aa r X i v : . [ m a t h . L O ] O c t Principles Weaker than BD-N
Robert S. Lubarsky and Hannes DienerDept. of Mathematical SciencesFlorida Atlantic UniversityBoca Raton, FL [email protected] Mathematik, Fak. IVEmmy-Noether-Campus, Walter-Flex-Str. 3University of Siegen57068 Siegen, [email protected] 27, 2018
Abstract
BD-N is a weak principle of constructive analysis. Several interestingprinciples implied by BD-N have already been identified, namely the clo-sure of the anti-Specker spaces under product, the Riemann PermutationTheorem, and the Cauchyness of all partially Cauchy sequences. Herethese are shown to be strictly weaker than BD-N, yet not provable in settheory alone under constructive logic. keywords: anti-Specker spaces, BD-N, Cauchy sequences, partially Cauchy,Riemann Permutation Theorem, topological models
AMS 2010 MSC:
BD-N, first identified in [13] with a pre-history in [12], has turned out to bean important foundational principle, being equivalent to many statements inanalysis having to do with continuity [14, 15, 16]. It is also weak, or deep, in thatit holds in all major traditions of mathematics, including classical mathematics,intuitionism, and Russian constructivism (based on computability). As sucha central and weak principle, it is reasonable to guess that most particularconsequences of it would either imply it or be outright provable. A moment’ssober reflection would lead one to realize that BD-N is not an atom, or co-atom,in the Heyting algebra of statements, and that it would almost certainly just bea matter of time before natural intermediate statements were found. This notereports on exactly that. 1he first example is the closure of the anti-Specker spaces under products.(For background, see [2, 3, 4].) Douglas Bridges showed such closure underBD-N [8], speculating that they were equivalent. It was shown in [19] that infact such closure does not imply BD-N. This still leaves open the question as towhether this closure is provable outright. We show below it is not.The next example is the Riemann Permutation Theorem. For background,see [5, 6], where it is shown, among other things, that BD-N implies the RPT.We show that RPT does not itself imply BD-N, in a very similar manner to theanti-Specker property, and also that it is not itself provable.Our final example has to do with a weakening of the definition of a Cauchysequence, which was identified by Fred Richman (notes), who called such se-quences partially Cauchy. He showed that under BD-N, all partially Cauchysequences are Cauchy. A similar kind of sequence, an almost Cauchy sequence,has also been identified [7]. There it was shown that, under Countable Choice,all almost Cauchy sequences are Cauchy iff BD-N holds. It is shown below thatthe Cauchyness of all partially Cauchy sequences does not imply BD-N, andalso that the Cauchyness of all partially Cauchy sequences is not itself provableon the basis of set theory alone.We speculate that there is an intricate and interesting world of unprovableprinciples strictly weaker than BD-N. To investigate this, several tasks needfulfilling. For one, we would like to see just how interesting the ones discussedhere are, by how many interesting statements they’re equivalent with. We wouldalso like to see other such intermediate statements. Of course, we need to knowwhether these intermediate properties are themselves mutually inequivalent,and, if so, what implications might hold under additional hypotheses, such asCountable Choice. As for an independence result in the general case, the obviousplace to look first would be the models contained in this paper. These topologicalmodels have a claim at being the generic models for their specific purposes.As such, they naturally tend to keep principles not intended to be falsifiedtrue. So, for instance, you’d expect that, in the model falsifying RPT, partiallyCauchy sequences would still be Cauchy, unless of course the latter assertionimplied RPT. We do not attempt a thorough analysis of all of these issues here,preferring to leave this for future work.Regarding the methods employed, any independence result of the form “Adoes not imply B” is shown here by providing a model of A in which B is false.These models are all topological models, which works essentially like forcingfrom classical set theory when you leave out that part of the basic theory whereyou mod out by the double negation, the purpose of which is only to modelclassical instead of just constructive logic, clearly a move which is anathemato our purposes. For background on topological models, see [10, 11] and theaddendum to [18], or the brief discussion before theorem 2.3 below.It would be interesting to see how these issues would play out with realiz-ability models. The first models discovered falsifying BD-N were of this kind[1, 9, 