the body of [5]: DG is part of intuitionistic mathe-matics, as claimed by the authors. This claim is not without merit: DG is indeedinconsistent with the Transfer axiom of Nonstandard Analysis, and this axiomis essentially the nonstandard version of comprehension. By way of an example,
Transfer restricted to Π -formulas translates to the ‘Turing jump functional’ ∃ , School of Mathematics, University of Leeds, UK & Depart of Mathematics, TUDarmstadt, Germamy
E-mail address : [email protected] .2010 Mathematics Subject Classification.
Key words and phrases.
Nonstandard Analysis, higher-order arithmetic, intuitionism.
A NOTE ON NON-CLASSICAL NONSTANDARD ARITHMETIC as defined in Section 3.1.2.
However , ‘non-classical’ does not necessarily imply‘intuitionistic’, and we shall study the following two questions in this paper.(Q1) In the spirit of
Reverse Mathematics , what is the minimal fragment of
Transfer that is inconsistent with DG ?(Q2) What other (intuitionistic) axioms are inconsistent with DG ?The main results of this paper constitute answers to the questions (Q1) and (Q2),and we now discuss them in some more detail.Regarding (Q1), we refer to [24–26] for an introduction and overview of ReverseMathematics (RM for short). We shall consider the parameter-free Transfer princi-ple studied in [3], and related axioms. We will identify the axiom of extensionality(relative to the standard world) as the real culprit: this axiom follows from Transfer and is inconsistent with DG , while other axioms implied by Transfer , even involvingthe Turing jump functional, are consistent with DG .Regarding (Q2), we show that DG is inconsistent with a number of (non-classical)axioms which one would categorise as intuitionistic , i.e. part of Brouwer’s intuition-istic mathematics . The most blatant example is the statement that, relative tothe standard world, all functionals on the Cantor space are (uniformly) continuous.In the course of investigating (Q1) and (Q2), one eventually stumbles uponthe fact that the Dinis-Gaspar system allows one to define a (highly elementary) standard part map , as discussed in Section 3.3. Since such a map is not available inNelson’s internal set theory, and external in Robinson’s approach, the Dinis-Gasparsystem thus pushes the boundary of what still counts as ‘Nonstandard Analysis’.As to the structure of this paper, we briefly discuss the importance of continuityin intuitionism in Remark 1.1. The formal system DG from [5] and associatedprerequisites are sketched in Section 2. Our main results may be found in Section 3,which provide fairly definitive answers to questions (Q1) and (Q2). We formulatethe conclusion to this paper in Section 4.Next, we point out the (intimate) relationship between Brouwer ’s intuitionisticmathematics and continuity, lest the reader believe the above is merely pedantry. Remark 1.1 (Intuitionism and continuity) . L.E.J. Brouwer is the founder of in-tuitionism , a philosophy of mathematics which later developed into the first full-fledged school of constructive mathematics . The latter is an umbrella term forapproaches to mathematics in which ‘there exists x ’ is systematically interpretedas ‘we can compute/construct x ’ (and similarly for the other logical symbols).Under this new interpretation of the logical symbols, certain laws do not makeany sense, and are therefore rejected; the most (in)famous one being the law ofexcluded middle P ∨ ¬ P . The resulting logic is intuitionistic logic , and we refer to[1, 28] for an introduction to the various approaches to constructive mathematics.Brouwer proved in 1927 (see [11, p. 444] for an English translation) that everytotal (in the intuitionistic sense) function on the unit interval is (uniformly) con-tinuous, a result which seems to contradict classical mathematics. The core axiomsfor intuitionistic mathematics indeed include a ‘continuity’ axiom (called WC-N in[28] and BP in [1]) which contradicts classical mathematics, and can be used toprove the aforementioned (uniform) continuity theorem by Brouwer. Since ‘intuitionism’ is the first keyword of [5], we take ‘intuitionistic’ to mean ‘part ofBrouwer’s intuitionistic mathematics’. We discuss this choice in more detail in Remark 2.8.
NOTE ON NON-CLASSICAL NONSTANDARD ARITHMETIC 3
The previous is well-known, but is mentioned since we want to stress the follow-ing: a very low bar a logical system has to clear to deserve the moniker ‘intuition-istic’, is to be consistent with the aforementioned continuity theorem and axiom.As it turns out, this does not seem to be the case for the Dinis-Gaspar system.Finally, despite (or perhaps better: ‘because’) the above criticism, we do be-lieve that DG has some has some interesting features. For instance, DG is a sortof non-classical analogue of the Fernand-Oliva bounded functional interpretation(BFI hereafter; see [7]). Now, the BFI refutes extensionality, so results such asTheorem 3.8 and its corollaries are, in this light, rather natural. Regarding BFIand related matters, [8, Section 3] is also highly informative.2. Preliminaries
We introduce the Dinis-Gaspar system DG , and some preliminaries.2.1. Internal set theory and its fragments.
In this section, we discuss Nelson’s internal set theory , first introduced in [18], and the Dinis-Gaspar system DG from[5]. The system DG is an extension of a fragment of Nelson’s system with (non-classical) axioms pertaining to majorizability .In Nelson’s syntactic approach to Nonstandard Analysis ([18]), as opposed toRobinson’s semantic one ([21]), a new predicate ‘st( x )’, read as ‘ x is standard’ isadded to the language of ZFC , the usual foundation of mathematics. The notations( ∀ st x ) and ( ∃ st y ) are short for ( ∀ x )(st( x ) → . . . ) and ( ∃ y )(st( y ) ∧ . . . ). A formula iscalled internal if it does not involve ‘st’, and external otherwise. The three externalaxioms schemes Idealisation , Standard Part , and
Transfer govern the new predicate‘st’; they are respectively defined as: (I) ( ∀ st fin x )( ∃ y )( ∀ z ∈ x ) ϕ ( z, y ) → ( ∃ y )( ∀ st x ) ϕ ( x, y ), for internal ϕ . (S) ( ∀ x )( ∃ st y )( ∀ st z ) (cid:0) [ z ∈ x ∧ ϕ ( z )] ↔ z ∈ y (cid:1) , for any ϕ . (T) ( ∀ st x ) ϕ ( x, t ) → ( ∀ x ) ϕ ( x, t ), where ϕ is internal, t captures all parametersof ϕ , and t is standard.The system IST is (the internal system)
ZFC extended with the aforementionedthree external axioms; the former is a conservative extension of
