aa r X i v : . [ m a t h . L O ] O c t Separating the Fan Theorem and Its Weakenings
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] 18, 2018
Abstract
Varieties of the Fan Theorem have recently been developed in reverseconstructive mathematics, corresponding to different continuity princi-ples. They form a natural implicational hierarchy. Some of the implica-tions have been shown to be strict, others strict in a weak context, andyet others not at all, using disparate techniques. Here we present a familyof related Kripke models which separates all of the as yet identified fantheorems. keywords: fan theorems, Kripke models, forcing, non-standard models,Heyting-valued models
AMS 2010 MSC:
To be able to talk about fans, Cantor space, and similar objects properly, we willstart by introducing some notation. The space of all infinite binary sequences,endowed with the standard topology (wherein a basic open set is given by a finitebinary sequence), will be denoted by 2 N ; the set of all finite binary sequenceswill be denoted by 2 ∗ . The concatenation of u, v ∈ ∗ will be denoted by u ∗ v .For α ∈ N and n ∈ N , the first n elements of α form a finite sequence denotedby αn . A subset B ⊆ ∗ is called a bar if ∀ α ∈ N ∃ n ∈ N ( αn ∈ B ) , and a bar is called uniform if ∃ n ∈ N ∀ α ∈ N ∃ m n ( αm ∈ B ) . Notice that if a bar B is closed under extensions, that is if ∀ u ∈ ∗ ( u ∈ B = ⇒ ∀ v ∈ ∗ u ∗ v ∈ B ) , ∃ n ∈ N ∀ α ∈ N ( αn ∈ B ) . Not all of the bars we consider will be closed under extensions.There are currently four versions of Brouwer’s Fan Theorem in common use.All of them enable one to conclude that a given bar is uniform. The differencesamong them lie in the definitional complexity demanded (as an upper bound)of the bar in order for the theorem to apply to it, which ranges from the verystrongest requirement to no restriction on the bar at all. A bar C ⊂ ∗ is decidable if it is decidable as a set: ∀ u ∈ ∗ u ∈ C ∨ u C. A bar C ⊂ ∗ is called a c -bar if there exists a decidable set C ′ ⊂ ∗ such that u ∈ C ⇐⇒ ∀ v ∈ ∗ ( u ∗ v ∈ C ′ ) . A bar B ⊂ ∗ is called a Π -bar if there exist a decidable set S ⊂ ∗ × N suchthat u ∈ B ⇐⇒ ∀ n ( u, n ) ∈ S . (The Π n -nomenclature alludes to the arithmetical hierarchy in computabilitytheory.) We can now state four commonly used versions of the Fan Theorem.FAN ∆ : Every decidable bar is uniform.FAN c : Every c -bar is uniform.FAN Π : Every Π -bar is uniform.FAN full : Every bar is uniform.Notice that every decidable bar can be taken to be closed under extensions;that is, the closure of a decidable bar under extension is still decidable. If thereis no restriction on the definability of a bar, then every bar can be taken to beso closed, by working with the closure of any given bar. Every c -bar is alreadyclosed under extension. In contrast, Π -bars seemingly cannot be replaced bytheir closures while remaining Π .By way of motivation, these principles were developed within reverse con-structive mathematics, because they are equivalent with certain continuity prin-ciples. In particular, over a weak base theory, FAN ∆ is equivalent with theassertion that every uniformly continuous, positively valued function from [0,1]to R has a positive infimum [8], FAN c with the uniform continuity of everycontinuous f : 2 N → N [2], and FAN Π with the uniform equicontinuity of everyequicontinuous sequence of functions from [0,1] to R [6].The following implications hold trivially [2, 5] and over a weak base theory:FAN full = ⇒ FAN Π = ⇒ FAN c = ⇒ FAN ∆ . One naturally wonders whether any of the implications can be reversed, includ-ing whether FAN ∆ is outright provable in constructive set theory. Some suchnon-implications have already been determined. • It is well-known (see [1] for instance) that FAN ∆ is not provable, viarecursive realizability. That is, there is an infinite (Turing) computable2ub-tree of 2 ∗ with no infinite computable branch, which fact translates toa failure of FAN ∆ under IZF (Intuitionistic ZF, the constructive correlateto classical ZF) via recursive realizability, and also to the independence ofWKL (Weak K¨onig’s Lemma) over RCA in reverse mathematics [10]. • Berger [3] shows that FAN ∆ does not imply FAN c over a very weak basesystem. His argument is in its essence a translation of the reverse mathe-matics proof that WKL does not imply ACA [10], by coding the Turingjump into a c -bar. In order for this argument to work, he must be in acontext in which the existence of the Turing jump is not outright provable,hence the use of a weak base system. • Fourman and Hyland [7] present a Heyting-valued, almost topological,model in which FAN full fails; they do not address which fragments of theFan Theorem might hold, since these distinctions were not available at thetime. We show below that FAN Π holds in their model, separating theleft-most pair of principles in the diagram above.We are not aware of any prior proofs separating FAN c and FAN Π .The goal of this paper is to separate all of these principles via a uniformtechnique. This has several benefits. For one, it separates FAN c and FAN Π .For another, it separates FAN ∆ and FAN c over full IZF. That is new becauseBerger’s argument still leaves open the possibility that IZF would allow thatimplication to go through; independence of FAN c over IZF + FAN ∆ meansthat it does not. In addition, since the arguments employed rather handilyprovide four separation results, they seem to provide a flexible tool that mightbe useful elsewhere. This seems not to be the case for the other techniques thathave been used. It could well be the case, for instance, that realizability couldproduce all of the results discussed here. But no one has been able to do this yet.As for the Fourman-Hyland argument, they also show in the same work thatall topological models satisfy FAN full . So for the separations of interest here,topological models are just out. To be sure, variants of topological models,along the lines used by Fourman and Hyland, might still do the trick. Butbefore coming up with the arguments below that’s exactly what we tried, andgot nowhere. In short, we cannot say that the techniques used here get youanything that could not be gotten by other means, but at least it seems to beeasier to use. Beyond that, it could be the case that the proofs here really arein some sense the right ones for these results. In the face of the perfectly nicerealizability and Heyting-valued models that provide some of these separations,we are not at this point making that claim. While the constructions below arenatural enough, they are not so compelling as to seem canonical. Nonetheless,since they seem to work so well, it might be that with further reflection anddevelopment, it turns out that proofs along these lines are the way to go for alarge class of problems.As for what the techniques employed actually are, we would like to providesome motivation for how we happened upon them. Since it seemed that re-alizability and Heyting algebras weren’t working, we turned to the only otherkind of model we know of, Kripke models. To build a tree we could control,along with its paths, over set theory with full Separation and Collection, weturned to forcing. In order to have the trees be decidable, yet not completely3inned down, as required by the theories in question, we were forced to usenon-standard integers, to provide non-standard levels on the trees.Since this is a paper about constructive mathematics, a word about themeta-theory used is in order. It is classical through and through. We work inZFC. Presumably most if not all of the arguments are fully constructive, as inso many mathematical papers in all fields. We did not check, and so have noidea.In the next section we discuss the Fan Theorem in topological and re-lated models, including giving a proof that the Fourman-Hyland model satisfiesFAN Π . The following sections provide the advertised separation results, goingright-to-left in the diagram above. We then close with some questions. To make this paper somewhat self-contained, we repeat the proof that explainswhy the construction afterwards is more complicated than just a topologicalmodel.