17]. Each and every one of them also exhibits a separation of the kindproved here, depending on which among the anti-Specker closure property, RPT,and the partially Cauchy property hold in it. The extra challenge presented by2his context is that realizability models seem not to be the canonical models forthese properties, and they’re less flexible to deal with. By way of illustration,most of the topological models presented here and in [19] were not that difficultto come up with; in contrast, it seems completely unclear how to concoct a real-izability model for the same purposes. As another illustration, as argued for in[19], topological models seem to be canonical for their purposes, in part becauseground model properties tend to persist into the topological models, except ofcourse for those that imply the property purposely being falsified. For instance,the three properties considered here are all true in the topological model of notBD-N, as predicted. In contrast, in the realizability models at hand, all bets areoff as to which of those three hold there. The experts do not have a clear expec-tation of this outcome, and furthermore, at least in the one case tried (RPT inextensional K realizability), they have not been able to prove one way or theother whether it holds. On the other hand, many of these models are naturallyoccurring in and of themselves. Hence it would be nice to know which of theprinciples under consideration hold where, in order to understand these modelsbetter, as well as the computational content of the principles themselves.The paper is organized as followed. Anti-Specker spaces are discussed insec. 2. It was shown already in [19] that their closure under Cartesian productdoes not imply BD-N; here we see that such closure can fail under standardset theory (IZF). The reason we work over IZF is twofold. It is the closestconstructive correlate to ZFC, the de facto gold standard in mathematics, andit is strictly stronger than the other theories commonly considered, such asCZF and BISH, so that an independence result of IZF implies the same overthese others. The following two sections are about the Riemann PermutationTheorem, first that it does not imply BD-N, and then that it can fail even underIZF. The two sections after that show the corresponding results for the assertionthat all partially Cauchy sequences are Cauchy. An anti-Specker space for our purposes is a metric space X such that, when youenlarge X by adding a single point ∗ at a distance of 1 away from every x ∈ X ,then every countable sequence through X ∪ {∗} which is eventually apart fromevery point of X is eventually equal to ∗ . (Actually, there are various such anti-Specker properties, sometimes inequivalent, parametrized by how the space X is extended. Since we consider here only this one version, we suppress mentionof this choice in the notation and terminology.) Anti-Speckerhood is a form ofcompactness. As such, one might reasonably expect anti-Specker spaces to beclosed under Cartesian product. We produce a topological space T such thatthe (full) model over T falsifies such closure. Definition 2.1
Let T consist of ω -sequences ( z n ) such that finitely many en-tries are pairs of real numbers h x n , y n i and the rest are ∗ , which is taken by onvention to equal h∗ , ∗i (so every entry has both projections). We give thetopology by describing a sub-basis. An open set in the sub-basis is given by thefollowing information. The positive information is a finite sequence α n ( n < N ) ,each entry of which is either ∗ or a pair of finite open intervals h I n , J n i . A se-quence ( z n ) satisfies this positive constraint if z n = ∗ whenever α n = ∗ and z n ∈ I n × J n otherwise ( n < N ). The negative information is an assignment toeach of finitely many closed and bounded sets C i ( i ∈ I, I an index set) in R ofa natural number M i . This negative information is satisfied by ( z n ) if, for all n > M i , z n C i (where ∗ 6∈ R ). (Notice that the empty set is given by theintersection of two sub-basic open sets with incompatible positive information.) An open set is said to be in normal form if the following conditions hold.For one, for m, n < N, either h I m , J m i = h I n , J n i or I m × J m and I n × J n aredisjoint (where X is the closure of X ). Also, I is a singleton – that is, thenegative information has only one closed set – and that unique closed set C isa (necessarily finite) rectangle. Finally, for m, n < N I m × J n ⊆ C . (Implicitly,when reference is made to h I n , J n i when α n = ∗ , that clause does not apply.) CI × J ( x , y ) I × J = I × J ( x , y ) = ( x , y ) I × J ( x , y )Figure 1: Open set in normal form containing the point(( x , y ) , ( x , y ) , ( x , y ) , . . . ) with ( x , y ) = ∗ . Lemma 2.2
The opens in normal form constitute a sub-basis.