ZFC for the internallanguage, as proved in [18].In [2, 5, 9, 10], the authors study G¨odel’s system T extended with versions 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 arithmetic inall finite types and the axiom of extensionality . We refer to [2, § finiteness central to the latter is replaced by the notion of strong majorizability . Thelatter notion and the associated system DG is introduced in the next paragraph,assuming familiarity with the higher-type framework of G¨odel’s T .The system DG , a conservative extension of E-HA ω , is based on the Howard-Bezem notion of strong majorizability . We first introduce the latter and relatednotions. For a more extensive background on strong majorizability, see [17, § The superscript ‘fin’ in (I) means that x is finite, i.e. its number of elements are bounded bya natural number. A NOTE ON NON-CLASSICAL NONSTANDARD ARITHMETIC
Definition 2.1 (Majorizability) . The strong majorizability predicate ‘ ≤ ∗ ’ is in-ductively defined as follows: • x ≤ ∗ y is x ≤ y ; • x ≤ ∗ ρ → σ y is ( ∀ u )( ∀ v ≤ ∗ ρ u ) (cid:0) xu ≤ ∗ σ yv ∧ yu ≤ ∗ σ yv (cid:1) .An object x ρ is called monotone if x ≤ ∗ ρ x . The quantifiers (˜ ∀ x ρ ) and (˜ ∃ y ρ ) rangeover the monotone objects of type ρ , i.e. they are abbreviations for the formulas( ∀ x )( x ≤ ∗ x → . . . ) and ( ∃ y )( y ≤ ∗ y ∧ . . . ).The system DG is defined as follows in [5, § E-HA ω st is thelanguage of E-HA ω extended with a new ‘standardness’ predicate st σ for every finitetype σ . The typing of the standardness predicate is usually omitted. Definition 2.2 (Standard quantifiers) . We write ( ∀ st x τ )Φ( x τ ) and ( ∃ st x σ )Ψ( x σ )as short for ( ∀ x τ ) (cid:2) st( x τ ) → Φ( x τ ) (cid:3) and ( ∃ st x σ ) (cid:2) st( x σ ) ∧ Ψ( x σ ) (cid:3) . A formula A is‘internal’ if it does not involve st, and external otherwise. The formula A st is definedfrom A by appending ‘st’ to all quantifiers (except bounded number quantifiers).Regarding the previous definition, we often say that ‘the formula A st is theformula A relative to the standard world’. Definition 2.3. [Basic axioms] The system
E-HA ω st is defined as E-HA ω + T ∗ st + IA st ,where T ∗ st consists of the following axiom schemas.(a) x = σ y → (st σ ( x ) → st σ ( y ));(b) st σ ( y ) → ( x ≤ ∗ σ y → st σ ( x ));(c) st σ ( t ), for each closed term t of type σ ;(d) st σ → τ ( z ) → (st σ ( x ) → st τ ( zx )).Items (a)-(d) are called the standardness axioms , and (b) is singled out regularlybelow. The external induction axiom IA st is the following schema for any Φ:Φ(0) ∧ ( ∀ st n )(Φ( n ) → Φ( n + 1)) → ( ∀ st n )Φ( n ) . ( IA st )The system DG is then defined as E-HA ω st plus the following non-basic axioms. Definition 2.4. [Non-basic axioms] • Monotone Choice mAC ω : For any Φ, we have(˜ ∀ st x )(˜ ∃ st y )Φ( x, y ) → (˜ ∃ st f )(˜ ∀ st x )( ∃ y ≤ ∗ f ( x ))Φ( x, y ) . • Realization R ω : For any Φ, we have( ∀ x )( ∃ st y )Φ( x, y ) → (˜ ∃ st z )( ∀ x )( ∃ y ≤ ∗ z )Φ( x, y ) . • Idealisation I ω : For any internal φ , we have:(˜ ∀ st z )( ∃ x )( ∀ y ≤ ∗ z ) φ ( x, y ) → ( ∃ x )( ∀ st y ) φ ( x, y ) • Independence of premises IP ω ˜ ∀ st : For any internal φ and any Ψ:[(˜ ∀ st x ) φ ( x ) → (˜ ∃ st y )Ψ( y )] → (˜ ∃ st z )[(˜ ∀ st x ) φ ( x ) → (˜ ∃ y ≤ ∗ z )Ψ( y )] • Nonstandard Markov’s principle M ω : For any internal φ, ψ , we have[(˜ ∀ st x ) φ ( x ) → ψ ] → (˜ ∃ st y )[( ∀ x ≤ ∗ y ) φ ( x ) → ψ ] • Majorizability axiom
MAJ ω : ( ∀ st x )( ∃ st y )( x ≤ ∗ y ) NOTE ON NON-CLASSICAL NONSTANDARD ARITHMETIC 5
Other axioms are mentioned in [5, § DG . Thevariables are not specified for I ω in [5], and we have chosen the version from IST .We have also added ‘nonstandard’ to the description of the axiom M ω to distinguishit from the semi-constructive axiom MP , known as ‘Markov’s principle’, as follows:( ∀ f ) (cid:2) ¬¬ (cid:2) ( ∃ n ) f ( n ) = 0 (cid:3) → ( ∃ n ) f ( n ) = 0 (cid:3) . ( MP )Now, M ω implies MP with all quantifiers relative to ‘st’, which explains the name.2.2. Notations in DG . In this section, we introduce notations relating to DG .First of all, we will use the usual notations for rational and real numbers andfunctions as introduced in [16, p. 288-289] (and [25, I.8.1] for the former). Definition 2.5 (Real numbers and related notions in
RCA ω ) . • Natural numbers correspond to type zero objects, and we use ‘ n ’ and‘ n ∈ N ’ interchangeably. Rational numbers are defined as signed quotientsof natural numbers, and ‘ q ∈ Q ’ and ‘ < Q ’ have their usual meaning. • Real numbers are coded by fast-converging Cauchy sequences q ( · ) : N → Q ,i.e. such that ( ∀ n , i )( | q n − q n + i ) | < Q n ). We use Kohlenbach’s ‘hatfunction’ from [16, p. 289] to guarantee that every f defines a real number. • We write ‘ x ∈ R ’ to express that x := ( q · ) ) represents a real as in theprevious item and write [ x ]( k ) := q k for the k -th approximation of x . • Two reals x, y represented by q ( · ) and r ( · ) are equal , denoted x = R y , if( ∀ n )( | q n − r n | ≤ n − ). The inequality ‘ < R ’ is defined similarly. • Functions F : R → R mapping reals to reals are represented by Φ → mapping equal reals to equal reals, i.e. ( ∀ x, y )( x = R y → Φ( x ) = R Φ( y )). • Sets of type ρ objects X ρ → , Y ρ → , . . . are given by their characteristicfunctions f ρ → X , i.e. ( ∀ x ρ )[ x ∈ X ↔ f X ( x ) = f ρ → X ≤ ρ → Remark 2.6 (Equality) . The system DG includes equality between natural num-bers ‘= ’ as a primitive. Equality ‘= τ ’ for type τ -objects x, y is then defined as:[ x = τ y ] ≡ ( ∀ z τ . . . z τ k k )[ xz . . . z k = yz . . . z k ] (2.1)if the type τ is composed as τ ≡ ( τ → . . . → τ k → ≤ τ ’ is just(2.1) with ‘ ≤ ’, i.e. binary sequences are given by f ≤
1, which we also denote as‘ f ∈ C ’ or ‘ f ∈ N ’. We define ‘approximate equality ≈ τ ’ as follows:[ x ≈ τ y ] ≡ ( ∀ st z τ . . . z τ k k )[ xz . . . z k = yz . . . z k ] (2.2)with the type τ as above. The system DG includes the axiom of extensionality :( ∀ ϕ ρ → τ )( ∀ x ρ , y ρ ) (cid:2) x = ρ y → ϕ ( x ) = τ ϕ ( y ) (cid:3) . ( E ρ → τ )for all finite types. We write ( E ) for the collection of all axioms ( E ρ → τ ).Finally, we introduce some notation to handle finite sequences nicely. Notation 2.7 (Finite sequences) . We assume the usual coding of finite sequencesof objects of the same type. We denote by ‘ | s | = n ’ the length of the finite sequence s = h s ρ , s ρ , . . . , s ρn − i , where |hi| = 0, i.e. the empty sequence has length zero. Forsequences s, t of the same type, we denote by ‘ s ∗ t ’ the concatenation of s and t ,i.e. ( s ∗ t )( i ) = s ( i ) for i < | s | and ( s ∗ t )( j ) = t ( | s | − j ) for | s | ≤ j < | s | + | t | . Fora finite sequence s , we define sN := h s (0) , s (1) , . . . , s ( N − i for N < | s | . For asequence α → ρ , we also write αN = h α (0) , α (1) , . . . , α ( N − i for any N . A NOTE ON NON-CLASSICAL NONSTANDARD ARITHMETIC
Finally, we discuss our use of the term ‘intuitionistic’.