Proposition 1. (Fourman-Hyland [7] ) In any topological model FAN full holds.Proof.
Let T be a topological space, and suppose T (cid:13) “ B ⊆ ∗ is a bar closed under extension.”Then, in particular, for any external sequence α ∈ N (that is, one from theground model)(1) T = J ∃ n αn ∈ B K = [ n ∈ N J αn ∈ B K . Let A u denote the open set J the bar { w | u ∗ w ∈ B } is uniform K . If T (cid:13) “ B is uniform,” then choose some p / ∈ A () . Define a set T r = { u ∈ ∗ | p A u } . Since A u = A u ∗ ∩ A u ∗ for any u ∈ ∗ , T r is a tree (i.e. closedunder restriction) with no terminal nodes. Since in addition () ∈ T r (that is,
T r is non-empty),
T r is infinite. Thus, by Weak K¨onig’s Lemma, there existsan infinite path β in T r . By the definition of
T r, p / ∈ A βn for all n ∈ N . NowEquation 1 yields the existence of n ∈ N such that p ∈ J βn ∈ B K ;but this contradicts J βn ∈ B K ⊂ A βn .This suggests that if we are looking for models in which some form of theFan Theorem fails we need to “delete points”. This was done in [7], section4, where they consider K ( T ), the coperfect open sets of a topological space T .This can be viewed as the equivalence classes of the open sets of T , under whichan open set is identified with its smallest coperfect superset. In this setting,removing a point from an open set does not change the set. Definition 1.
A Heyting algebra is connected if A ∨ B = ⊤ and A ∧ B = ⊥ implies that either A = ⊤ or A = ⊥ . K ([0 , × [0 , Proposition 2. If H is a connected Heyting algebra, then H (cid:13) FAN Π . Proof.
Suppose H (cid:13) “ B is a Π -bar, given say by S : u ∈ B iff ∀ n ∈ N ( u, n ) ∈ S. ” Since H (cid:13) “ S is decidable,” for any u ∈ ∗ and n ∈ N ,H (cid:13) “( u, n ) ∈ S ∨ ( u, n ) S. ”By the connectedness of H either H (cid:13) “( u, n ) ∈ S ” or H (cid:13) “( u, n ) S. ” Sodefine a set ˜ B ⊂ ∗ in the metatheory by u ∈ ˜ B ⇐⇒ ∀ n ∈ N H (cid:13) “( u, n ) ∈ S. ”˜ B is itself a bar, as follows. Let α ∈ N be arbitrary. If αn / ∈ ˜ B for all n ∈ N then for all n there exist i n such that J ( αn, i n ) ∈ B K = ⊥ . Thus J ∀ m ∈ N ( αn, m ) ∈ B K = ⊥ for any n ∈ N , and therefore J ∃ n ∈ N ∀ m ∈ N ( αn, m ) ∈ B K = ⊥ ;a contradiction to B being a bar internally. Hence ˜ B is a bar externally, andtherefore, working with a classical metatheory (or simply the Fan Theorem),it is uniform. So there exists N such that for all u of length N some initialsegment of u is in ˜ B . Then it is easy to see, that this same N witnesses theuniformity of B internally. Corollary 3.
FAN Π does not imply FAN full (over IZF).Proof. In [7] it shown that Ω (cid:13) FAN full . ∆ is not Provable As discussed in the introduction, recursive realizability shows that IZF does notprove FAN ∆ . However, we do not see how to adapt that, or the Heyting-valuedmodel from the previous section, to the other desired separation results. Hencewe are hoping not merely to provide here a different model falsifying FAN ∆ asa technical exercise, but rather to provide a technique more flexible than thosereferenced, to produce the other separation results. Of course, if this really is aflexible technique, it should work for the known separations too.We will build a Kripke model, working within ZFC. To construct a bar, itwill be crucial to control what paths exist. This is most easily done with ageneric set, in the sense of forcing. Definition 2.