Proof:
Given ( z n ) ∈ O extend the positive information to include all of ( z n )’snon- ∗ entries. Then for h x m , y m i = h x n , y n i shrink I m × J m and I n × J n to4e equal; for h x m , y m i 6 = h x n , y n i shrink I m × J m and I n × J n to satisfy thedisjointness condition. Furthermore, if n > M i then I n × J n must be shrunkento be disjoint from C i . Then enclose all of the C i ’s by one rectangle C , alsolarge enough to cover each I m × J n , and assign to C the length of the positivesequence.Let G be the generic. To help make this paper somewhat self-contained, thebasics of topological models include that the universe of the extension consistsexactly of terms, which are sets of the form {h O i , σ i i | i ∈ I } , where O i is anopen set, σ i inductively a term, and I an index set. When each O i hereditarilyis the entire space, then the term is the canonical image ˆ x of a ground modelset x . The generic G is the term {h O, ˆ O i | O an open set of T } , which in thiscase can be identified with a sequence ( g n ). Let X be the set of reals from thefirst components of the g n ’s, and Y the reals from the second components. Theorem 2.3 T (cid:13) “X and Y are anti-Specker spaces, and X × Y is not.” Proof:
Clearly, T (cid:13) “( g n ) is a sequence through X × Y ∪ {∗} .” By consideringthe normal opens, ( g n ) is eventually apart from each point in X × Y . In greaterdetail, suppose ( z n ) ∈ O (cid:13) ( x, y ) ∈ X × Y. Then some neighborhood of ( z n )forces x to be in some I m and y to be in some J n . Let U be a normal opensubset of that neighborhood containing ( z n ). If U ’s positive information haslength N , then U (cid:13) “Beyond N ( g n ) is apart from ( x, y ).”Also, no open set forces ( g n ) eventually to be ∗ , because the closed sets inthe negative information are finite. That is, given any open set O and naturalnumber k , there is member of O with a non- ∗ entry beyond slot k .Hence ( g n ) witnesses that X × Y is not an anti-Specker space.All that remains to show is that X and Y are anti-Specker spaces. We willshow this for X , the case for Y being symmetric.To this end, suppose O (cid:13) “( a n ) is a sequence through X ∪ {∗} eventuallyapart from each point in X .” For ( z n ) ∈ O we must find a neighborhood of ( z n )forcing a place beyond which ( a n ) is always ∗ . First extend (i.e. shrink) O sothe positive information α contains all of ( z n )’s non- ∗ entries. Then we claimwe can extend again to force an integer K beyond which (i.e. for k > K ) a k is apart from each x n in ( z n )’s non- ∗ entries, all the while keeping ( z n ) in theopen set. That is, for each α n of the form h I n , J n i , a k is forced to be at leastsome fixed rational distance r n away from the real approximated by I n . Todo this, iteratively extend the open set to have this property for each h I n , J n i individually.Then extend again by shrinking I n (to an interval we will still call I n , recy-cling notation) so that I n has length less than r n . This forces a k to be apart fromthe entire interval I n ; even more, a k is forced not to be in some open intervalcontaining I n ’s endpoints, some extension of I n both upwards and downwards.We call such a lengthened interval a forbidden zone.5 I × J ( x , y ) I × J = I × J ( x , y ) = ( x , y ) I × J ( x , y )forbidden zone forbidden zone( x, y ) v ℓ Figure 2: The same open set as before, with forbidden zones and path P.Finally, extend yet again to an open set U ∋ ( z n ) in normal form, withpositive information given by α of length N and negative information given by C . The claim is that U (cid:13) “For k > K a k = ∗ ”. If not, for some fixed k > K and l let some extension V of U force “ a k = x l ”. Call V ’s positive information β . Notice that, by the construction above, l > N . That means we can change β l without violating U ’s positive information. Pick some v = ( v n ) ∈ V . Let P be a path in R starting at v l , ending at some ( x, y ) with x in some forbiddenzone, and avoiding C ; this is possible, because C is just a finite rectangle. Foreach p ∈ P let v ( p ) be identical with v except that v l is replaced by p . Noticethat v ( p ) ∈ U , because by avoiding C we’re also not violating U ’s negativeinformation. So around each v ( p ) is an open subset of U forcing either that a k is ∗ or that it’s not. Let P ∗ be { p | some neighborhood of v ( p ) forces “ a k = ∗} and P x be { p | some neighborhood of v ( p ) forces “ a k = ∗} . Since positive infor-mation is given by open sets, whatever some neighborhood of some v ( p ) forces,6he same neighborhood will force the same thing for all v ( q ) where q is in someneighborhood of p . In other words, both P ∗ and P x are open. Since paths in R are connected, one of those is empty and the other is P . Since v l ∈ P x , P x = P .Recall that P ends at some ( x, y ) with x in a forbidden zone. This contradictsthe choice of K , and so completes the proof. At some point in this paper, it should be stated what the Riemann PermutationTheorem actually is.
