We now present a pair of results that explore how anti-Specker properties may be used to recover information about how functions of a certain kind are (or are not) structured with respect to the location of their zeroes [BDMJ13c]. Accordingly, for functionsffrom a metric spaceXinto a normed
spaceY, we shall be concerned with thezero set
Zf≡
x∈X: f(x) =0 .
We move to a more general setting than that of the holomorphic functions, but nevertheless retain some aspect of their form. In particular notice that, provided holomorphic functions are indeed differentiable in the sense of Bishop and Bridges [BB85, pp. 130–131], we can apply a strong location-of-zeroes result of theirs to prove the following proposition.
Proposition 6.25: BISH ` LetΩ⊆Cbe an open region,KbΩbe compact, andBbe a border
forK. Suppose thatf:Ω→Cis a holomorphic function withm(f,B)> m(f,K). Thenfsatisfies
(∗) For every > 0, we can findδ > 0 such that, for eachz ∈K: if|f(z)| < δ, then
|z−ζ|< for some zeroζ∈Zf.
Proof. Sincem(f,B) > 0 andm(f,K) 6= m(f,B),Corollary 5.3of [BB85, p. 153] shows that
m(f,K) =0. We are now able to applyTheorem 5.11of the same to find pointsz1, . . . ,zn ∈K and a functiongdifferentiable onKsuch thatµ≡m(g,K)>0 and
f(z) = (z−z1)· · ·(z−zn)g(z)
forz∈K. Fix any >0, and observe that
z−z1 · · · z−zn 6 f(z) µ .
Accordingly, if we takeδ≡µ· 12n, we get
z−z1 · · · z−zn < 12 n
whenever|f(z)|< δ. This rules out the possibility of having|z−zi|> 12for alli; hence we are
able to find an indexkwith|z−zk|< .
We will henceforth focus our attention upon functions that exhibit a pointwise variant of the property(∗). LetXbe a metric space andYa normed space. We say thatf:X→Yiszero-stable(on
X) if, for each pointx∈Xand >0, there existsδ >0 such that ifkf(x)k< δ, thenρ(x,ζ)<
for someζ∈Zf: that is,
(∀x∈X)(∀ >0)(∃δ >0)hf(x)
< δ =⇒ (∃ζ∈Zf)ρ(x,ζ)<
i
.
We will also consider how we may attain a stronger generalisation of(∗), in its original uniform formulation. The functionfis said to beuniformly zero-stable(onX) if, for each >0, we can find δ >0 such that, for eachx∈X: ifkf(x)k< δ, thenρ(x,ζ)< for someζ∈Zf. That is,
(∀ >0)(∃δ >0)(∀x∈X)hf(x) < δ =⇒ (∃ζ∈Zf) ρ(x,ζ)< i .
In the classical setting, it is straightforward to show that every functionf:X→Yis zero-stable.
For, givenx∈Xand >0, we have eitherf(x) =0 orf(x)6=0. In the former case, setδ=1, and chooseζ=xto satisfy the consequent of
f(x)
< δ =⇒ (∃ζ∈Zf)ρ(x,ζ)< ; in the latter case, chooseδ=kf(x)k>0 to falsify the antecedent.
In fact, we can use a sequential compactness argument to obtain the following more interesting result.
Proposition 6.26: CLASS ` LetX be a compact metric space, and letf be a sequentially
continuous mapping ofXinto a normed spaceY. Thenfis uniformly zero-stable.
Proof. Fix any >0, and suppose
(6.1) (∀δ >0)(∃x∈X)hf(x)
< δ∧(∀ζ∈Zf)ρ(x,ζ)>
i
.
