Realizability Models Separating Various Fan Theorems
aa r X i v : . [ m a t h . L O ] O c t Realizability Models Separating Various FanTheorems
Robert S. Lubarsky ⋆ and Michael Rathjen Dept. of Mathematical SciencesFlorida Atlantic UniversityBoca Raton, FL 33431, USA
[email protected] Dept. of Pure MathematicsUniversity of LeedsLeeds LS2 9JT, England [email protected]
Abstract.
We develop a realizability model in which the realizers arethe reals not just Turing computable in a fixed real but rather the reals ina countable ideal of Turing degrees. This is then applied to prove severalseparation results involving variants of the Fan Theorem.
Keywords: realizability, Kripke models, Fan Theorem, Weak K¨onig’sLemma, Weak Weak K¨onig’s Lemma
AMS 2010 MSC:
Certain constructions in computability theory lend themselves well to realiz-ability, the latter being based on an abstract notion of computation. A coarseexample of this is the notion of a Turing computable function itself, as the col-lection of Turing machines makes an applicative structure and so provides anexample of realizability. This model is closely tied to Turing computation, nat-urally enough, and so provides finer examples. Consider Weak K¨onig’s Lemma,WKL, which is among constructivists more commonly studied in its contrapos-itive form, the Fan Theorem FAN. (For background on realizability, the FanTheorem, and constructive mathematics in general, there are any of a numberof standard texts, such as [3,23,25].) Kleene’s well-known eponymous tree is acomputable, infinite tree of binary sequences with no computable path. In thecontext of reverse mathematics, this shows that RCA does not prove WKL.Within the realizability model, the same example shows IZF does not proveFAN.Perhaps a word should be said on the choice of the ground theory. For theclassical theory, it’s RCA , while for the constructive, it’s IZF. The former isnotably weak, the latter strong. Why is that? And why those theories in partic-ular? This is not the place to discuss the particular choice of RCA . As for IZF, ⋆ I would like to thank Wouter Stekelenburg, Thomas Streicher, and Jaap van Oostenfor useful discussion during the development of this work. its use is of secondary importance. The point is that much of reverse classicalmathematics is to show the equivalence of various principles, for which a weakerbase theory provides a stronger theorem. Even for independence results, whichon the surface would be better over a stronger base, are often of the form thata weaker theory does not imply a stronger one, where the latter easily impliesthe former; clearly here, the base theory is the weaker of the two. Also for thosecases where the independence result desired is an incomparability, the principlesin question are all weak set existence principles, weak in the sense that they use atiny fragment of ZF, and hence a weak base theory is needed, to keep the theoriesin question from being outright provable. In contrast, reverse constructive math-ematics studies not set existence principles, but rather logical principles. Insteadof fragments of ZF, the subject is fragments of Excluded Middle. Especially whendiscussing independence, when a stronger base theory gives a stronger theorem,in order to highlight that it really is the logic that is up for grabs, and not setexistence, strong principles of set theory are taken as the base theory. IZF isused here, since it is the simplest and most common constructive correlate toZF, the classical standard. Even for equivalence theorems, there would still be atendency to work over IZF, since the degree to which a result depended on theIZF axioms is the degree to which the result is ultimately classical. In practice,if any theorem needs less than full IZF, what is actually used could be read offfrom the proof anyway.Returning to realizability, the picture is not quite so rosy when it comesto other constructions. A case in point is the distinction between WKL andWWKL, Weak Weak K¨onig’s Lemma. WWKL states that for every binary tree(of finite, 0-1 sequences) with no path there is a natural number n such that atleast half the sequences of length n are not in the tree. This principle has beenstudied in reverse mathematics, both classical [22] and constructive [20]. Yu andSimpson [26] showed that WWKL does not imply WKL (over RCA ). That’snot so simple as merely taking the computable sets; while that would falsifyWKL, as discussed above, it invalidates WWKL too. What they do is to extendthe computable sets by a carefully selected real R (implicitly closing underTuring reducibility) which provides a path through all the “bad” trees while notdestroying the Kleene tree counter-example. That’s not enough, though, becausethe construction of a counter-example to WWKL relativizes. So while R killsoff the bad computable trees, it introduces new bad trees of its own. Hencethe construction must be iterated: R , R , ... In the end, the union of the (realscomputable in the) R n ’s suffices.The statement of WWKL carries over just fine to a constructive setting,where we will call it the Weak Fan Theorem W-FAN, as well as the questionof whether W-FAN implies FAN. For better or worse, the construction thoughdoesn’t. One might first think to use the Yu-Simpson set of reals as the set ofrealizers. An immediate problem is that we need an applicative structure: therealizers need to act on themselves. It is immediate and routine to view thesereals as functions from the naturals to the naturals – that’s a trivial identificationthese days. It doesn’t help