Let the forcing partial order P be the set of appropriate labelingsof finitely many nodes from 2 ∗ . A labeling of nodes assigns to each one eitherIN, OUT, or ∞ , with the following restrictions. Any node labeled IN has nodescendant, the idea being that once a node gets into the eventual bar so are allof its descendants automatically, so nothing more need be said. Any descendantof a node labeled OUT must be labeled IN or OUT. Finally, for any node labeled ∞ , if both children are labeled, then at least one of them must be labeled ∞ . G be a generic through the condition that labels hi with ∞ . By straight-forward density arguments, any node labeled OUT by G has a uniform bar aboveit (or below it, depending on how you draw your trees) all labeled IN, and everynode labeled ∞ has a path through it always labeled ∞ , in fact a perfect set ofsuch.Let B = { α ∈ ∗ | for some n G ( α ↾ n ) = IN } . B ∈ M [ G ] is the interpretation σ GB of the term σ B = {h p, ˆ α i | for some n p ( α ↾ n ) = IN } . (As usual, the functionˆ . is the canonical injection of the ground model into the terms: ˆ x = {h∅ , ˆ y i | y ∈ x } .) Because of these latter ∞ -paths, B is not a bar. However, we mightreasonably think that if we no longer had access to the distinction between theOUT and the ∞ nodes, we might no longer be able to build a path avoiding B .This intuition is confirmed by the next proposition. Definition 3.
The shadow forcing Q is the set of functions from finite subtreesof ∗ to { IN, OUT } such that any node labeled IN has no descendant. Equiv-alently, Q is the sub-partial order of P beneath the condition labeling hi withOUT (together with the condition which labels hi IN, which has no extension).The canonical projection proj Q of P onto Q replaces all occurrences of ∞ withOUT. The canonical projection of the terms of P ’s forcing language to those of Q ’s, ambiguously also called proj Q , acts by applying proj Q to the conditions thatappear in the terms, hereditarily. (Notice that Q term are also P terms.) Notice that a P -filter projects to a Q -filter. If G is a generic P -filter, thenproj Q ( G ) will not be Q -generic, because in Q the terminal conditions are dense.Still, proj Q ( G ) induces an interpretation σ proj Q ( G ) of the terms σ of Q . Theseinterpretations are in M [ G ], as they are easily definable from σ and G ; alterna-tively, σ proj Q ( G ) = ( proj − ′′ Q σ ) G . For any P -filter G, proj Q ( σ B ) proj Q ( G ) = B : the induced interpretation ofthe projection of B is just B itself. Effectively, B as a P -term is already a Q -term. Proposition 4. If σ is a Q -term and p (cid:13) P “ proj − ′′ Q σ is an infinite branchthrough ∗ ,” then p (cid:13) P “ proj − ′′ Q σ goes through σ B .”Proof. By standard forcing technology, it suffices to extend p to some conditionforcing “ proj − ′′ Q σ goes through σ B ,” as then it will be dense beneath p to forceas much, and so will happen generically.First extend p so that every sequence in 2 ∗ of length 2 n − for some n eitheris labeled OUT or ∞ or has a proper initial segment labeled IN. Then extendagain by adjoining both children to all nodes of length 2 n − , and labeling them ∞ whenever possible (otherwise IN or OUT). For a technical reason soon tobecome clear, we must extend yet again. This time have the domain include alllength k descendants of the length n nodes not labeled IN, and label them sothat every length n node labeled ∞ has a unique descendant of length k labeled ∞ , and, most importantly, for each pair of nodes α and β of length k labeled ∞ , there is some i with α ( i ) = 1 and β ( i ) = 0 . One way of doing this is to let s be the number of nodes of length n labeled ∞ , to let k be n + s , and to buildthe ∞ -labeled descendant of the j th such node by adjoining to it j − s − j k being labeled OUT.Extend one last time to q (cid:13) proj − ′′ Q σ (ˆ k ) = ˆ α for some fixed α , where asusual ˆ x is the standard term for the internalization of the set x . Moreover,6 should force the equality in the strong sense that for each j < k there is aterm τ and a condition r ≥ q with h r, τ i ∈ proj − ′′ Q σ and q (cid:13) τ = h ˆ j, ˆ α (ˆ j ) i ;even further, if α ( j ) = 1 then q forces a particular element to be in τ ’s secondcomponent.If q labels some initial segment of α IN then we’re done.If q labels α OUT then it is dense beneath q that all descendants of α ofsome fixed length are labeled IN, and again we’re done.If q labels α ∞ then let q alt be identical to q except that all descendants of α ↾ n labeled ∞ by q are labeled OUT by q alt . Observe first that q alt extends p .Then note that, because proj Q ( q alt ) = proj Q ( q ), the strong forcing facts positedof q hold for q alt as well: for the same τ and j as above, q alt (cid:13) τ ∈ proj − ′′ Q σ and q alt (cid:13) “ τ is an ordered pair with first component ˆ j ,” and if q forced τ ’s secondcomponent to be non-empty, q alt also forces it to be non-empty, containing thesame term as for q . The difference between q and q alt , from σ ’s point of view, isthat q alt has more extensions than q : there are conditions extending q alt whichbar the tree beneath α , which is not so for q . That means that it is possiblefor extensions of q alt to force sets into Q -terms that no extension of q could.In the case of proj − ′′ Q σ (ˆ k ), though, such opportunities are limited. That termis already forced by p to be a function with domain k ; for each j < k there isalready a fixed term forced to stand for h j, ( proj − ′′ Q σ (ˆ k ))( j ) i ; if that functionvalue at j was forced by q to be 1 then it must retain a member and so is alsoforced by q alt to be 1. The only change possible is that something formerlyforced to be empty (i.e. be 0) could now be forced by some extension to havean element (i.e. be 1). Recall, though, the construction of q on level k : if proj − ′′ Q σ (ˆ k ) is ever forced by some r ≤ q alt to be some β = α by flipping some0’s to 1’s, by α ’s distinguished 1 r cannot label β ∞ . So r can be extended sothat all extensions of β of a certain length are labeled IN, forcing proj − ′′ Q σ tohit σ B . Of course, any extension of q alt forcing proj − ′′ Q σ (ˆ k ) to be α works thesame way as such an r does, since q alt already labels α OUT. In either case wehave an extension of p forcing proj − ′′ Q σ go through σ B .Even though we have just seen that B is a bar relative to the Q -paths, wewill perhaps surprisingly have occasion to consider weaker situations, where B islarger and hence even easier to hit. The case of interest is if we were to changesome ∞ ’s in G to OUTs, thereby allowing uniform bars above those nodes.Notice that if α ’s sibling is not labeled ∞ , then α ’s label could not consistentlybe changed from ∞ , as then α ’s parent, labeled ∞ , would then have bothchildren not labeled ∞ . Such considerations do not apply when α = hi . Definition 4. H is a legal weakening of G if H can be constructed by choosingfinitely many nodes labeled ∞ by G , changing those labels (to either IN or OUT),also changing the labeling of finitely many descendants of those nodes from ∞ or OUT to OUT or IN in such a way that each node labeled OUT has a uniformbar above it labeled IN, and then eliminating all descendants of nodes labeled IN.Furthermore, this must be done in such a manner that H is a filter through P (avoiding, for instance, the problem posed just before this definition). Notice that the difference between H and G can be summarized in one con-dition p , which contains the new bars, all labeled IN, and all of their ancestors.Hence we use the notation G p to stand for this H : to build G p , make theminimal change to each condition in G in order to be consistent with p .7 emma 5. If G p is a legal weakening of G then G p is generic through p . Remark 6.