Theorem 3.1 (Riemann) If every permutation of a series of real numbers con-verges, then the series converges absolutely.
For a constructive analysis of the issues involved with convergence of series,see [5, 6]. These include a proof that BD-N implies RPT, as well as that absoluteconvergence follows from merely having a bound on the partial sums of theabsolute values, which we use implicitly below.In [19] a topological model falsifying BD-N is presented, as well as a proofthat, in that model, the anti-Specker spaces are closed under products. It ispredictable that the proofs that other properties slightly weaker than BD-N holdin the same model would be very similar, and also true. To make the currentpaper somewhat self-contained we will describe the underlying topological spaceagain; the argument afterwards that RPT holds should seem familiar to anyonefamiliar with the anti-Specker closure section from [19].The points in T be the functions f from ω to ω with finite range, that is,enumerations of finite sets. A basic open set p is (either ∅ or) given by anunbounded sequence g p of integers, with a designated integer stem( p ), beyondwhich g p is non-decreasing. f ∈ p if f ( n ) = g p ( n ) for n < stem( p ) and f ( n ) ≤ g p ( n ) otherwise. Notice that p ∩ q is either empty (if g p and g q through theirstems are incompatible) or is given by taking the larger of the two stems, thefunction up to that stem from the condition with the larger stem, and thepointwise minimum beyond that. Hence these open sets do form a basis. It issometimes easier to assume that g p (stem( p )) ≥ max { g p ( i ) | i < stem( p ) } . Theintuition is that once a certain value has been achieved there’s nothing to begained anymore by trying to restrict future terms from being that big. It isnot hard to see that that additional restriction does not change the topology.So whenever more convenient, a basic open set can be taken to be of this morerestrictive form. Theorem 3.2 T (cid:13) RPT.
Proof:
Suppose f ∈ p (cid:13) “For every permutation σ the series ( a σ ( n ) ) converges.”It suffices to find a neighborhood of f forcing an upper bound for Σ | a n | . We7 ωg p stem( p ) . . .Figure 3: An artist’s impression of an open set p .assume as usual that p is basic open and that g p (stem( p )) ≥ sup(rng( f )). So itsuffices to extend p to r forcing a bound for Σ | a n | , without altering the stemor the value g p (stem( p )) (i.e. stem( p ) = stem( r ) and g p = g r at their commonstem), as f will then be in r . We can also assume that each a n is rational, as a n could be replaced by a rational number (using Countable Choice, which holdsin this model [19]) sufficiently close that convergence will not be affected. Definition 3.3
A finite sequence of integers σ of length at least stem( p ) is compatible with p if for all i < stem( p ) σ ( i ) = g p ( i ) and for all i with stem( p ) ≤ i < length ( σ ) σ ( i ) ≤ g p ( i ) . For σ compatible with p, p ↾ σ isthe open set q ⊆ p such that stem( q ) = length ( σ ) , for i < stem( q ) g q ( i ) = σ ( i ) , and otherwise g q ( i ) = g p ( i ) . The following lemma is analogous in statement and proof to lemma 3.3 from[19], the proof of which was an extrapolation of some lemmas from an earliersection, which themselves were just extensions of the basic lemma about thismodel. All of which is meant to explain why the proof will not be repeated here.
Lemma 3.4
There is an open set q ⊆ p , with stem( q ) = stem( p ) and g q (stem( q )) = g p (stem( p )) , which determines the values of a n in the following sense: for every n ∈ N there is a length i n (increasing as a function of n ) such that, for all σ oflength i n compatible with q , q ↾ σ forces a (rational) value for a n , say r σ . Let q be as in the lemma. The members of q naturally form a tree T r q : thenodes are those finite sequences compatible with q , and the members of q arethose paths through the tree with bounded range. At height j ≥ stem( q ) of T r q , the amount of branching is g q ( j ) + 1. The nodes at height i n determinethe value of a n . We will have use for subsets of q the members of which haveranges that are uniformly bounded. (Such subsets are, of course, not open.)8hese subsets can be given as the set of paths through a subtree T r of T r q witha fixed bound on the ranges of its nodes, as follows. Definition 3.5
A tree
T r ⊆ T r q is bounded if there is a J such that for all σ ∈ T r and j < length ( σ ) σ ( j ) < J. The following is the analogue of [19]’s lemma 3.5.