This gives us a sequence(xn)n>1 of points in Xsuch that, for eachn, kf(xn)k < 2−n, and
ρ(xn,ζ)>for eachζ∈Zf. Use the sequential compactness ofXto find a subsequence(xnk)k>1
that converges to some pointx∈X. Sincekf(xnk)k →0, we must havekf(x)k=0 by sequential
continuity; that is,x∈Zf. But by our definition of(xn), this now gives usρ(xnk,x)>for all
k∈N+— a contradiction. So (6.1) must be false, and uniform zero-stability now follows.
Constructively, the situation is somewhat less straightforward. While holomorphic functions still exhibit zero-stability on appropriate domains (in light ofProposition 6.25), continuous functions
in our more general setting need not. Indeed, even the claim that functions in a class as restricted as the quadratic polynomials of a real variable are zero-stable entailsLPO.
To see this, fix anyα>0, and consider the quadratic functionf: [0, 1]→Rdefined byf(x) =x2+α.
Suppose thatfis zero-stable. In particular, we can apply zero-stability at 0 to obtain a number δ >0 such that ifα=|f(0)|< δ, then there existsζ∈[0, 1]withf(ζ) =0 and|ζ|<1. Now, either α >0 orα < δ. In the latter case: ifα >0, we getf(ζ) =ζ2+α >0 for all realζwith|ζ|<1,
a contradiction; hence,α= 0. So zero-stability allows us to decide whetherα >0 orα =0, a decision which is equivalent toLPO[BV06, p. 29].
With this in mind, we willassume(pointwise) zero-stability in the results that follow. The focus,
then, is upon how we can combine zero-stability with anti-Specker properties to obtain structural information about the functions in question. We begin by showing howASltdallows us to pass from zero-stability to uniform zero-stability, providedZfis inhabited and separable. One can see by comparison withProposition 6.26that this is another instance in which an anti-Specker property
allows us to recapture some of the power of sequential compactness.
Proposition 6.27: BISH+ASltd ` LetXbe a compact metric space, and letfbe a zero-stable,
continuous mapping ofXinto a normed spaceYsuch thatZfis inhabited and separable. Thenfis uniformly zero-stable onX.
Proof. Fix any >0. SinceZfis inhabited, we can useTheorem 4.9of [BB85, p. 98] to construct a strictly decreasing sequence of numbersηn∈(0, 2−n)for which the sets
Sn≡
x∈X: f(x) 6ηn are compact. Now let(zn)n>1be a dense sequence inZf, and write
Zn≡
z1,z2, . . . ,zn ⊆Zf
for eachn. SinceZnis finite and therefore located inX, we can form the set
Pn ≡
ρ(x,Zn) : x∈Sn ⊂R.
As in the proof of Proposition 6.20,Pn is totally bounded: we thus see that supPn exists. This allows us to construct a binary sequence(λn)n>1such that
λn =0 =⇒ supPn > 12, and
λn =1 =⇒ supPn < .
LetX∪{ω}be a one-point extension ofX, and use(λn)to define a sequence(xn)n>1 in this one-point extension as follows:
v ifλn=0, pickxn ∈Snso thatρ(xn,Zn)> 12; v ifλn=1, setxn =ω.
We now show that(xn)is eventually bounded away from each point ofX. Fix anyx∈X. The zero-stability offgives us a numberα >0 such that
(6.2) f(x) < α =⇒ (∃ζ∈Zf) ρ(ζ,x)< 14.
Two possibilities arise. Note that ifλn=1, we haveρ(x,xn)>ρ(ω,X)>0, which means thatxn is immediately bounded away fromx; hence, we need only consider the situation whereλn=0.
v Ifkf(x)k < α, then (6.2) gives us a zeroζ∈Zfwithρ(ζ,x)< 14. ChooseN∈N+such thatρ(ζ,zN)< 14−ρ(ζ,x), and setδ = 14. Then for alln>Nwithλn = 0, we have
zN ∈Znand thusρ(xn,zN)> 12; it follows that
ρ(xn,x)>ρ(xn,zN) − ρ(zN,ζ) +ρ(ζ,x)
> 12− 14= 14=δ.
v Ifkf(x)k >0, chooseN ∈N+so thatη
N < 12kf(x)k, and then use continuity to choose
δ >0 so thatkf(x) −f(x0)k< 12kf(x)kwheneverx0 ∈Xis a point withρ(x,x0)< δ. Then
for alln>Nwithλn=0, we havexn∈Sn, and so
f(x) −f(xn) > f(x) − f(xn) >f(x) −ηn >f(x) −ηN >f(x) − 12 f(x) = 12 f(x) . Henceρ(x,xn)>δ.