Notice that if p labels the empty sequence IN or OUT then p = G p is a terminal condition in P , trivially satisfying the lemma.Proof. Let D be dense beneath p . Notice that G ↾ dom ( p ) is a condition in P contained in G . It is not hard to define the notion of projection beneath p , proj p , by making the minimal changes in a condition necessary to be compatiblewith p . We claim that proj − ′′ p D is dense beneath G ↾ dom ( p ). To see this,let q ≤ G ↾ dom ( p ). Extend proj p ( q ) to r ∈ D . The only way r can extend proj p ( q ) is by labeling extensions α of nodes which are unchanged by proj p : if α ∈ dom ( r ) \ dom ( proj p ( q )) then, for α ↾ n ∈ dom ( q ) , q ( α ↾ n ) = proj p ( q )( α ↾ n ). Extend q to q r by labeling those same extensions the same way: for α ∈ dom ( r ) \ dom ( proj p ( q )) q r ( α ) = r ( α ). We have that proj p ( q r ) = r , hence q r ∈ proj − ′′ p D . So proj − ′′ p D is dense beneath G ↾ dom ( p ), hence contains a memberof G , say q . Then proj p ( q ) is in both D and G p .We can now describe the ultimate Kripke model. Recall that G is generic for P over M and labels the empty sequence with ∞ . The bottom node ⊥ of theKripke model consists of the Q -terms, with membership (not equality!) as inter-preted by proj Q ( G ). Let N be an ultrapower of M [ G ] using any non-principalultrafilter on ω , with elementary embedding f : M [ G ] → N . This necessarilyproduces non-standard integers. Let H be the set of legal weakenings of f ( G ),as defined in N , which induce the same B on the standard levels of 2 ∗ , whichrestriction is definable only in M [ G ]. That is, any standard node labeled ∞ by G can only be changed to OUT by the legal weakening. H will index thesuccessors of ⊥ . At the node indexed by f ( G ) p , the universe will be the Q -termsof N as interpreted by proj Q ( f ( G ) p ). Regarding the embeddings from ⊥ , fora Q -term σ ∈ M , f ( σ ) is an f ( Q )-term in N , so send σ to f ( σ ). If f ( G ) p isa terminal condition in P , then the node indexed by f ( G ) p is terminal in theKripke ordering. Else iterate. That is, suppose f ( G ) p is non-terminal. Thestructure at its node can be built in N . As an ultrapower of M [ G ], N internallylooks like f ( M )[ f ( G )]; internally, f ( G ) is f ( P )-generic over the ground model f ( M ). The structure at node f ( G ) p could be built in f ( M )[ f ( G ) p ], where,by the previous lemma, f ( G ) p is generic through f ( P ) , and also non-terminal.Hence the construction just described, using an ultrapower and legal weaken-ings to get additional nodes, can be performed in f ( M )[ f ( G ) p ] just as above.Continue through ω -many levels. We will ambiguously use f to stand for any ofthe elementary embeddings, including compositions of such (making f a sort-ofpolymorphic transition function). Notice that the construction relativizes: theKripke structure from node f ( G ) p onwards is definable in f ( M )[ f ( G ) p ] just asthe entire structure is definable in M [ G ].This defines a Kripke structure interpreting membership. Equality at anynode can now be defined as extensional equality beyond that node in this struc-ture, inductively on the ranks of the terms, even though the model is not well-founded, thanks to the elementarity present. That is, working at ⊥ , suppose σ and τ are terms of rank at most α , and we have defined equality at ⊥ forall terms of rank less than α . Moreover, suppose (strengthening the inductiveassumption here) that this definability was forced in M by the empty condition ∅ . At node f ( G ) p the structure is definable over f ( M )[ f ( G ) p ], and, by elemen-tarity, in f ( M ), ∅ (cid:13) “Equality in the Kripke model is unambiguously definable8or all terms of rank less that f ( α ).” So at that node we can see whether thereis a witness to f ( σ ) and f ( τ ) being unequal. If there is such a witness at anynode f ( G ) p , then σ and τ are unequal at ⊥ , else they are equal at ⊥ . Thisextends the definability of equality to all terms of rank α . Hence inductivelyequality is definable for all terms. Proposition 7. ⊥ 6 (cid:13)
FAN ∆ . Proof.