Remark 2.8.
As already pointed out in Footnote 1, we interpret ‘intuitionistic’ tomean ‘part of Brouwer’s intuitionistic mathematics’, due to the first keyword of [5]being ‘intuitionism’. This term also has a more loose interpretation, not uncommonin functional interpretations, meaning that the system is based on intuitionisticlogic, i.e. does not include all of classical logic. These kind of systems are sometimesmore correctly called semi-intuitionistic . Due to the topic of [5], the authors seemto have had the second meaning in mind. Nonetheless, our results below show thatboth senses of the term ‘intuitionistic’ are not completely compatible with DG , i.e.there is good reason to claim that DG is ‘merely’ non-classical.I thank the referee for pointing out the content of the previous remark.3. Main results
Our main results fall into three main categories, as follows.(i) In answer to (Q1), we show in Section 3.1 that DG is inconsistent withcertain (very) weak fragments of Transfer , but (oddly) not with others.(ii) In answer to (Q2), we show in Section 3.2 that DG is inconsistent withcertain intuitionistic axioms, relative to the standard world; we also showthat DG does prove weak K¨onig’s lemma , relative to the standard world.(iii) Inspired by these answers to (Q1) and (Q2), we show in Section 3.3 that DG involves a highly elementary standard part map .In light of the first two items, it seems that DG is not really a system of intuitionistic arithmetic (but non-classical nonetheless), while the third item shows that DG already pushes the boundary of what still counts as ‘Nonstandard Analysis’.3.1. Non-classical aspects of the Dinis-Gaspar system.
We provide a partialanswer to question (Q1) from Section 1.1 by showing that DG is inconsistent withvarious weak fragments of Transfer , including parameter-free Transfer from [3], andthe Turing jump functional ∃ from e.g. [16], relative to the standard world.3.1.1. Parameter-free Transfer.
We show that various extensions of DG , also involv-ing intuitionistic axioms, are inconsistent with parameter-free Transfer as follows. Principle 3.1 ( PF-TP ∃ ) . For internal ϕ ( x ) with all free variables shown, we have ( ∃ x ) ϕ ( x ) → ( ∃ st x ) ϕ ( x ) . (3.1)To be absolutely clear, (standard) parameters are not allowed in ϕ ( x ) as in (3.1).In contrast to richer fragments of Transfer , PF-TP ∃ is weak: when added to(fragments of) the classical system from [2], one obtains a conservative extension,by [3, § PF-TP ∃ yields a smooth developmentof the (classical) Reverse Mathematics of Nonstandard Analysis.We point out that certain fragments of the axiom of choice (including
QF-AC , as in the next theorem) are widely accepted in constructive and intuitionistic math-ematics (see e.g. [1]). We also recall Markov’s principle MP introduced after Defi-nition 2.4 and note that MP is rejected in intuitionistic mathematics ([28, p. 237]). Theorem 3.2.
The system DG + PF-TP ∃ + QF-AC , + MP is inconsistent. NOTE ON NON-CLASSICAL NONSTANDARD ARITHMETIC 7
Proof.
Recall that DG includes the axiom of extensionality ( E ), which implies( ∀ Y , f , g )( ∃ N )[ f N = gN → Y ( f ) = Y ( g )] , by Markov’s principle MP . Applying QF-AC , , we obtain Φ → such that( ∀ Y , f , g )( ∃ N ≤ Φ ( Y, f, g ))[ f N = gN → Y ( f ) = Y ( g )] . Applying
PF-TP ∃ , there is standard such Φ , yielding that( ∀ st Y , f , g )[ f ≈ g → Y ( f ) = Y ( g )] , (3.2)since Φ ( Y, f, g ) is standard for standard inputs. Note that (3.2) is ( E ) st , i.e. theaxiom of ‘standard extensionality’. Now consider the functional Y defined as: Y ( f ) := ( ∃ n ≤ N + 1)( f ( n ) = 0)1 otherwise , (3.3)where N is nonstandard. Since Y ≤ ∗
1, and the constant-one-mapping of typetwo is standard, Y is also standard by item (b) in the standardness axioms . Hence, Y satisfies (3.2) and now consider f := 11 . . . and g := f N ∗ . . . , whichsatisfy f ≈ g and Y ( f ) = 0 = 1 = Y ( g ). Note that g is standard by theaforementioned item (b), as g ≤ ∗ (cid:3) Next, we show that the previous proof also goes through using a fragment ofMarkov’s principle MP , called weak Markov’s principle ( WMP for short; see [12]).Most importantly for us,
WMP is accepted in intuitionistic mathematics (but notin Bishop’s constructive mathematics by the results in [14]). We will actually usethe following version of WMP , defined in [13]:( ∀ Y , f , g )( Y ( f ) = Y ( g ) → f = g ) . ( SE )Since ‘ x = y ’ is generally a stronger statement than ‘ ¬ ( x = y )’ in constructivemathematics, SE is said to express strong extensionality . Corollary 3.3.
The system DG + PF-TP ∃ + QF-AC , + SE is inconsistent.Proof. Note that SE implies the following by considering the least such N :( ∀ Y , f , g )( ∃ N )( Y ( f ) = Y ( g ) → f N = gN )As for the theorem, one derives ( E ) st and Y yields a contradiction. (cid:3) The inconsistency in the theorem also pops up when combining
PF-TP ∃ with intuitionistic axioms, like the intuitionistic fan functional ([16, 27]) as follows.( ∃ Ω )( ∀ Y )( ∀ f, g ≤ f Ω( Y ) = g Ω( Y ) → Y ( f ) = Y ( g )) ( MUC ) Corollary 3.4.
The system DG + PF-TP ∃ + MUC is inconsistent.Proof.
Since
MUC is a sentence,
PF-TP ∃ guarantees the existence of a standard Ω as in the former. Now consider Y , f , g from the proof of the theorem and notethat Ω( Y ) is a standard number. Hence, since f ≈ g by definition, we have0 = Y ( f ) = Y ( g ) = 1, a contradiction. (cid:3) Note that SE is actually weaker than WMP , but SE ↔ WMP by [13, Thm. 11] in Bishop’sconstructive mathematics plus a non-trivial fragment of the axiom of choice.