Lemma 3.6
Let
T r ⊆ T r q be bounded. Then there is a bound B in the sensethat, for all σ ∈ T r of length some i n , Σ m ≤ n | r σ ↾ i m | ≤ B . Proof:
Say that τ ∈ T r is good if the conclusion of the lemma is satisfied for
T r ↾ τ (i.e. those nodes in T r extending τ ). Notice that if every immediateextension of τ is good then τ itself is good, by taking the maximum of thebounds witnessing the goodness of τ ’s extensions. So if the root of T r is bad(i.e. not good) then there is a path f through T r consisting of all bad nodes.Because
T r is bounded, f ∈ T .Define the permutation σ as follows. At stage n , we have inductively apermutation σ n of a natural number l n (with l = 0). By the choice of f , thereis an extension τ , with length some i k , of f ↾ i l n , such that Σ l n
0, and for all j there are nodes τ ∈ T r and ρ ∈ T r j extending τ such that, summing over the i m ’s between the lengths of τ and ρ , Σ | r ρ ↾ i m | is as close to ǫ as you want. Bya construction as in the previous lemma, a permutation σ could then be builtwith no condition forcing Σ n a σ ( n ) to converge. Hence B ∞ = B . Choose j sothat B j is within 1/2 of B . Let T r be T r j and B be B j .Continuing inductively, given T r s , build T r s +1 which allows nodes to takeon the value I + s past a certain point and has a lemma-induced least upperbound B s +1 no greater than 2 − s more than B s . Let T r ∞ be S s T r s . T r ∞ induces an open set which forces Σ n | a n | to be bounded by B + 1 . RPT may fail
We now present a topological model in which RPT is false. First we define theunderlying space T , which not surprisingly will involve reference to permuta-tions. By way of notation, for σ a permutation of ω , we think of σ ( n ) as the inte-ger in the n th slot. So applying σ to 0 , , , ... would produce σ (0) , σ (1) , σ (2) , ... .Applying σ to ( a n ) produces the sequence a σ (0) , a σ (1) , ... , for which we use thenotation ( a σ ( n ) ). Definition 4.1
Let T be the set of sequences ( a n ) which are eventually 0 andwhich sum to 0.An open set O is given in part by a finite sequence I n ( n < N ) of intervalsfrom R , thought of as approximations to an initial segment of ( a n ) ; that is,in order for ( a n ) to be in O , it is necessary that a n ∈ I n . Also, finitely manypermutation σ are given. Each such σ is associated with finitely many pairs ǫ, M , with ǫ > and M ∈ N . For ( a n ) to be in O , it must also be the casethat the partial sums Σ mn =0 a σ ( n ) ( m > M ) are less than ǫ in absolute value. Inwords, after permuting ( a n ) by σ , the series must have converged to within ǫ by M . Theorem 4.2 T (cid:13) ¬ RPT.