It is immediate that B is a bar: any node is internally of the form f ( M )[ f ( G ) p ]; by the lemma, f ( G ) p is always f ( P )-generic; by the proposition,no path given by a Q -term can avoid the interpretation of the term for B asgiven by an f ( P )-generic. Moreover, B is decidable, as f ( G ) p agrees with G on the standard part of 2 ∗ , the only part that exists at ⊥ , and that argumentrelativizes to all nodes. However, B is not uniform at any non-terminal node,since f ( G ) p , when non-terminal, has labels of ∞ at every level.What remains to show is that our model satisfies IZF. In order to do this,we will need to get a handle on internal truth in the model. This is actuallyunnecessary for most of the IZF axioms, but for Separation in particular we willhave to deal with truth in the model. When forcing, this is done via the forcingand truth lemmas: M [ G ] | = φ iff p (cid:13) φ for some p ∈ G, where (cid:13) is definable in M . Since our Kripke model is built in M [ G ], statements about it are statementswithin M [ G ], and so are forced by conditions in G . The problem is that theKripke model internally does not have access to G , but only to B . In detail,Separation for M [ G ] is proven as follows: given φ and σ , it suffices to consider {h q, τ i | for some h p, τ i ∈ σ, q ≤ p and q (cid:13) φ ( τ ) } . The problem we face is thatthat set seems not to be in the Kripke model, even if σ is. What we need toshow is that if σ and φ ’s parameters are Q -terms then that separating set isgiven by a Q -term.Recall that proj Q operates by replacing all occurrences of ∞ by OUT. Definition 5. p ∼ p ′ if proj Q ( p ) = proj Q ( p ′ ) . Definition 6. p (cid:13) ∗ φ , for φ in the language of the Kripke model, i.e. when φ ’sparameters are Q -terms, inductively on φ : • p (cid:13) ∗ σ ∈ τ if, for some h q, ρ i ∈ τ, q ≥ Q proj Q ( p ) and p (cid:13) ∗ σ = ρ. • p (cid:13) ∗ σ = τ if for all p ′ ∼ p , p ′′ ≤ P p ′ , and h q, ρ i ∈ σ, if proj Q ( p ′′ ) ≤ Q q then there is a p ′′′ ≤ P p ′′ such that p ′′′ (cid:13) ∗ ρ ∈ τ , and symmetrically. • p (cid:13) ∗ φ ∧ θ if p (cid:13) ∗ φ and p (cid:13) ∗ θ . • p (cid:13) ∗ φ ∨ θ if p (cid:13) ∗ φ or p (cid:13) ∗ θ . • p (cid:13) ∗ φ → θ if for all for all p ′ ∼ p and p ′′ ≤ P p ′ , if p ′′ (cid:13) ∗ φ then there isa p ′′′ ≤ P p ′′ such that p ′′′ (cid:13) ∗ θ. • p (cid:13) ∗ ∃ x φ ( x ) if, for some Q -term σ, p (cid:13) ∗ φ ( σ ) . • p (cid:13) ∗ ∀ x φ ( x ) if for all for all p ′ ∼ p , p ′′ ≤ P p ′ , and Q -term σ , there is a p ′′′ ≤ P p ′′ such that p ′′′ (cid:13) ∗ φ ( σ ) . Lemma 8. If p ∼ p ′ then p (cid:13) ∗ φ iff p ′ (cid:13) ∗ φ. roof. For the cases ∈ , = , → , and ∀ , that is built right into the definition of (cid:13) ∗ .The other cases are a trivial induction. Lemma 9. If q ≤ P p and p (cid:13) ∗ φ then q (cid:13) ∗ φ. Proof.
Inductively on φ . For ∈ , use that proj Q is monotone. The cases ∧ , ∨ , and ∃ are trivial inductions. For the remaining cases, suppose q ′′ ≤ P q ′ , q ′ ∼ q, and q ≤ P p. Then q ′′ ≤ P q ′ ↾ dom ( p ) ∼ p , and use that p (cid:13) ∗ φ . Proposition 10. ⊥ | = φ iff p (cid:13) ∗ φ for some p ∈ G. Proof.
Inductively on φ.σ ∈ τ : ⊥ | = σ ∈ τ iff there are p ∈ G and h q, ρ i ∈ τ such that proj Q ( p ) ≤ Q q and ⊥ | = σ = ρ . Inductively ⊥ | = σ = ρ iff there is an r ∈ G ∗ -forcing the same.In one direction, using lemma 9, p ∪ r suffices, in the other we have p = r . σ = τ : Suppose p ∈ G and p (cid:13) ∗ σ = τ . By taking p ′ equal to p in thedefinition of (cid:13) ∗ , for every member ρ of either σ or τ , it is dense to ∗ -force ρ to be in the other set. By the genericity of G some such p ′′′ will be in G , andso inductively ρ will end up in the other set. This shows that σ and τ havethe same members at ⊥ . Regarding a future node f ( G ) p ′′ , because f ( G ) p ′′ is alegal weakening of f ( G ), p ′′ ↾ dom ( p ) ∼ p , so again it is dense for any memberof σ or τ to be forced into the other, so they have the same members at node f ( G ) p ′′ . Hence ⊥ (cid:13) ∗ σ = τ. Conversely, suppose for all p ∈ G p (cid:13) ∗ σ = τ. That means there are p ′ ∼ p, p ′′ ≤ P p ′ , and ρ forced by p ′′ into σ (without loss of generality), but p ′′ hasno extension ∗ -forcing ρ into τ . For every natural number n the set D n = { q | for some k > n, dom ( q ) ⊆ k , and all binary sequences of length k either arelabeled ∞ by q or some initial segment is labeled IN by q } is dense. Hencecofinally many levels of G are in D . Observe that if q is in D ∩ G and q ′ ∼ q then any extension of q ′ can be extended again to induce a legal weakening of G . In N , by overspill choose p ∈ f ( G ) to be in f ( D ). Choose p ′′ ≤ P p ′ ∼ p and ρ as given by the case hypothesis. Extend p ′′ to p ′′′ so that f ( G ) p ′′′ is a legalweakening of f ( G ). Since p ′′′ has no extension ∗ -forcing ρ into τ , inductively atnode f ( G ) p ′′′ ρ is not a member of τ . Hence ⊥ 6| = σ = τ.