A NOTE ON NON-CLASSICAL NONSTANDARD ARITHMETIC
We note that the inconsistency of DG with much stronger fragments of Transfer is proved in [5, Theorem 29]. In particular,
Transfer for Π -formulas is used in thelatter, which readily translates to the Turing jump functional ∃ in the systemsfrom [2]. As it happens, we study ∃ in the next section.We do not know whether the classical contraposition of (3.1) also leads to in-consistency, but we now show that it implies the fan theorem, as follows.( ∀ T ≤ (cid:2) ( ∀ α ≤ ∃ m )( αm T ) → ( ∃ n )( ∀ β ≤ βn T ) (cid:3) ( FAN )The variable ‘ T ’ is reserved for trees, while ‘ T ≤
1’ means that T is a binary tree. Principle 3.5 ( PF-TP ∀ ) . For internal ϕ ( x ) with all free variables shown, we have ( ∀ st x ) ϕ ( x ) → ( ∀ x ) ϕ ( x ) . (3.4)To be absolutely clear, (standard) parameters are not allowed in ϕ ( x ) as in (3.4). Theorem 3.6.
The system DG + QF-AC , + PF-TP ∀ proves FAN .Proof.
We first prove
FAN st . If ( ∀ st f ≤ ∃ st n )( αn T ), then we have ( ∀ f ≤ ∃ st n )( αn T ) since all binary sequences are standard by item (b) of the non-standard axioms. Applying R ω , we obtain ( ∃ st k )( ∀ f ≤ ∃ n ≤ k )( αn T ),and FAN st follows. The latter immediately implies that( ∀ st T ≤ , G ) (cid:2) ( ∀ st α ≤ ∃ m ≤ G ( α ))( αm T ) → ( ∃ st n )( ∀ β ≤ βn T ) (cid:3) . Now drop the ‘st’ predicates inside the square brackets and apply
PF-TP ∀ . Theresulting formula then yields FAN , thanks to
QF-AC , . (cid:3) Note that
QF-AC , is ‘innocent’ in that it is included in the base theory ofhigher-order Reverse Mathematics (see [16]). Next, we show that PF-TP ∀ leads tothe ‘full’ Heine-Borel compactness of the Cantor space (for uncountable covers), asin: ( ∀ G )( ∃h β , . . . , β k i )( ∀ α ≤ ∃ i ≤ k )( α ∈ [ β i G ( β i ))]) . ( HBU C )Intuitively, any functional G gives rise to the ‘canonical’ cover ∪ f ∈ C [ f G ( f )] of theCantor space, and HBU C tells us that the latter always has a finite sub-cover. Theorem 3.7.
The system DG + PF-TP ∀ proves HBU C .Proof. Since every binary sequence is standard in DG , we have ( ∀ α ≤ ∃ st β ≤ α ≈ β ), i.e. the nonstandard compactness of the Cantor space. However, theusual proof that the latter is equivalent to HBU C (see [3,23]) does not go through in DG due to the weak conclusion of R ω . Instead, we prove (3.5), which immediatelyyields HBU C via PF-TP ∀ .( ∀ st G )( ∃h β , . . . , β k i )( ∀ α ≤ ∃ i ≤ k )( α ∈ [ β i G ( β i ))]) (3.5)To prove (3.5), fix nonstandard N and define β i := σ ∗
00 where σ is the i -th binarysequence of length N . Then h β , . . . , β N i is as required for (3.5), as every β i isstandard by item (b) in the nonstandard axioms, and hence G ( β i ) is standard.Indeed, ( ∀ α ≤ ∃ i ≤ N )( β i ≈ α ), we are done. (cid:3) As to concluding remarks, Benno van den Berg has suggested ‘ ϕ ( x ) ≡ ( ∃ y )( x ≤ ∗ y )’ to show that PF-TP ∀ does not lead to a conservative extension of DG .Secondly, by [19, Cor. 6.7] and [20, Thm. 3.3], the Turing jump functional ∃ from Section 3.1.2 and HBU C give rise to ATR , and it is a natural question whetherthe same holds for the system DG + PF-TP ∀ + QF-AC , + ( ∃ ). NOTE ON NON-CLASSICAL NONSTANDARD ARITHMETIC 9
Thirdly, the axiom
HBU C is extremely hard to prove: by [20, § k -CA ω does not prove HBU C (for all k ), where the former is RCA ω plus the existence of S k , a functional which decides the truth of Σ k -formulas (only involving type oneparameters). Hence, DG + PF-TP ∀ is a rather peculiar system.Fourth, since DG proves that all binary sequences are standard, the followingfragment of Transfer , introduced in [19], follows trivially:( ∀ st Y ) (cid:2) ( ∃ f ≤ Y ( f ) = 0) → ( ∃ st f ≤ Y ( f ) = 0) (cid:3) ( WT )Note that WT is quite strong: working in the systems from [2], WT gives riseto a functional κ which computes a realiser for HBU C , but not vice versa (see[19, Theorem 6.17]). In fact, the combination of κ and ∃ (from the next section),gives rise to full second-order arithmetic by [19, Rem. 6.13].3.1.2. The Turing jump functional.
We show that the Dinis-Gaspar system is in-consistent with ( ∃ ) relative to the standard world . The axiom ( ∃ ) is given by:( ∃ ϕ )( ∀ f )[( ∃ n )( f ( n ) = 0) ↔ ϕ ( f ) = 0] . ( ∃ )Note that Π - TRANS → ( ∃ ) st by the proof of Corollary 3.9, where Π - TRANS is:( ∀ st f ) (cid:2) ( ∀ st n )( f ( n ) = 0) → ( ∀ n )( f ( n ) = 0) (cid:3) , (3.6)i.e. ( ∃ ) st follows from a fragment of Transfer using M ω . Theorem 3.8.
The system DG + ( ∃ ) st is inconsistent.Proof. Define the (standard) functional Z → as Z ( f )( n ) = 0 if f ( n ) = 0, and 1otherwise. Since Z ( f ) ≤ ∗
1, the binary sequence Z ( f ) is standard for any input f ,due to item (b) of the standardness axioms . Hence, ( ∃ ) st immediately yields:( ∃ st ϕ )( ∀ f )[( ∃ st n )( f ( n ) = 0) ↔ ϕ ( f ) = 0] , (3.7)by taking ϕ := ϕ ◦ Z for ϕ as in ( ∃ ) st . Clearly, (3.7) implies( ∀ f )[ ϕ ( f ) = 0 → ( ∃ st n )( f ( n ) = 0)] , and applying IP ω ˜ ∀ st yields( ∀ f )( ∃ st m )[ ϕ ( f ) = 0 → ( ∃ n ≤ m )( f ( n ) = 0)] , while applying R ω yields:( ∃ st k )( ∀ f )( ∃ m ≤ k )[ ϕ ( f ) = 0 → ( ∃ n ≤ m )( f ( n ) = 0)] . (3.8)Now let k be a standard number as in (3.8) and define f as f ( i ) = 1 for i ≤ k +1,and 0 otherwise. Clearly, ϕ ( f ) = 0 by (3.7), but this contradicts (3.8). (cid:3) The following corollary also follows from the proof of [5, Theorem 29].
Corollary 3.9.
The system DG + Π - TRANS is inconsistent.Proof.