Proof:
The generic G induces the generic sequence of reals ( g n ), with O (cid:13) “ g n ∈ I n . ” Also, T (cid:13) “( g n ) is total,” since, for every ( a n ) ∈ T and k , theopen set determined by any ( I , ..., I k ) , with a n ∈ I n , and no σ ’s, forces “ g k isdefined.” The generic sequence ( g n ) will witness the failure of RPT.First, we want to see that for every ground model permutation σ, Σ g σ ( n ) converges. Notice that for every ( a n ) ∈ T and ǫ > M such that theopen set determined by associating ǫ and M to σ contains ( a n ), for the simplereasons that Σ a n converges to 0 and that ( a n ) is eventually 0: just choose M so large so that all non-0 entries of ( a n ) have already occurred in a σ ( n ) by the M th entry there. It follows immediately that T (cid:13) “Σ g ˆ σ ( n ) = 0 . ”As for arbitrary permutations, suppose O (cid:13) “ σ is a permutation.” We claimthat no extensions of O can force different values for any σ ( n ); if that is so,then O itself forces all of the values of σ ( n ). To see the claim, let ( a n ) and ( b n )be two members of O . Consider the continuous family of sequences ( c n ) t := t ( b n ) + (1 − t )( a n ) , ≤ t ≤ . Notice that ( c n ) = ( a n ), ( c n ) = ( b n ), and,for all t , ( c n ) t ∈ O , since the constraints imposed by O are linear. For anyvalue of t of t , some neighborhood of ( c n ) t forces a value for σ ( n ). Any suchneighborhood forces the same value for all ( c n ) t for t in a neighborhood of t ;that is, those t ’s that force any fixed value for σ ( n ) form an open set. Since[0,1] is connected, all ( c n ) t ’s must have neighborhoods forcing the same value for σ ( n ). Hence the values of σ ( n ) are all determined by O . As the forcing relationis definable in the ground model, these values form a ground model permutation.Since all permutations are equal locally to ground model permutations, by theprevious paragraph, Σ g σ ( n ) converges for all σ .10t remains only to show that T (cid:13) “( g n ) diverges absolutely.” Consider any( a n ) ∈ O . There is a K such that, for any σ which is a part of O ’s definition, thepartial sums beyond K of the permuted sequence are 0: for k > K, Σ kn =0 a σ ( n ) =0 . (It suffices to take K = max { σ − ( n ) | a n = 0 and σ is constrained by O } .)Choose some i > σ ”( K ) (the image of K under σ ), and change a i to be δ , where δ is less than all of the ǫ -constraints imposed by O . Iterate to find another safespot j , and change a j to be − δ . This can be iterated to get the sum of theabsolute values to be as big as you want. Hence O does not force any bound onthe sum of the absolute values. Following a definition of Fred Richman (private notes), we say that a sequence ofreals x n is partially Cauchy if, for all increasing h , lim n diam( x n , x n +1 , ..., x h ( n ) ) =0 . (The diameter of a set in a metric space is the supremum of the distancesbetween members of the set, taken two at a time, if this supremum exists. If theset is finite, as it is here, the supremum does exist.) Richman showed, amongother things, that, under BD-N, every partially Cauchy sequence is Cauchy. Inthis section we show that BD-N is not necessary for this, in that the latter resultdoes not imply BD-N. In the next section, we show that the result in questionis not provable in basic set theory alone.Let T be the space from [19], reviewed in section 3 above, the model overwhich falsifies BD-N. As in the other cases, we have: Theorem 5.1 T (cid:13) “Every partially Cauchy sequence is Cauchy.” Proof:
Suppose p (cid:13) “( x n ) is partially Cauchy.” In a personal communication,Fred Richman studied several notions of Cauchyness, all akin to partiality, andshowed essentially that any sequence which is Cauchy in any sense (partially,weakly, almost) is the sum of a Cauchy sequence (in as strong a sense as youlike) and a rational sequence which is Cauchy in the same sense as the startingsequence. His proof uses Countable Choice, which is no problem here, as T (cid:13) Dependent Choice (see [19]), which implies Countable Choice, and is otherwisestraightforward. The upshot of this is that we can assume that each x n isrational.For every f ∈ p we must find a neighborhood q of f forcing ( x n ) to beCauchy. So let T (cid:13) ǫ > . It suffices to assume ǫ is rational, so we do not haveto deal with conditions forcing ǫ to have an approximate value. We assume asusual that p is basic open and that g p (stem( p )) ≥ sup(rng( f )). So it suffices toextend p to r forcing an appropriate value N for ǫ , without altering the stem or11he value g p (stem( p )) (i.e. stem( p ) = stem( r ) and g p (stem( p )) = g r (stem( r ))),as f will then be in r .As in section 3 above, we state without proof: Lemma 5.2
There is an open set q ⊆ p , with stem( q ) = stem( p ) and g q (stem( q )) = g p (stem( p )) , which determines the values of x n in the following sense: for every n ∈ N there is a length i n (increasing as a function of n ) such that, for all σ oflength i n compatible with q , q ↾ σ forces a (rational) value for x n , say r σ . With terminology and notation as in section 3 above, we have the followinganalogue of lemma 3.6.