φ ∧ θ : Trivial. φ ∨ θ : Trivial. φ → θ : Suppose p ∈ G and p (cid:13) ∗ φ → θ . At any node f ( G ) p ′ , if f ( G ) p ′ | = φ then inductively choose p ′′ ∈ f ( G ) p ′ such that p ′′ (cid:13) ∗ φ. Without loss ofgenerality p ′′ can be taken to extend p ′ . Since f ( G ) p ′ indexes a node in themodel, p ′ ↾ dom ( p ) ∼ p , so p ′′ ≤ P p ′ ↾ dom ( p ) ∼ p . By the case assumptionthere is a p ′′′ extending p ′′ with p ′′′ (cid:13) ∗ θ. By the genericity of f ( G ) p ′ there issuch a p ′′′ in f ( G ) p ′ . So inductively f ( G ) p ′ | = θ. At node ⊥ the argument iseven simpler, as p ′ can be chosen to be p . So ⊥ | = φ → θ. Conversely, suppose for all p ∈ G that p (cid:13) ∗ φ → θ. That means there are p ′ ∼ p and p ′′ ≤ P p ′ with p ′′ (cid:13) ∗ φ but no extension of p ′′ ∗ -forces θ . As inthe = case above, in N , by overspill choose p ∈ f ( G ) to be in f ( D ). Choose p ′′ ≤ P p ′ ∼ p as given by the case hypothesis. Extend p ′′ to p ′′′ so that f ( G ) p ′′′ is a legal weakening of f ( G ). Inductively f ( G ) p ′′′ | = φ, but since p ′′′ has noextension ∗ -forcing θ , inductively f ( G ) p ′′′ = θ . Hence ⊥ 6| = φ → θ. ∃ x φ ( x ): Trivial. 10 x φ ( x ): Suppose p ∈ G and p (cid:13) ∗ ∀ x φ ( x ). For any node f ( G ) p ′ andany σ in the universe there, p ′ ≤ P p ′ ↾ dom ( p ) ∼ p , so there is a p ′′ ≤ P p ′ such that p ′′ (cid:13) ∗ φ ( σ ) . By genericity there is such a p ′′ in f ( G ) p ′ . Inductively f ( G ) p ′ | = φ ( σ ) . So every element at node f ( G ) p ′ satisfies φ there. At node ⊥ the argument is even easier, since p ′ can be chosen to be p . Hence ⊥ | = ∀ x φ ( x ) . Conversely, suppose for all p ∈ G that p (cid:13) ∗ ∀ x φ ( x ) . That means there are p ′ ∼ p, p ′′ ≤ P p ′ , and Q -term σ such that p ′′ has no extension ∗ -forcing φ ( σ ).As in the cases of = and → above, in N , by overspill choose p ∈ f ( G ) to be in f ( D ). Choose p ′′ ≤ P p ′ ∼ p and σ as given by the case hypothesis. Extend p ′′ to p ′′′ so that f ( G ) p ′′′ is a legal weakening of f ( G ). Since p ′′′ has no extension ∗ -forcing φ ( σ ), inductively f ( G ) p ′′′ = φ ( σ ). Hence ⊥ 6| = ∀ x φ ( x ) . Theorem 11. ⊥ | = IZF
Proof.
Empty Set and Infinity are witnessed by ∅ and ˆ ω respectively. Pairingis witnessed by {h∅ , σ i , h∅ , τ i} , and Union by {h q ∪ r, ρ i | for some τ h q, τ i ∈ σ and h r, ρ i ∈ τ } . Extensionality holds because that’s how = was defined.For ǫ -Induction, suppose ⊥ | = “( ∀ y ∈ x φ ( y )) → φ ( x ) . ” If it were not thecase that ⊥ | = “ ∀ x φ ( x )”, then at some later node G p there would be a term σ with f ( G ) p = φ ( σ ). The restricted Kripke model of node f ( G ) p and itsextensions is definable in a model of ZF, say N , which is a finite iteration of theultrapower construction, and so is itself a model of ZF. Hence, in N, σ can bechosen to be such a term of least V -rank, say κ . Then at all nodes after f ( G ) p ,by elementarity, it holds that f ( κ ) is the least rank of any term not satisfying φ . So all members of σ , being of lower rank, satisfy φ at whatever node theyappear. By the induction hypothesis, σ must also satisfy φ , contradicting theassumption that some term does not satisfy φ. For the powerset of σ take all sets with members of the form h q, τ i , where h p, τ i ∈ σ and q ≤ Q p .It is easy to give a coarse proof of Bounding. The Kripke model can be builtin M [ G ]. Given a σ at ⊥ , Bounding in M [ G ] can be used to bound the rangeof φ on σ at ⊥ . Also, the set of nodes is set-sized, so there are only set-manyinterpretations of f ( σ ) at the other nodes, so the range of φ on them can alsobe bounded. Since the standard ordinals are cofinal through the ordinals in allof the iterated ultrapowers, by picking κ large enough, ˆ V κ suffices for boundingthe range of φ on σ .For Separation, given φ and σ , let Sep φ,σ be {h proj Q ( p ) , τ i | for some h q, τ i ∈ σ with p ≤ q we have p (cid:13) ∗ φ ( σ ) } . By lemmas 8 and 10, this works.Although this model does not satisfy FAN ∆ , it does satisfy ¬¬ FAN ∆ , as theterminal nodes are dense. Admittedly this is a rather weak failure of FAN ∆ . Inthe final section, we will address the issue of getting stronger failures of FAN ∆ . ∆ does not imply FAN c We will need a tree similar to that of the last proof. In fact, we will need twotrees: the c − bar C , and the decidable set C ′ from which C is defined. (Both canbe viewed either as 2 ∗ with labels or as subtrees of 2 ∗ .) Mostly we will focuson C . Because FAN c refers to eventual membership in a tree, the difference11etween IN and OUT nodes is no longer relevant: the bar is uniform beneathany OUT node. So we can describe the forcing in terms similar to those before,and with some simplifications introduced. The forcing partial order P will bethe set of appropriate labelings of finitely many nodes from 2 ∗ . A labeling ofnodes assigns to each one either IN or ∞ , with the following restrictions. Anynode labeled IN has no descendant, the idea being that once a node gets into theeventual bar so are all of its descendants automatically, so nothing more needbe said. For any node labeled ∞ , if both children are labeled, then at least oneof them must be labeled ∞ . Let G be P -generic through the condition labelingthe empty sequence with ∞ .As before, we will need to look at weaker trees, ones with bigger bars. Definition 7. H is a legal weakening of G if H can be constructed by choosingfinitely many nodes labeled ∞ by G , whose siblings are also labeled ∞ by G , andchanging those labels to IN and eliminating all descendants. As before, each legal weakening H can be summarized by one forcing condi-tion p , which consists of those nodes changed by H and their ancestors, labeledas in G . H is then the set of conditions in G each minimally changed to beconsistent with p . Hence we refer to H as G p . Lemma 12. If G p is a legal weakening of G then G p is generic through p .Proof. As in the corresponding lemma in the previous section.