Note that Y as in (3.3) is standard, while Π - TRANS guarantees that Y behaves just like ϕ in ( ∃ ) st . Indeed, M ω implies MP st , i.e. Markov’s principlerelative to the standard world, and Π - TRANS thus implies( ∀ st f ) (cid:2) ( ∃ n )( f ( n ) = 0) → ( ∃ st n )( f ( n ) = 0) (cid:3) , which is trivially equivalent to (3.6) in classical logic. (cid:3) Let TJ ( f, ϕ ) be the formula in square brackets in ( ∃ ). By [3, Theorem 4.4], wehave Π - TRANS ↔ [( ∃ st ϕ )( ∀ st f ) TJ ( f, ϕ ) + ( E ) st ] over a weak classical system.Surprisingly, only the final conjunct gives rise to inconsistency, which is implicit inthe proof of Theorems 3.2 and 3.20. Corollary 3.10. DG + ( ∃ st ϕ )( ∀ st f ) TJ ( f, ϕ ) is consistent if E-HA ω + ( ∃ ) is.Proof. Using the same trick involving Z as in the theorem, ( ∃ st ϕ )( ∀ st f ) TJ ( f, ϕ )is equivalent to ( ∃ st ϕ )( ∀ f ) TJ ( f, ϕ ). The latter follows from ( ∃ ) by taking such ϕ and defining ϕ ( f ) = 1 if ϕ ( f ) = 0, and 0 otherwise. Since ϕ ≤ ∗
1, this functionalis standard by item (b) of the standardness axioms. As ( ∃ ) is internal, the theoremnow follows from the soundness theorem as in [5, Theorem 16], since E-HA ω st and E-HA ω prove the same internal formulas. (cid:3) Corollary 3.11.
The system DG + ( E ) st is inconsistent. It should be noted that DG is even inconsistent with the rule version of theaxiom ( E ) st . Indeed, DG proves that Y , f , g from the proof of Theorem 3.2 arestandard and satisfy f ≈ g . However, a proof of Y ( f ) = Y ( g ), say obtainedby the aforementioned rule, then leads to a contradiction. Corollary 3.12.
The system DG + QF-AC , + ( ∃ ) + PF-TP ∃ is inconsistent.Proof. Using
QF-AC , , ( ∃ ) readily implies( ∃ ϕ , Ψ )( ∀ f ) (cid:2) (( ∃ n )( f ( n ) = 0) → ϕ ( f ) = 0) ∧ ( ϕ ( f ) = 0 → f (Ψ( f )) = 0) (cid:3) , where Ψ( f ) is the least such n if existent. By PF-TP ∃ , there is standard such Ψ ,upon which we obtain Π - TRANS , a contradiction by Corollary 3.9. (cid:3)
In conclusion, while DG is inconsistent with a number of fragments of Transfer ,the inconsistency is really due to the axiom of extensionality relative to the standardworld, and not e.g. the Turing jump functional as in Corollary 3.10. Since the axiomof extensionality is not rejected in constructive (esp. intuitionistic) mathematics,all we can say is that these results suggest that DG is non-classical .Furthermore, DG proves ¬ ( ∃ ) st by Theorem 3.8, and classically ¬ ( ∃ ) is equiv-alent to the continuity of all functionals on the Baire space ([16, Prop. 3.7]); asimilar equivalence involving SE holds constructively by [13, Thm. 26]. However,these equivalences use Grilliot’s trick (see [16, p. 287]) and hence require the axiomof extensionality (in some form or other). Thus, to derive (intuitionistic) continu-ity theorems from ¬ ( ∃ ) st , one would need standard extensionality , which leads toinconsistency by Corollary 3.11. Thus, DG is definitely non-classical , but not reallyintuitionistic.3.1.3. Arithmetical comprehension.
We show that the Dinis-Gaspar system is in-consistent with
ACA relative to the standard world . The axiom ACA is:( ∀ f ≤ ∃ g ≤ ∀ n )[( ∃ m )( f ( n, m ) = 0) ↔ g ( n ) = 0] . ( ACA )Our formulation of arithmetical comprehension as in ACA makes use of functions,while the version used in RM (see [25, II]) makes use of sets. These versionsare equivalent in light of [25, II.3]. We single out ACA lest anyone believe theinconsistency in Theorem 3.8 is due to the presence of third-order objects. Theorem 3.13.
The system DG + ACA st0 is inconsistent.
NOTE ON NON-CLASSICAL NONSTANDARD ARITHMETIC 11
Proof.
Since all binary sequences are standard in DG , ACA st0 implies that for all f ≤
1, there is standard g ≤ ∀ st n )[( ∃ st m )( f ( n, m ) = 0) → g ( n ) = 0] ∧ ( ∀ st k )[ g ( k ) = 0 → ( ∃ st l )( f ( k, l ) = 0)] . The second conjunct yields ( ∃ st h )( ∀ st k )[ g ( k ) = 0 → ( ∃ l ≤ h ( k ))( f ( k, l ) = 0) dueto IP ω ˜ ∀ st and mAC ω . Thus, ( ∀ f ≤ ∃ st h )( ∃ st g ≤ A ( f, g, h ), where A ( f, g, h ) is( ∀ st n )[( ∃ st m )( f ( n, m ) = 0) → g ( n ) = 0] ∧ ( ∀ st k )[ g ( k ) = 0 → ( ∃ l ≤ h ( k ))( f ( k, l ) = 0)] . Since realisation R ω also applies to external formulas, we obtain( ∃ st h )( ∀ f ≤ ∃ h ≤ ∗ h )( ∃ st g ≤ A ( f, g, h ) (3.9)Now define f ( n, m ) as 0 if m > h ( n ), and 1 otherwise, where h is as in (3.9).For this f , (3.9) provides g , which satisfies by definition:( ∀ st n ) (cid:2) ( ∃ st m )( f ( n, m ) = 0) → ( g ( n ) = 0) → ( ∃ m ≤ h ( n ))( f ( n, m ) = 0) (cid:3) , which contradicts the definition of f , and we are done. (cid:3) It is tempting, but incorrect , to apply the reasoning from the previous proof to( ∀ f ≤ ∃ st n )( f ( n ) = 0) → ( ∃ st m )( f ( m ) = 0)] . (3.10)Indeed, IP ω ˜ ∀ st does not allow pulling the underlined quantifier in (3.10) to the front.Finally, while DG proves the non-classical ¬ ( ACA st0 ), we show in Section 3.2.1that it does prove the classical
WKL st , i.e. the latter does not lead to inconsistency.3.1.4. Non-classical continuity.
We show that relative to the standard world, ex-tensional functions on the Cantor space are automatically continuous on C . Wealso show that they are nonstandard continuous as follows:( ∀ st f ∈ C )( ∀ g ∈ C )( f ≈ g → Y ( f ) = Y ( g )) . (3.11)Using M ω and R ω , one readily shows that (3.11) implies ‘epsilon-delta’ continuityrelative to the standard world, and the latter implies (3.11) using item (b) of thenonstandard axioms of DG . Note that uniform nonstandard continuity is (3.11)with the leading ‘st’ dropped. Theorem 3.14.