Lemma 5.3
Let
T r ⊆ T r q be bounded, and δ > be rational. Then there is anatural number k such that, for all m, n ≥ k, m < n, and σ m ⊆ σ n of lengths i m and i n respectively, | x σ m − x σ n | < δ. Proof:
Say that τ ∈ T r is good if the conclusion of the lemma is satisfied for
T r restricted to τ . Notice that if every immediate extension of τ is good then τ itself is good, by taking the maximum of the k ’s witnessing the goodness of τ ’s extensions. So if the root of T r is bad (i.e. not good) then there is a path f through T r consisting of all bad nodes. Because
T r is bounded, f is a memberof the topological space T .Define the function h as follows. Given k , let m and n be as given by thebadness of f ↾ k (i.e. there are nodes σ m ⊆ σ n in the tree beneath f ↾ k with | x σ m − x σ n | ≥ δ ) . Let h ( k ) be at least as big as that n (and, for k >
0, biggerthan h ( k − h witnesses that ( x n ) is not partially Cauchy, as anyneighborhood of f must contain all of T r restricted to some initial segment of f . To complete the proof, let T r be the subtree of T r q of all nodes with valuesless than or equal to I := g q (stem( q )). Apply the lemma with T r for T r and ǫ/ δ . Let k be the integer produced by the lemma. Let T r ⊆ T r q extend T r by allowing nodes that may take on the value I + 1 at positions beyond i k .Again apply the lemma, with T r for T r and ǫ/ δ , to produce k , whichcan be taken to be larger than k . More generally, at stage s , let T r s ⊆ T r q extend T r s − by allowing the value I + s − i k s − , and let k s > k s − be the result of applying the lemma to T r s and ǫ/ s .To finish the definition of r , we must just give g r , and show that beyond N := k r forces the values of ( x n ) to be within ǫ of one another. As motivation,consider x N itself, as compared with x m for some larger m (larger than N ). Ifthe value of x m is determined by some node in T r , we’re golden – even betterthan golden, x m being within ǫ/ x N . But once we go into T r , all bets areoff. Hence we want to restrict T r r to equal T r at least for nodes up to length i k . If m ≤ k , then x m is determined by T r , and we’re done. For m > k ,at least we can bound | x N − x k | by ǫ/
2, and work on bounding | x k − x m | by ǫ/
4, which would suffice. While working on the latter, we can now afford to12e in the tree
T r . By continuing to expand the tree in which we work in thisfashion, we can guarantee that g r be unbounded, while still remaining within ǫ of x N .So let g r between i k s and i k s +1 have the value I + s −
1. This makes g r be unbounded, and forces the values of ( x n ) beyond N to be within ǫ of oneanother, by the argument sketched in the previous paragraph. As usual, in the coming topological model the generic will be a partially Cauchysequence which is not Cauchy. We start by defining the underlying topologicalspace.
Definition 6.1
Let T be the space of all Cauchy sequences ( x n ) . A basic openset is given by finitely many pieces of information. One is a finite sequence ofintervals I n ( n < k ) . A sequence ( x n ) satisfies the requirements I n ( n < k ) if forall k < n x n ∈ I n . In addition, to each of finitely many functions h and rationalnumbers ǫ > is associated a natural number n h,ǫ . A sequence ( x n ) satisfiesthat requirement if for all n ≥ n h,ǫ diam( x n , x n +1 , ..., x h ( n ) ) < ǫ . The basic opensets as given are closed under intersection, and so form a basis. Theorem 6.2 T (cid:13) “Not every partially Cauchy sequence is Cauchy.” Proof:
The generic induces a sequence ( g n ) of reals. For every ( x n ) ∈ T ,ground model function h from N to itself, and rational ǫ >