Definition 8.
Terms are defined inductively (through the ordinals) as sets ofthe form {h B i , σ i i | i ∈ I } , where I is any index set, σ i a term, and B i a finiteset of truth values. A truth value is a symbol of the form b + or b ′ or ¬ b ′ , for b ∈ ∗ a finite binary sequence. Definition 9.
Let C be the term {h{ b + } , ˆ b i | b ∈ ∗ } , and C ′ be {h{ b ′ } , ˆ b i | b ∈ ∗ } . In our final model, (the interpretation of) C will be the c -bar induced by(the interpretation of) C ′ , and C will not be uniform, thereby falsifying FAN c .Furthermore, we will show that FAN ∆ holds in this model.We can now describe the ultimate Kripke model. Recall that G is genericfor P over M and labels the empty sequence with ∞ . The bottom node ⊥ ofthe Kripke model consists of the terms. At ⊥ , b + counts as true iff G ( b ) =IN, b ′ always counts as true, and ¬ b ′ never counts as true. Later nodes willhave different ways of counting the various literals as true. At any node, for σ = {h B i , σ i i | i ∈ I } , if each member of some B i counts as true, then at thatnode σ i ∈ σ . This induces a notion of extensional equality among the terms.One way of viewing this is at any node to remove from a term σ any pair h B i , σ i i if some member of B i is not true at that node. Then each remaining h B i , σ i i can be replaced by σ i . Equality is then as given by the Axiom of Extensionalityas interpreted in the model.As for what the other nodes in the model are, there are two different kinds.As in the last section, let N be an ultrapower of M [ G ] using any non-principalultrafilter on ω , with elementary embedding f : M [ G ] → N . This necessarilyproduces non-standard integers. In N , any forcing condition p which induces alegal weakening of f ( G ) will index a successor node to ⊥ . At the node indexed12y p , the universe will be the terms of N as interpreted by f ( G ) p . That is, b + is true if f ( G ) p ( b ) = IN, b ′ is always true, and ¬ b ′ never. Regarding theembeddings from ⊥ , for a term σ ∈ M , f ( σ ) is a term in N , so send σ to f ( σ ).In addition, definably over M [ G ], any non-standard c ∈ ∗ with f ( G )( c ) = ∞ also indexes a node. At such a node c , b ′ counts as true iff b = c , ¬ b ′ countsas true iff b = c , and b + counts as true iff b c ( b is not an initial segment of c ). Note that at ⊥ any b ′ refers only to a standard b ; for some b ′ to be declaredfalse at a later node c , b would have to equal c , and c indexes a node only if c isnon-standard. Hence there is no conflict with the Kripke structure: once b ′ isdeemed true, it remains true. Similarly with b + : G p is a fattening of G . Hencemembership, being based on finitely many truth values, is monotone.Any node indexed by such a c ∈ ∗ is terminal in the Kripke ordering. Also,among nodes of the other kind, there is one trivial condition p , the one with p ( hi ) = IN. This is also a terminal node, where each b + and each b ′ is true. Atany other node, iterate. That is, suppose p is not the preceding condition. Themodel at p can be built in N . As an ultrapower of M [ G ], N internally lookslike f ( M )[ f ( G )]. The structure at node p could be built in f ( M )[ f ( G ) p ], where f ( G ) p is generic through f ( P ) (and non-trivial). Hence the construction justdescribed, using an ultrapower and legal weakenings and non-standard binarystrings to get additional nodes, can be performed in f ( M )[ f ( G ) p ] just as above.This provides immediate successors to nodes indexed by (non-trivial) p ’s. Iterate ω -many times.The picture is that, at ⊥ , C looks like G , that is, those nodes G assigns to beIN. This tree gets fatter at later nodes that are legal weakenings. At terminalnodes c , C is everything but the branch up to c . At most nodes C ′ looks likeeverything; at node c , where c is non-standard relative to its predecessor, wefind the one thing not in C ′ , namely c .What we need to show is that this model satisfies IZF and FAN ∆ , andfalsifies FAN c . Lemma 13. ⊥ 6| = FAN c .Proof. It is easy to see that C is the c − set induced by C ′ : once b is forced into C , none of its descendants index terminal nodes, so no descendant is forced outof C ′ ; similarly, if b is not forced into C , say at node p , then G p ( b ) = ∞ , andin N some non-standard extension c of b will also be labeled ∞ by f ( G ) p , andthat c will index a node at which c is not in C ′ . Clearly, C is not uniform, and C ′ is decidable. So it remains only to show that ⊥ | = C is a bar.Suppose σ is forced to be an infinite binary path at some node. If thatnode is a terminal node, C contains cofinitely many members of 2 ∗ , and socertainly intersects σ . Else without loss of generality we can assume the nodeis ⊥ . Then, for some p ∈ G, p (cid:13) “ ⊥ | = σ is an infinite binary path.” If it isnot dense beneath p to force the standard part of σ (that is, σ applied to thestandard integers) to be in the ground model, then extensions q and r of p forceincompatible facts about σ . The only incompatible facts about σ are of theform b ⌢ ∈ σ and b ⌢ ∈ σ . The positive parts of q and r (that is, q − ( IN ) and r − ( IN )) induce a legal weakening of G . That is, there is a canonical conditioninpart( q, r ), with domain dom ( q ) ∪ dom ( r ), that returns IN on any node thateither q or r returns IN on, as well as on any node if inpart( q, r ) returns IN onboth children, else OUT. Because terms use only positive (i.e. IN) information,at the node f ( G ) inpart ( q,r ) , both b ⌢ b ⌢ σ . (More coarsely and13erhaps more simply, at the node induced by the trivial condition sending theempty sequence to IN, the same conclusion holds for the same reason.) Hence ⊥ could not have forced σ to be a path in the first place. Therefore p forces σ on the standard binary tree to be in the ground model. It is easy to see thatgenerically G labels some node in σ IN.