though. If the realizers are those functions, well, those functions act on naturals, not on functions. What we would need would be integercodes for those functions. The realizers from this point of view would be thatset of naturals, which as need be could be taken to be functions. But here’s theproblem: what code do you give a function which showed up in some increasingtower? If you were looking at those functions computable in some fixed oracle,you could consider all the naturals as each coding such a machine. If insteadyour oracle is continually changing, it’s not clear what to do and still maintainan applicative structure. The fixes we tried did not work, as we were warned.The same issue comes up with another distinction around FAN, namely thedistinction between FAN and WKL. In order to show their inequivalence, onemight want to come up with a model of FAN falsifying WKL. This has alreadybeen done using K realizability, Kleene’s notion of functional realizability. Itis another matter to prove this theorem via K realizability. The problem is asabove: every Turing degree has an infinite binary tree with no path of the sameor lesser degree (the Kleene tree relativized to that degree). One could take aset of degrees, any of the trees of which have a path in some other degree. Theproblem here is how to turn that into a realizability structure.The goal of this paper is a realizability model in which the realizers codefunctions from the natural numbers to themselves with no highest Turing degreeamong them. As corollaries to this method we get the two results just cited. The main idea here is to build a Kripke model, and then within that a realiz-ability model, which has sometimes been called relative realizability [25]. Thiskind of construction was apparently first suggested by de Jongh [10]. Variantshave been used by several people: having the Kripke partial order consist of onlytwo points and the realizers be at ⊥ certain computable objects which are theninjected into a full set of realizers [1,2], or using instead of Kripke semanticseither double-negation [14] or a kind of Beth [24] semantics. For more detail onall of this, see the last two sections of [25].To help keep things simple, we assume ZFC in the meta-theory. For most ofthis work, neither classical logic nor the Axiom of Choice is necessary, but wewill not be careful about this.Let the underlying Kripke partial order be ω <ω . Let M be the full Kripkemodel built on that p.o. Intuitively, that means throw in all possible sets. Moreformally, a set in the model is a function f from ω <ω to the sets of the model(inductively) which is non-decreasing (i.e. if σ ⊆ τ then f ( σ ) ⊆ f ( τ )). Equalityand membership are defined by a mutual induction. On general principles, M | =IZF. Moreover, the ground model V has a canonical image in M: given x ∈ V ,let ˇ x be such that ˇ x ( σ ) = { ˇ y | y ∈ x } . We often identify x with ˇ x , the contexthopefully making clear whether we’re in V or in M. For slightly more detail onthe full model, defined over any partial order, see for instance [17].Within M, we will identify a special set R of natural numbers, based on aprior sequence R n of reals ( n ∈ ω ). We assume the R n are of strictly increasing Turing degree. At node ⊥ = hi , R looks empty: ⊥ 6| = s ∈ R ; equivalently, R ( hi ) = ∅ . Suppose inductively R ( σ ) is defined, where σ has length n . Let R in list all the reals that differ from R n in finitely many places. Let R ( σ ⌢ i ) be R ( σ ) ∪ {h n, s i | s ∈ R in } . In words, at level n + 1, beneath each node on level n ,put into the n th slice of R all of the finite variations of R n , spread out amongall the successors. So R is a kind of join of the R n ’s, just not all at once.Because R is (in M) a real, it makes sense to use R as an oracle for Turingcomputation. At ⊥ , if a computation makes any query s ∈ R of R , there are somenodes at which s is in R and others where it is not, so the oracle cannot answerand the computation cannot continue. This follows from the formal model oforacle computability: a run of an oracle machine is a tuple of natural numberscoding a correct computation; the rule for extending a tuple when the last entryis an oracle call is that the next entry must contain the right answer; if thereis, at a node, no right answer, then there can be no extending tuple. Hence theonly convergent computations are those that make no oracle calls, and the only R -computable functions are the computable ones. More generally, at node σ oflength n , any query of the form h k, s i ∈ R with k ≥ n will be true at some futurenodes and false at others, hence unanswerable at σ . The computable functionsat σ are those computable in R n − .In M, let App be the applicative structure of the indices of functions com-putable in R (using, of course, the standard way of turning such indices into anapplicative structure). In M, let M[App] be the induced realizability model. Ongeneral principles, M[App] | = IZF. The natural numbers of M[App] can be iden-tified with those of M, so any set of such in M[App] can be identified with one inM. Furthermore, at any node, a decidable real in one structure corresponds to adecidable real in the other, and that can be identified with a real in the ambientclassical universe. Henceforth these various reals will not even be distinguishednotationally. For instance, if σ | = “ M [ App ] | = “ T ⊆ ω is decidable” ” then wemight refer to the real T in V.For notational convenience, we will abbreviate σ | = “ M [ App ] | = φ ” as σ | = App φ. Lemma 1.