In either case,(xn)is eventually bounded away fromx. The limited anti-Specker property now gives us an indexkfor whichxk=ω, whenceλk =1. Then, for eachx∈Xwithkf(x)k< ηk, we
havex∈Skand thereforeρ(x,Zk)< . That is, there exists a zeroζ∈Zk⊆Zfwithρ(x,ζ)< ; writingη≡ηk, we have shown that
(∀ >0)(∃η >0)(∀x∈X)hf(x) < η =⇒ (∃ζ∈Zf) ρ(x,ζ)< i.
Note that this result also applies in the case whereZf islocatedin X. For,Proposition 2.2.4 of [BV06, p. 39] shows thatXis separable, and located subsets of a separable metric space are
themselves separable.
For our next result, we introduce the following notion: a subsetSof a metric spaceXis said to be
countably isolatedif there exists a one-one enumeration(sn)n>1ofSthat is eventually bounded
away from each of its own terms. It is straightforward to show that, ifXis sequentially compact and fis a sequentially continuous mapping fromXinto a normed space, thenZfcannot be countably isolated. The following proposition uses the non-Specker property to help recapture this idea in a semi-constructive setting.
Proposition 6.28: BISH+AS¬ ` LetXbe a compact metric space, and letfbe a zero-stable,
continuous mapping ofXinto a normed spaceY. ThenZfis not countably isolated.
Proof. Suppose thatZfis countably isolated, and let(zn)n>1be a one-one enumeration ofZf that is eventually bounded away from each termzk. Fix any pointx∈X: we show that(zn)is eventually bounded away fromx. Use the zero-stability offatxto find a sequence of positive
distances(ηn)n>1such that, for eachn,
(6.3)
f(x)
< ηn =⇒ (∃ζ∈Zf)ρ(x,ζ)<2−n. Now define a binary sequence(λn)n>1such that
λn=0 =⇒ f(x) < ηn, and λn=1 =⇒ f(x) >0.
Observe that ifλk = 1 for somek ∈ N+, it follows that(zn)is bounded away fromx. For if
kf(x)k > 0, the continuity offallows us to computeδ > 0 such thatkf(x0)k > 0 whenever ρ(x,x0)< δ. Then for eachn,kf(zn)k =0 and soρ(x,zn)>δ.
With this in mind, assume thatλ1=0 (for ifλ1=1, we are done). Construct a sequence(ζn)n>1
inXas follows: for eachn∈N+,
v ifλm=0 for allm6n, use (6.3) to chooseζn ∈Zfsuch thatρ(x,ζn)<2−n; v ifλm=1 for some minimalm6n, setζn=ζm−1.
Then(ζn)is a Cauchy sequence and therefore converges to some limitζ∈X. Since each term of (ζn)is a zero off, we havef(ζ) =0 by (sequential) continuity. That is,ζis equal to a term of(zn); hence(zn)must be eventually bounded away fromζ.
Accordingly, findN0∈N+andδ >0 such thatρ(zn,ζ)>2δfor alln
>N0. Consider the distance d≡ρ(ζ,x).
v Ifd < δ, then for alln>N0:
ρ(zn,x)>ρ(zn,ζ) −ρ(ζ,x)>2δ−δ=δ.
v Ifd > 0, then we can find an indexN1such that 2−N1 < 1
2dandρ(ζn,ζ) 6 12dfor all
n>N1. Then for all suchn, we have
(6.4) ρ(ζn,x)>ρ(ζ,x) −ρ(ζn,ζ)>d− 12d= 12d.