The system DG proves that any Y satisfying ( E ) st is also non-standard ( uniformly ) continuous on the Cantor space.Proof. Suppose Y satisfies ( E ) st , which immediately yields:( ∀ f, g ∈ C )( ∃ st N )( f N = gN → Y ( f ) = Y ( g )) . (3.12)using M ω and the fact that all binary sequences are standard in DG . Applying R ω to (3.12) yields that Y is nonstandard (uniformly) continuous. (cid:3) Note that, by the proof Theorem 3.2, there are plenty (standard) functionals Y that are not standard extensional as in ( E ) st .Theorem 3.14 can be interpreted as saying that DG has intuitionistic features(in that ‘more’ functionals are continuous than in classical mathematics), but thefollowing corollary shows that something ‘much more non-classical’ is going on. Afunctional Y is near-standard if ( ∀ st f )( ∃ st n )( Y ( f ) = n ), as defined in [21, p. 93]. Corollary 3.15.
The system DG proves that for any near-standard Y satisfying ( E ) st , there is standard Z such that ( ∀ f ∈ C )( Z ( f ) = Y ( f )) . Proof.
First of all, since all binary sequences are standard, ( ∀ f ∈ C )( ∃ st n )( Y ( f ) = n ) follows from the near-standardness of Y , and applying R ω yields a standardupper bound n for Y on the Cantor space. Fix nonstandard N and define Z ( f )as Y ( f N ∗ . . . ) if f N is a binary sequence, and n otherwise. Then Z ( f ) = Y ( f )for f ∈ C by standard extensionality, and Z ≤ ∗ n implies that Z is standard. (cid:3) By the theorem, standard extensionality implies continuity relative to the stan-dard world. Now, as discussed in [2, 22], one can naturally interpret the standard-ness predicate ‘st( x )’ as ‘ x is computationally relevant’ using the systems from[2]. With this interpretation in mind, Corollary 3.15 expresses that relative to‘st’, continuity implies being computable (in some sense). However, intuitionis-tic mathematics, the continuity axiom WC-N in particular, refutes
Church’s thesis CT , where the latter expresses that all sequences are computable (in the sense ofTuring), and the former implies Brouwer’s continuity theorem (see [28, p. 211]).We can even prove a stronger consequence of Theorem 3.14, as follows. Corollary 3.16.
The system DG + ( ∃ ) proves that for any near-standard Y andstandard g , there is standard Z such that ( ∀ f ≤ g )( Z ( f ) = Y ( f )) .Proof. Use ( ∃ ) to define Z ( f ) as Y ( f ) if f ≤ g and 0 otherwise. Then Z isstandard in the same way as in the corollary: since g is standard, f ≤ g is too. (cid:3) By the previous, any functional Y is automatically standard if it is near-standard on C , and zero elsewhere.3.2. Non-intuitionistic aspects of the Dinis-Gaspar system.
We show thatthe Dinis-Gaspar system does not qualify as a system of intuitionistic mathematicsfor the following reasons:(i) The system DG proves, relative to the standard world, the weak K¨onig’slemma , which is rejected in constructive mathematics (Section 3.2.1).(ii) The system DG is inconsistent with the axiom, relative to the standardworld, all functions are ( epsilon-delta ) continuous on the Baire space (Sec-tion 3.2.2).(iii) The system DG is inconsistent with the axiom schema, relative to the stan-dard world, called Kripke’s scheme (Section 3.2.3).Regarding the occurrence of ‘relative to the standard world’ in the previous items,we recall the following regarding the standard objects in internal set theory.For example, the set N of all natural numbers, the set R of all realnumbers, the real number π , and the Hilbert space L ( R ) are allstandard sets, since they may be uniquely described in conventionalmathematical terms. Every specific object of conventional mathe-matics is a standard set . It remains unchanged in the new theory.([18, p. 1166], emphasis in original)We note that all closed terms of DG are standard, and presumably every objectwhich may be constructed (in some sense or other from constructive mathematics)will be standard. Moreover, even in the classical system from [2], the standard ob-jects yield (copious) computational/constructive content, as detailed in [22]. Thus,the standard world should be the focus of our attention, if we are interested incomputational/constructive content. NOTE ON NON-CLASSICAL NONSTANDARD ARITHMETIC 13
Weak K¨onig’s lemma.
We show that the Dinis-Gaspar system proves, rela-tive to the standard world, weak K¨onig’s lemma and the latter’s uniform version.Recall that the variable ‘ T ’ is reserved for trees, and we denote by ‘ T ≤
1’ that T is a binary tree. Then WKL is just the classical contraposition of
FAN , and( ∃ Ψ)( ∀ T ≤ (cid:2) ( ∀ n )( ∃ β ≤ βn ∈ T ) → ( ∀ m )(Ψ( T ) m ∈ T ) (cid:3) ( UWKL )the uniform version. As
WKL is (constructively) equivalent to a fragment of thelaw of excluded middle (see [12]), it is rejected in constructive mathematics.
Theorem 3.17.
The system DG proves WKL st and UWKL st .Proof. Let f be the sequence that is constant 0. Let T be a standard binary treesuch that ( ∀ st n )( ∃ σ ≤ σn ∈ T ), i.e. T is infinite relative to the standard world.We immediately obtain:( ∀ st n )( ∃ σ ≤ ∀ m ≤ n )( σm ∈ T ) , and applying I ω (since ‘ ≤ ∗ ’ is ‘ ≤ ’ by definition) yields ( ∃ σ ≤ ∀ st n )( σn ∈ T ).Since σ ≤ ∗
1, item (b) of the nonstandard axioms implies that σ is a standardbinary sequence, and WKL st follows. To obtain UWKL st , fix nonstandard N anddefine Φ → as follows: Φ( T ) is σ ∗ f where σ ∈ T is the left-most binary sequenceof maximal length | σ | ≤ N , if it exists, and f otherwise. Since Φ ≤ ∗ →
1, thisdefines a standard functional, and we are done. (cid:3)
Kohlenbach shows in [15] that
RCA ω ⊢ UWKL ↔ ( ∃ ) crucially depends on theaxiom of extensionality. Assuming DG is consistent, we do not have access to ( E ) st by the proof of Theorem 3.8, and hence ( ∃ ) st does not follow from UWKL st in DG ,i.e. the previous theorem does not lead to a contradiction. Moreover, the first partof the theorem, involving a classical system, has been proved in [4], and the proofin the latter seems to go through in our (semi-)intuitionistic setting.Moreover, WKL is (constructively) equivalent to ( ∀ x ∈ R )( x ≥ ∨ x ≤
0) and tothe fact that every real in [0 ,
1] has a binary representation (see [12]). As expected, DG also proves versions of the latter, relative to the standard world. Theorem 3.18.
The system DG proves that every real in the unit interval has astandard binary approximation, i.e. ( ∀ x ∈ [0 , ∃ st f ∈ C ) (cid:0) x ≈ P ∞ n =0 f ( n )2 n (cid:1) , and ( ∃ st Φ )( ∀ x ∈ R )(Φ( x ) = 0 → x / ∧ Φ( x ) = 1 → x ' . (3.13) Proof.