For σ of length n and R a real, σ | = App “ X is decidable” iff X isTuring computable in R n − .Proof. The statement “ X is decidable” is ∀ m ∈ ω m ∈ X ∨ m X. A realizer r of the latter would be a function that, on input m , decides whether m is in orout of X . If in the course of its computation r asked the oracle any question ofthe form h k, s i with k ≥ n then the computation would not terminate at σ . So r can access only R n − , making X computable in R n . The converse is immediate. Lemma 2. If σ | = App “ X is an infinite branch through the binary tree” then σ | = App “ X is decidable.”Proof. To be an infinite branch means for every natural number m there is aunique node of length m . The realizer that X is an infinite branch has to producethat node given m . Often people are concerned about the use of various choice principles. Theindependence results presented here are that much stronger because DependentChoice holds in our models.
Proposition 1. hi | = App
DCProof.
The same proof that DC holds in standard Kleene K realizability workshere. When adapting the classical results to the current setting, we need an additionalstipulation. All of the trees, and hence principles, we consider will be decidable:for all binary sequences b and trees T , either b ∈ T or b T . So, for instance,instead of the full Fan Theorem FAN, we will be considering the Decidable FanTheorem: if a decidable tree in { } N has no infinite path, then the tree is finite.Also, Weak FAN, also known as WWKL, when applied to decidable trees, wouldread: if a decidable tree in { } N has no infinite path, then there is an n suchthat at least half of { } n is not in the tree.This brings us to an annoying point about notation. Decidable FAN has beenreferred to in various places as D-FAN, FAN D , ∆ -FAN, and FAN ∆ . So notationfor Weak Decidable FAN could be any of those, with a “W” stuck in somewhere.To make matters worse, even though the statement of Weak FAN is a weakeningof FAN and not of WKL, the name WWKL for it is already established in theclassical literature, and so one could make a case to stick with it, and insertdecidability (“D” or “ ∆ ”) somewhere in there. These same considerations applyto other variants of FAN, whether already identified (c-FAN, Π -FAN) or not.Whatever we do here will likely not settle the matter. Still, we have to choosesomething. It strikes us as confusing to distinguish between FAN and WKL, andthen call a variant of FAN by a variant of WKL. Also, what if somebody somedaywants to study the contrapositive of “WWKL”? Hence we stick with the nameW-FAN. As for how to get in the decidability part, we choose the option that’sthe easiest to type: W-D-FAN.Returning to the matter at hand, Yu-Simpson [26] construct a sequence X n of reals of increasing Turing degree such that: i ) if T is a tree computable in X n the branches through which form a set ofpositive measure, then a path through T is computable in X n +1 , and ii ) no path through the Kleene tree is computable in any X n .We apply the construction from the previous section, with R n set to X n .From this, the following lemmas are pretty much immediate. Lemma 3. hi | = App
W-D-FANProof.