If it were the case thatλm = 0 for allm6N1, we would haveρ(ζN1,x) <2−N1 < 12d, contradicting (6.4) forn=N1. Henceλk=1 for somek6N1.
In either case, we see that(zn)is eventually bounded away from the pointx; sincexwas arbitrary,
(zn)is a Specker sequence inX. This is a contradiction in view ofAS¬.
For future work, it would be interesting to know whether this result can be adapted to rule out the countable isolation ofZfin somepositivesense, ideally using the limited anti-Specker property (but possibly falling back upon the full anti-Specker property instead).
We thus arrive at the end of our investigation. We have identified and classified a family of several weak semi-constructive principles, theanti-Specker properties, and subsequently illustrated where
and how these principles may be used. Notably, they may sometimes be used in the stead of the highly nonconstructive property ofsequential compactness; however, even when not applied to
compactness results, they often allow us to recover information about such things as the structure of certain classes of functions.
Prominently, we have seen how thelimitedandincreasing anti-Specker properties,ASltdandAS↑,
allow us to establish a pair of Heine-Borel compactness results — one of which linksASltd to
Brouwer’s fan theorem for detachable bars,FT∆. Subsequently, we gave new, direct proofs to
illuminate the equivalence of thenon-Specker propertyAS¬and a family of weak, negative fan-
theoretic principles. Alongside these results, we have classified the similar-in-principlelimit-stability propertyLSPand several recursively-valid antitheses of intuitionistic principles.
In a similar capacity, we have examined the omniscience principlesWLPOandLLPO, presenting
direct proofs that explore their relationships withFT∆,AS¬andLSP.
As well as allowing one to prove results not attainable withinBISHalone, our anti-Specker properties
may also be used to streamline proofs of known results. In particular, we have seen how they may be used to recover a version of themaximum modulus theoremfor holomorphic functions, based upon
a power-series argument rather than the use of Cauchy’s integral formula (as was the approach of [BB85]).
Finally, we showed howASltdandAS¬can be used to recover information about the structure of zeroes for so-calledzero-stablefunctions.
In conclusion, then: these anti-Specker properties are a valuable aid in semi-constructively estab- lishing important results of analysis. But as suggested throughout the preceding chapters, there remains plenty of further work to be done. This is typically indicated at the relevant places; however, a common theme concerns our many results of the form
BISH+P ` Q
(Corollary 3.10is a prominent example). In many cases, it would be desirable to expand upon
these results either by reversing the implication (thereby showing thatPandQare equivalent over BISH, or maybe overBISHplus some additional principle) or by giving a model ofBISHin which Qholds butPdoes not (thereby showing that the implication is strict).