Fix nonstandard N and define Φ as: Φ( x ) = 0 if [ x ]( N ) ≤ N , and 1otherwise. Note that Φ ≤ ∗ x − / x , and given the first n suchbits b , . . . , b n − , then Φ( x − ( n +1 + P n − i =0 b i i +1 )) yields the n + 1-th bit. (cid:3) There are a number of other theorems (constructively) equivalent to
WKL by[12], like e.g. the intermediate value theorem. As expected, one can also establishthese theorems relative to ‘st’ inside DG , but we do not go into details.It is well-known that WKL is inconsistent with the aforementioned axiom
Church’sthesis CT ([1, p. 68]). Since DG proves WKL st , one expects DG to be inconsistentwith CT relative to the standard world. Let ‘ ϕ e,s ( n ) = m ’ be the (primitive recur-sive) predicate expressing that the Turing machine with index e and input n haltsafter at most s steps with output m . Then Church’s thesis is defined as follows.( ∀ f )( ∃ e )( ∀ n , m ) (cid:2) ( ∃ s )( ϕ e,s ( n ) = m ) ↔ f ( n ) = m (cid:3) . ( CT ) Theorem 3.19.
The system DG proves ¬ CT st .Proof. Suppose CT st holds. Fix nonstandard N and define (standard by definition) f ≤ f ( e ) = 1 if ( ∃ s ≤ N )( ϕ e,s ( e ) = 0), and 0 otherwise. Then thereis standard e such that ( ∃ st s )( ϕ e ,s ( e ) = m ) ↔ f ( e ) = m for any standard m .However, f ( e ) = 1 implies by definition ( ∃ s ≤ N )( ϕ e ,s ( e ) = 0), a contradiction.Similarly, f ( e ) = 0 implies by definition ( ∀ s ≤ N )( ∀ n )( ϕ e ,s ( e ) = n → n = 0),a contradiction. Since we obtained a contradiction in each case, CT st is false. (cid:3) Intuitionistic continuity.
We show that DG is inconsistent with certain ax-ioms, relativised to the standard world, of intuitionistic mathematics.First of all, we consider the continuity principle BCT C ≡ ( ∀ Y ) cont C ( Y ), whichexpresses that all functionals are (epsilon-delta) continuous on the Cantor space,as given by the following formula:( ∀ f ≤ ∃ N )( ∀ g ≤ f N = gN → Y ( f ) = Y ( g )) . ( cont C ( Y ))Secondly, we consider the principle weak continuity for numbers ( ∀ α )( ∃ n ) A ( α, n ) → ( ∀ α )( ∃ n , m )( ∀ β )[ αn = βm → A ( α, m )] ( WC-N )for any formula A in the language of finite types. Let WC-N be the restriction of WC-N to quantifier-free formulas, and recall the axiom SE from Section 3.1.1. Theorem 3.20.
The systems DG + ( BCT C ) st , DG + ( WC-N ) st , and DG + SE st areinconsistent.Proof. For the first part, consider Y , f , g as in the proof of Theorem 3.2 and notethat f ≈ g contradicts ( BCT C ) st . For the second part, take A ( α, n ) ≡ ( Y = n )and note that ( WC-N ) st implies that Y is epsilon-delta continuous on C , relativeto the standard world. For the third part, note that SE st implies ( E ) st . (cid:3) Note that Y is not sequentially continuous relative to the standard world, i.e. therestriction of BCT C to sequential continuity does not change the previous theorem.Moreover, due to M ω , there is no difference between LPO st and the weaker WLPO st ,i.e. the associated notion of nondiscontinuity ([12, Thm. 3]) is not relevant here.As an aside, SE follows from WMP by [13, Thm. 11], which in turns is provablein (constructive) recursive mathematics (see [12, Prop. 13]). Hence, DG is alsoinconsistent with theorems of recursive mathematics, relative to the standard world.As another aside, we prove that DG is consistent (or even outright proves) certaintheorems of intuitionistic mathematics. Indeed, a consequence of BCT C (togetherwith FAN ) is that all functions on C are bounded. Theorem 3.21.
The system DG proves ( ∀ st Y )( ∃ st N )( ∀ st f ≤ Y ( f ) ≤ N ) ; thesystem DG + PF-TP ∀ proves ( ∀ Y )( ∃ N )( ∀ f ≤ Y ( f ) ≤ N ) .Proof. For standard Y , since all binary sequences are standard, we have ( ∀ f ≤ ∃ st n )( Y ( f ) ≤ n ), and R ω finishes the first part. For the second part, drop allbut the leading ‘st’ and apply PF-TP ∀ . (cid:3) The previous implies that DG + PF-TP ∀ is inconsistent with recursive mathemat-ics, as the latter involves unbounded functionals on 2 N (see [1, p. 70]). In particular, DG + PF-TP ∀ + CT is inconsistent, which also follows from Theorem 3.6 if we inaddition add QF-AC , to the system. NOTE ON NON-CLASSICAL NONSTANDARD ARITHMETIC 15
Finally, we show that the Dinis-Gaspar system is inconsistent with a classical continuity principle. Our motivation is to exclude an incorrect interpretation ofthe results in the previous two sections. Indeed, one could say that DG is slightlyclassical (as it proves WKL st ) and therefore Theorem 3.20. As it turns out, DG isinconsistent with ( BCT C ) st restricted to continuous functionals.Thus, define CCT C ≡ ( ∀ st Y )( cont C ( Y ) → [ cont C ( Y )] st ), which expresses thatall functionals which are (epsilon-delta) continuous on C , are also continuous in thisway relative to the standard world . Note that CCT C readily follows from Transfer . Theorem 3.22.
The system DG + CCT C is inconsistent.Proof. Consider the standard objects Y , f , g as in the proof of Theorem 3.2 andnote that f ≈ g contradicts CCT C as cont C ( Y ). (cid:3) One could replace the antecedent of
CCT C with more restrictive internal formu-las, but the end result would still be the same.3.2.3. Kripke’s scheme.
We show that DG is inconsistent with a fragment of Kripke’sscheme relative to the standard world. This is not that surprising since DG includesnonstandard Markov’s principle M ω , which implies MP st , i.e. Markov’s principle MP relative to the standard world. Indeed, Markov’s principle MP is rejected inintuitionistic mathematics, which was first established by Brouwer using an axiomscheme nowadays called Kripke’s scheme (see [6, p. 244] for details). The ‘strong’form of this scheme is formulated as follows by Dummett in [6]. Principle 3.23 ( KS ∗ ) . For any formula A , we have (˜ ∃ β ≤ A ↔ ( ∃ n )( β ( n ) = 1)) . We consider the following special case of KS ∗ :( ∀ α ≤ ∃ β ≤ ∀ m ) (cid:2) ( ∀ k ) α ( k, m ) = 0 ↔ ( ∃ n )( β ( n, m ) = 0) (cid:3) . ( KS ∗ )By [28, § MP and the Kripke schema imply the law ofexcluded middle, which is a similar result to what is obtained in the followingproof. Theorem 3.24.