0, there is a neigh-borhood O of ( x n ) assigning a value to n h,ǫ . Then O (cid:13) “ if n ≥ n h,ǫ thendiam( g n , g n +1 , ..., g h ( n ) ) < ǫ ”. Furthermore, in this model, all functions from N to itself are ground model functions, by the same argument as for permutationswith respect to the RPT. Hence the generic sequence is partially Cauchy.To see that the generic sequence is not itself Cauchy, let O be an open setand N an arbitrary natural number, which without loss of generality is less than k , the length of O ’s sequence of intervals. We will show that O does not forcethat beyond N the values of the generic always stay within 1 of each other,which suffices.To simplify on notation (and thinking), we can strengthen (i.e. shrink) O byreducing to one h (by taking the pointwise maximum of the finitely many h ’s)and one ǫ (by taking the smallest). To be sure, this summary requirement doesnot capture all of the actual requirements present before this simplification, asthere may have been demands made on intervals starting at n < n h,ǫ . But thosedemands are only finite in number, and can be satisfied by choosing I n ( n < n h,ǫ )to be sufficiently small.Pick a sequence of values x n of length j := max( n h,ǫ , k ) which is an initialsegment of a member of O . Extend that finite sequence to have a value just13nder x j − + ǫ/ j through h ( j ). Extend again to have a valuejust under x h ( j ) + ǫ/ h ( j ) + 1 through h ( h ( j ) + 1). Extend again,by adding almost another ǫ/ s , un-til h ( s ). Continue this process for at least 2 /ǫ + 1-many steps, at which pointpick the Cauchy sequence which is constant from that point on. The upshot is,beyond N the sequence has increased by more than 1. References [1] Michael Beeson, “The nonderivability in intuitionistic formal systems oftheorems on the continuity of effective operations,”
Journal of SymbolicLogic , v. 40 (1975), p. 321-346[2] Josef Berger and Douglas Bridges, “A Fan-theoretic equivalent of the an-tithesis of Speckers Theorem,”
Proc. Koninklijke Nederlandse Akad.Wetenschappen (Indag. Math., N.S.), v. 18 (2007), p. 195-202[3] Josef Berger and Douglas Bridges, “The anti-Specker property, a Heine-Borel property, and uniform continuity,”
Archive for MathematicalLogic , v. 46 (2008), p. 583-592[4] Douglas Bridges, “Constructive notions of equicontinuity,”
Archive forMathematical Logic , v. 48 (2009), p. 437-448[5] Josef Berger and Douglas Bridges, “Rearranging series constructively,”
Journal of Universal Computer Science , v. 15 (2009), p. 3160-3168[6] Josef Berger, Douglas Bridges, Hannes Diener and Helmut Schwichtenberg,“Constructive aspects of Riemann’s permutation theorem for series,” sub-mitted for publicationnnnn[7] Josef Berger, Douglas Bridges, and Erik Palmgren, “Double sequences, al-most Cauchyness, and BD-N,”
Logic Journal of the IGPL , v.20 (2012),p. 349-354, doi: 10.1093/jigpal/jzr045[8] Douglas Bridges, “Inheriting the anti-Specker property”,
DocumentaMathematica , v. 15 (2010), p. 9073-980[9] Douglas Bridges, Hajime Ishihara, Peter Schuster, and Luminita Vita,“Strong continuity implies uniformly sequential continuity,”
Archive forMathematical Logic , v. 44 (2005), p. 887-895[10] Robin J. Grayson, “Heyting-valued models for intuitionistic set theory,” in
Applications of Sheaves , Lecture Notes in Mathematics, vol. 753 (eds.Fourman, Mulvey, Scott), Springer, Berlin Heidelberg New York, 1979, p.402-414 1411] Robin J. Grayson, “Heyting-valued semantics,” in
Logic Colloquium ’82 ,Studies in Logic and the Foundations of Mathematics, vol. 112 (eds. Lolli,Longo, Marcja) , North-Holland, Amsterdam New York Oxford, 1984, p.181-208[12] Hajime Ishihara, “Continuity and nondiscontinuity in constructive mathe-matics,”
Journal of Symbolic Logic , v. 56 (1991), p. 1349-1354[13] Hajime Ishihara, “Continuity properties in constructive mathematics,”
Journal of Symbolic Logic , v. 57 (1992), p. 557-565[14] Hajime Ishihara, “Sequential continuity in constructive mathematics,” in
Combinatorics, Computability, and Logic (eds. Calude, Dinneen, andSburlan), Springer, London, 2001, p. 5-12[15] Hajime Ishihara and Peter Schuster, “A Continuity principle, a versionof Baire’s Theorem and a boundedness principle,”
Journal of SymbolicLogic , v. 73 (2008), p. 1354-1360[16] Hajime Ishihara and Satoru Yoshida, “A Constructive look at the complete-ness of D ( R ),” Journal of Symbolic Logic ∼ streicher/THESES/lietz.pdf.gz;see also “Realizability models refuting Ishihara’s boundedness principle,”joint with Thomas Streicher, submitted for publication[18] Robert Lubarsky, “Geometric spaces with no points,” Journal of Logicand Analysis , v. 2 No. 6 (2010), p. 1-10, http://logicandanalysis.org/,doi: 10.4115/jla2010.2.6[19] Robert Lubarsky, “On the Failure of BD-N and BD, and an application tothe anti-Specker property,”