∂Ω,seeboundary #X,38 ∼Ω,seecomplement −Ω,seecomplement Γ(ζ,r),85 λ,17 |u|,17 kfkK,seenorm
b,seewell contained 2∗,seefan
2N+
,seeCantor space AC,seechoice AC!,seechoice ACω,seechoice anti-bdd,60 anti-FT?,53 anti-POS,65 anti-Specker property full,15 increasing,48 limited,27 anti-SPOS,61 anti-UCT,60 anti-USCT,61
AS,seeanti-Specker property AS↑,seeanti-Specker property AS↑¬,seenon-Specker property ASltd,seeanti-Specker property AS¬,seenon-Specker property
axiom of choice,seechoice
B(ζ,r),seeball
ball
closed,47,85 open,85 bar,17
c-bar,seec-subset
Π01-bar,18
uniform,17
BD-N,seeIshihara’s principle
BHK interpretation,2
BISH,seeBishop-style constructive mathematics
Bishop’s lemma,26
Bishop-style constructive mathematics,10
Blkd,77 blocked,17 Bolzano-Weierstraß principle,41 weak,41 border,92 boundary,92
boundedness principle,seeIshihara’s principle
Brouwerian counterexample,4
BWP,seeBolzano-Weierstraß principle
C,seeCantor set C,85 c-subset,18 Cantor set,26 Cantor space,25 choice, axiom of countable,8 dependent,8 full,8 unique,8 Church-Markov-Turing thesis,10
CLASS,seeclassical mathematics
classical mathematics,10 closed segment,seesegment
closed set,85
CMT thesis,seeChurch-Markov-Turing thesis
compactness,16
Heine-Borel,seeHeine-Borel property
sequential,16 complement,92
metric,93
complete binary fan,seefan
complete subfan,seesubfan
sequential,20 uniform,20 uniform sequential,20 convergence uniform,86 uniform absolute,86 countability,21
countable choice,seechoice
countable Heine-Borel property,seeHeine-Borel
property countably isolated,114
CRM,seeconstructive reverse mathematics csf,seesubfan
DC,seechoice
dependent choice,seechoice
detachability,12 distance to a subset,12
DNE,seedouble negation elimination
double negation elimination,2 dyadic rational number,58
-approximation,15
eventually bounded away from a point,14 from a subset,14 excluded middle, law of,2 extension,17 F,25 fan,17 complete binary,17 fan theorem,17 weak,59 finitism,3
FT?,seefan theorem FT¬¬? ,seefan theorem
G,26
Goldbach conjecture,5 has at most one limit,41
HB0,seeHeine-Borel property HBdi,seeHeine-Borel property
Heine-Borel property,16
countable, for disjoint intervals,49 countable, for intervals,32 holomorphy,86
inhabited set,12
INT,seeintuitionistic mathematics
intuitionistic logic,2 intuitionistic mathematics,10 Ishihara’s principle,21 least-upper-bound principle classical,13 constructive,13
LEM,seeexcluded middle
lesser limited principle of omniscience,seeomni-
science
limit-stability property,43 original,42
strong,44
limited principle of omniscience,seeomniscience LLPO,seeomniscience
locally nonconstant,89 locatedness,12
LPO,seeomniscience
LSP,seelimit-stability property LSP0,seelimit-stability property
m(f,K),88
Markov’s principle,7
metric complement,seecomplement
misses,17
MP,seeMarkov’s principle N,8
N+,8
non-Specker property,35 increasing,48 nonchoice,seechoice
nonconstant,87 nonuniform bar,53 norm,92
omniscience,6
lesser limited principle of,6 limited principle of,6 principle of,6
weak limited principle of,7 open set,85
path,17
Π01-subset,18 Π01cl-subset,19
positivity property,30 strong,60
principle of omniscience,seeomniscience
pseudoboundedness,21
R,12
restriction,17
ρ(·,X),seedistance to a subset
RUSS,seeRussian recursive mathematics
Russian recursive mathematics,10
s(t,δ,·),seespike function SC,44
segment,93
sequential compactness,seecompactness SLSP,seelimit-stability property Speck,seeSpecker property Speck↑,seeSpecker property
Specker property,14 increasing,66 Specker sequence,15 strong,15 Specker’s theorem,14 spike function,62
SPOS,seepositivity property StartsWith,77
strong positivity property,seepositivity property
subfan, complete,73 sup,seesupremum
support,62 supremum,12 totally bounded,16 trichotomy law,5
UCT,seeuniform continuity theorem
uniform absolute convergence,seeconvergence
uniform bar,seebar
uniform continuity theorem,21 uniform convergence,seeconvergence
uniform sequential continuity theorem,21 uniform zero-stability,seezero-stability
unique choice,seechoice
USCT,seeuniformly sequential continuity theo-
rem
WBWP,seeBolzano-Weierstraß principle
well contained,85 witness,61
WLPO,seeomniscience
zero set,109 zero-stability,110
uniform,110 Zf,seezero set
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