The system DG + ( KS ∗ ) st is inconsistent.Proof. Fix nonstandard N and fix standard α, β ≤ KS ∗ ) st ; let g ( m ) (resp. h ( m )) be the least k ≤ N such that α ( k, m ) = 0 (resp. β ( k, m ) = 0) if it exists,and N otherwise. Define (standard by definition) γ ≤ γ ( m ) = 0 if g ( m ) > h ( m ), and 1 otherwise. Then if we can prove the following:( ∀ st m ) (cid:2) ( ∀ st k ) α ( k, m ) = 0 ↔ ( ∃ st n )( β ( n, m ) = 0) ↔ γ ( m ) = 0 (cid:3) , (3.14)then we are done: M ω guarantees that (3.14) implies ACA st0 from Section 3.1.3, andTheorem 3.13 yields the desired contradiction. To prove (3.14), if for standard m ,we have ( ∃ st n )( β ( n, m ) = 0), then h ( m ) is standard, while g ( m ) is nonstandard(by the first equivalence in (3.14)), i.e. h ( m ) < g ( m ). Note that ( ∀ st k ) α ( k, m ) =0 implies ( ∀ k ≤ K ) α ( k, m ) = 0 for some nonstandard K using Idealisation I ω asusual. The reverse implication follows in the same way using M ω . (cid:3) Non-standard aspects of the Dinis-Gaspar system.
We show that thesystem DG includes a ‘standard part map’, a notion introduced in the next para-graph. As we will see, this raises the question to what extent DG (and the systemfrom [9]) can still be referred to as ‘Nonstandard Analysis’ or ‘internal set theory’.First of all, Robinson introduces the ‘standard part map’ ◦ in [21, p. 57]; thelatter maps any x ∈ [0 ,
1] to the (unique) standard ◦ x such that x ≈ ◦ x , and thelatter is called the ‘standard part’ of the former. However, in the Robinsonianframework, the standard part map is external .Secondly, in light of the previous, there is no hope of having access to this mapin Nelson’s IST : we are only given the
Standardisation axiom in which the standardpart of a real exists . Nonetheless, we show that DG does afford a standard partmap, and even a generalisation to functionals on the Cantor space. Theorem 3.25.
There is a term u (1 × → of G¨odel’s T such that DG proves: fornonstandard N and x ∈ [0 , , we have st ( u ( x, N )) and u ( x, N ) ≈ x .Proof. Let f be the constant zero function. Recall the functional Φ form Theorem3.18 and fix nonstandard N ; define v ( x, N ) as Ψ( x, N ) ∗ f if − N ≤ Q [ x ](2 N ) ≤ Q N , and f otherwise. Here, Ψ( x,
0) is h Φ( x − ) i and Ψ( x, n + 1) is Ψ( x, n ) ∗ h b i ,where b = Φ( x − ( n +1 + P n − i =0 Φ( x,n )( i )2 i +1 )). Since v ( x, N ) ≤ ∗
1, the former is standard(in the sense that st ( v ( x, N )) for any x ∈ [0 , P ∞ n =0 v ( x,N )( n )2 n +1 ≈ x by design. Define standard w → as w ( α )( n ) := P ni =0 α ( n )2 n , and note that u := w ◦ v is as required by the theorem. (cid:3) Recall that we (may) view any sequence as a real; since λx.v ( x, N ) ≤ ∗ → → ( λx.v ( x, N )), and the standard part map u := w ◦ v is thus standard in DG , a fairly ‘non-standard’ situation as discussed in Remark 3.28. Theorem 3.26.
There is s (2 × → in G¨odel’s T such that DG proves: for non-standard N and near-standard Y such that ( E ) st , we have st ( s ( Y, N )) ∧ ( ∀ f ∈ C )( s ( Y, N )( f ) = Y ( f )) .Proof. By the near-standardness of Y , and the fact that all binary sequencesare standard, we have ( ∀ f ∈ C )( ∃ st n )( Y ( f ) ≤ n ), and R ω implies ( ∀ f ∈ C )( ∃ n ≤ n )( Y ( f ) ≤ n ) for some standard n . Fix nonstandard N and define s ( Y, N )( f ) as Y ( f N ∗ . . . ) if f N is a binary sequence, and n otherwise. Then s ( Y, N )( f ) = Y ( f ) for f ∈ C by standard extensionality, and λf.s ( Y, N )( f ) ≤ ∗ n implies thatst ( λf.s ( Y, N )( f )), as required. (cid:3) Corollary 3.27.
The system DG + ( ∃ ) proves that there is Φ → such that fornear-standard Y , we have st (Φ( Y )) ∧ ( ∀ f ∈ C )(Φ( Y )( f ) = Y ( f )) .Proof. Use ∃ to define Φ( Y )( f ) as Y ( f ) if f ∈ C , and zero otherwise. Then Φ( Y )is standard in the same way as in the theorem. (cid:3) The previous theorem could be obtained for F : [0 , → R using Theorem 3.18,but this development would mostly be repetitive. We finish this section with an informal remark on just how unnatural the standard part maps of DG are. Remark 3.28.
The standard part maps of DG are quite unnatural from the pointof view of internal set theory for the following reason: the standard part of a real x ∈ [0 ,
1] is unique in IST , i.e. if x ≈ y ≈ z and the latter two are standard reals, NOTE ON NON-CLASSICAL NONSTANDARD ARITHMETIC 17 then y = z . Hence, if there were Φ : R → R such that Φ( x ) ≈ x ∧ st(Φ( x )) for any x ∈ [0 , ∀ x ∈ [0 , x ) ↔ x = Φ( x )). However, one ofthe central tenets of IST is that ‘st’ is not definable via an internal formula:To assert that x is a standard set has no meaning within conven-tional mathematics-it is a new undefined notion. ([18, p. 1165])These observations do not cause problems for DG of course: the uniqueness ofstandard parts in IST requires
Transfer anyway, while ‘ x = R y ∧ st( x )’ does notimply st( y ) in DG due to issues of representation of reals. Nonetheless, DG is onlyone basic step removed from being able to define ‘st ’ via an internal formula,something which goes against the very nature of IST . Although the frameworks areof course different, a similar case can be made for the Robinsonian approach.Now, the law of excluded middle is referred to as a ‘taboo’ in constructive math-ematics (see [1, I.3]). In light of the previous remark, those endorsing this kindof language should probably use heresy when referring to the above standard partmaps of DG in the context of Nonstandard Analysis and internal set theory.4. Conclusion
In the previous sections, we have provided fairly conclusive answers to questions(Q1) and (Q2) from Section 1.1. We isolated (very) weak fragments of
Transfer which are still inconsistent with DG , and we identified a number of axioms ofintuitionistic (and general constructive) mathematics which are inconsistent with DG when formulated relative to the standard world. We even established that DG allows for a highly elementary standard part map , a rather ‘non-standard’ featureof DG .These facts all suggest -in one way or another- that DG is indeed non-classical ,but does not really deserve the description intuitionistic . At the same time, sincea standard part map is not available in Nelson’s internal set theory, and external in Robinson’s approach, DG really pushes the boundary of what still counts as‘Nonstandard Analysis’ and ‘internal set theory’.In our opinion, the aforementioned problems trace back to one problematic ax-iom of DG , namely item (b) of the nonstandard axioms. Simply put, this axiom‘makes too many things standard’, an obvious example being the Cantor space.While this axiom may be necessary and/or useful for the connection to the boundedfunctional interpretation (see [5, §
6] and [9, § Acknowledgement 4.1.
This research was supported by the following fundingbodies: FWO Flanders, the John Templeton Foundation, the Alexander von Hum-boldt Foundation, and LMU Munch (via the
Excellence Initiative and CAS LMU).The author expresses his gratitude towards these institutions. I also thank thereferee for the many helpful suggestions.
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