At any node, a decidable tree T is computable in some X n . If in V themeasure of the branches through T were positive, then there would be a branchcomputable through X n +1 . So no node could force that there are no branchesthrough T . To compute a level at which half the nodes are not in T , just gothrough T level by level until this is found. Lemma 4. hi | = App ¬ D-FANProof.
The Kleene tree provides a counter-example.While we expect that even full W-FAN does not imply D-FAN, this modeldoes not satisfy W-FAN. To see that, recall that any path through the binary treeis decidable, hence computable in some X n . There are only countably many suchpaths. It is easy in V to construct a tree avoiding those countably many pathswith measure (of the paths) being as close to 1 as you’d like. The internalizationof such a tree in M[App] will not be decidable, but will be internally a tree withno paths. The distinction between FAN and WKL is a strange case. Their relation is thatWKL implies FAN, but not the converse. With some exaggeration, it seems asthough everyone knows that but no one has proven it. At the very introductionof non-classical logic, Brouwer himself must have realized this distinction, as hemade a conscious choice which variant of this class of principles he accepted.Moreover, while he accepted FAN (having proven it from Bar Induction), it iseasy to see that WKL implies LLPO, which Brouwer rejected. So while Brouwerdid not provide what we would today consider a model of FAN + ¬ WKL, wewould like to honor him in the style of the Pythagoreans by attributing thisresult to him, whatever may actually have been going through his mind.Such models have since been provided, for instance by Kleene, using hisfunctional realizability K [16,21]. However, in neither of those sources is itmentioned that WKL fails. In [5], both FAN and WKL are analyzed into con-stituent principles, it is shown that WKL’s components imply the correspondingFAN components, and it is nowhere stated that the converse does not hold. In[7], equivalents are given for what is there called FAN and WKL, although theirFAN is actually D-FAN, and it is at least asked how much stronger WKL is thanFAN. The one proof we have been able to find of some fan theorem not implyingWKL is in [19], where once again the fan principle used is D-FAN. For whatit’s worth, that argument, like ours, uses relative realizability [8], albeit with K realizability.Below we give a full proof that FAN does not imply WKL. We would beinterested in hearing of other extant proofs of such, and would find it amusingif there were none. What might be new here, if anything, is not the result itself,but rather the methodological point that this model is based on K . That is,while K is usually used to falsify even D-FAN, its variant below validates fullFAN. Theorem 1. (Brouwer) FAN does not imply WKL. Thanks are due here to Hannes Diener for first pointing this out to us and ThomasStreicher for useful discussion.
Proof.
Take a countable ω -standard model of WKL (see [22], ch. VIII). Thereis a sequence X n of reals of increasing Turing degree such that the reals in thismodel are exactly those computable in some X n . This induces a model M[App]as in section 2 above (with R n being set to X n ).To see that WKL fails, suppose to the contrary σ | = App “ n (cid:13) r W KL ”. Soif σ | = App “ e (cid:13) r T is an infinite binary tree” then, at σ, { n } ( e ) must computea path through T . But a path through a tree is not computable in the tree, asis standard, by considering the Kleene tree. This shows moreover that WKL fordecidable trees fails.To show FAN, suppose at some node σ in the Kripke partial order r realizesthat B is a bar. We must show how to compute a uniform bound on B . To do this,we will build a decidable subset C of B . Inductively at stage n , suppose we havedecided C on all binary sequences of length less than n . Consider each binarysequence ¯ b of length n (except if ¯ b extends something in C of shorter length, inwhich case what happens with ¯ b just doesn’t matter anymore). Consider the path P ¯ b which passes through ¯ b and is always 0 after that. Applying r to P ¯ b producesa sequence b + in B on P ¯ b . If b + has length at most n , include in C all extensionsof b + of length n , else just include b + in C . After doing this for all ¯ b of length n ,anything of length n not put into C is out of C . This procedure terminates onlywhen we have a uniform bound on C , hence on B . If this never terminates, wehave a decidable, infinite set of binary sequences not in C computable from r .Hence at any child τ of σ there will be an infinite path P avoiding C . Applying r to P produces an initial segment of P in B , say b . This computation itselfused only an initial segment of P , say c . Letting n be the maximum length of b and c , at stage n no initial segment of either has been put into C , by the choiceof P . So the procedure would consider the path through b and c which is all 0safterwards. This would then put the longer of b and c into C , contradicting thechoice of P . So this procedure must terminate, producing a bound for B .
1. The second construction was developed for an entirely different purpose.One way of stating FAN is that every bar is uniform. Weaker versions ofFAN can be developed by restricting the bars to which the assertion applies. Forinstance, Decidable FAN, D-FAN, states that for every decidable set B (i.e. forall b either b ∈ B or b B ), if B is a bar then B is uniform. Constructively,decidability is a very strong property; in fact, it is the strongest hypothesis ona bar yet to be identified. D-FAN is trivially implied by FAN; it has long beenknown that D-FAN is not provable in IZF (via the Kleene tree, described in anystandard reference, such as [3,23]; see [18] for a different proof). A somewhatmilder restriction on a bar B is that it be the intersection of countably manydecidable sets; that is, B is Π definable. Between decidable and Π bars arec-bars: if there is a decidable set B ′ such that b ∈ B iff for every c extending b c ∈ B ′ , then B is called a c-set, and if it’s a bar to boot then it’s called a c-bar. Often this definition seems at first unnatural and rather technical. All that matters at the moment is that this is a weaker condition than decidability: everydecidable bar is a c-bar. c-FAN is the assertion that every c-bar is uniform. Suchprinciples occur naturally in reverse constructive mathematics [15,4,12], and areall inequivalent [18].The first proof that D-FAN does not imply c-FAN, by Josef Berger [6], wentas follows. Classically, for X any collection of bars, X-FAN and X-WKL areequivalent (as contrapositives). Furthermore, the Turing jump of a real R canbe coded into a tree computable in R, so that c-WKL implies that jumps alwaysexist. Hence over RCA c-WKL implies ACA. D-FAN and WKL (which, in thesetting of limited comprehension, is just D-WKL) are equivalent. So if D-FANimplied c-FAN, constructively or classically, then WKL would imply ACA, whichis known not to be the case [22].A limitation of this argument is that it works over a very weak base theory. Itleaves open the question of whether D-FAN implies c-FAN over IZF. While thishas been settled [18], a question of method still remains open. Could Berger’sargument be re-cast to provide an independence results over IZF? The obviousplace to look seemed to be a model of WKL + ¬ ACA, using the functions there,which necessarily have no largest Turing degree, as realizers. Our analysis ofsuch a model did not achieve that goal. Is there another way of turning such amodel into a separation of D- and c-FAN?More generally, could there be any realizability model separating D- andc-FAN? All of the realizability models we know about either falsify D-FAN orsatisfy full FAN. Perhaps that’s because of the difficulty of realizing that some-thing is a bar. That is, to falsify any version of the FAN, one might well wantto provide a counter-example, which would be a non-uniform bar. If a bar isnot uniform, realizing the non-uniformity would typically be trivial, as nothingcould realize that it is uniform, which suffices. Realizing that a set is a bar isdifferent: given a binary path, you’d have to compute a place on that path andrealize that that location is in the alleged bar. If this set is decidable, that’s easy:continue along the path, checking each node on the way, until you’re in it. If theset cannot be assumed decidable, it is unclear to us how to realize that it’s abar. This is something we would like to see: a way of realizing a non-decidableset being a bar.2. The differences among D-FAN, c-FAN, Π -FAN, and FAN have to do withthe hypothesis; they apply to different kinds of bars. In contrast, the differencebetween FAN and W-FAN has to do with the conclusion, with whether the baris uniform or half-uniform, to coin a phrase. So these variants can be mixed andmatched. There are D-FAN, c-FAN, Π -FAN, FAN, and also W-D-FAN, W-c-FAN, W- Π -FAN, and W-FAN. Clearly any version of FAN implies its weakcousin. Other than that, we conjecture there is complete independence betweenthe variants of FAN and the variants of Weak FAN. This is, we conjecture W-FAN does not imply D-FAN, and conjoined with D-FAN does not imply c-FAN,and so on. Furthermore, we expect that D-FAN, while of course implying W-D-FAN, does not imply W-c-FAN, and c-FAN does not imply W-FAN, and so onfor other variants that might appear. References
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