Lemma 14. ⊥ | = FAN ∆ Proof.
By arguments similar to the above. If a set of nodes B is forced by p to be decidable, then no extensions of p can force incompatible facts about B .Hence B is in the ground model. If B were not a bar in the ground model,there would be a ground model path missing B . This path would also be in theKripke model. Hence B is a bar in the ground model. Since the ground modelis taken to be classical, B is uniform.Regarding getting IZF to be true, just as in the previous section, the prob-lem is that truth in the Kripke model is on the surface determined by forcingconditions in the ground model, to which the Kripke model has no access. Theessence is to capture truth at a node using those truth values that are allowedin the building of terms. Definition 10.
For a forcing condition p, B p = { b + | for some initial segment c of b, p ( c ) = IN } .For a set of truth values B, B + = B ∩ { b + | b ∈ ∗ } . Also, B is positive if B contains no truth value of the form ¬ b ′ . Definition 11. ¬ b ′ (cid:13) ∗ B iff c + ∈ B → c + b ′ , c ′ ∈ B → c = b, and ¬ c ′ ∈ B → c = b. σ ¬ b ′ = { σ ¬ b ′ i | for some h B i , σ i i ∈ σ, ¬ b ′ (cid:13) ∗ B i } .3. For φ ( σ , . . . , σ n ) in the language of the Kripke model, that is, with pa-rameters (displayed) terms, φ ¬ b ′ = φ ( σ ¬ b ′ , . . . , σ ¬ b ′ n ) .4. ¬ b ′ (cid:13) ∗ φ , for φ in the language of the Kripke model, if φ ¬ b ′ is true (i.e.in V ). Note that φ ¬ b ′ is a formula with set parameters. Definition 12. q ≤ W p ( q is a weakening of p as conditions) if for b ∈ dom ( p ) either p ( b ) = ∞ or for some initial segment c of b q ( c ) = IN. The idea behind this definition is the q may change some ∞ ’s to IN’s, as wellas extend the domain of p . Notice that ≤ W is a partial order, and inpart( p, q ),from lemma 14, is the greatest lower bound of p and q . Definition 13. p (cid:13) ∗ φ , for φ in the language of the Kripke model, i.e. when φ ’s parameters are terms, inductively on φ : • p (cid:13) ∗ σ ∈ τ if for some h B i , τ i i ∈ τ with B i positive, B + i ⊆ B p and p (cid:13) ∗ σ = τ i . • p (cid:13) ∗ σ = τ if i ) for all h B i , σ i i ∈ σ and q ≤ W p, if B i is positive and B + i ⊆ B q thenthere is an r ≤ q such that r (cid:13) ∗ σ i ∈ τ , and symmetrically between σ and τ , and ii ) for all b dom ( p ) , if for no initial segment c of b is c + in B p , then ¬ b ′ (cid:13) ∗ σ = τ . p (cid:13) ∗ φ ∧ θ if p (cid:13) ∗ φ and p (cid:13) ∗ θ . • p (cid:13) ∗ φ ∨ θ if p (cid:13) ∗ φ or p (cid:13) ∗ θ . • p (cid:13) ∗ φ → θ if i ) for all q ≤ W p if q (cid:13) ∗ φ then there is an r ≤ q such that r (cid:13) ∗ θ, and ii ) for all b dom ( p ) , if for no initial segment c of b is c + in B p , then ¬ b ′ (cid:13) ∗ φ → θ . • p (cid:13) ∗ ∃ x φ ( x ) if for some term σ p (cid:13) ∗ φ ( σ ) . • p (cid:13) ∗ ∀ x φ ( x ) if i ) for all terms σ and q ≤ W p, there is an r ≤ q such that r (cid:13) ∗ φ ( σ ) , and ii ) for all b dom ( p ) , if for no initial segment c of b is c + in B p , then ¬ b ′ (cid:13) ∗ ∀ x φ ( x ) . Proposition 15. If p (cid:13) ∗ φ and q ≤ W p then q (cid:13) ∗ φ. Proof.
Trivial induction on φ. Lemma 16. ⊥ | = φ iff p (cid:13) ∗ φ for some p ∈ G. Proof.
Inductively on φ.σ ∈ τ : ⊥ | = σ ∈ τ iff for some h B i , τ i i ∈ τ every member of B i is true at ⊥ and ⊥ | = σ = τ i . The former clause holds iff B i is positive and, for some p ∈ G, B i ⊆ B p . Inductively, the latter clause holds iff, for some q ∈ G, B q (cid:13) ∗ σ = τ i . Given such p and q , p ∪ q suffices. The converse direction is immediate. σ = τ : Suppose p ∈ G and p (cid:13) ∗ σ = τ. If q ∈ f ( P ) indexes a node then q ≤ W p . If q | = ρ ∈ σ then inductively there is a q ′ ∈ f ( G ) q , q ′ ≤ q, such that q ′ (cid:13) ∗ ρ = σ i ∧ σ i ∈ f ( σ ) for some h B i , σ i i ∈ f ( σ ) . By i ) of the case hypothesis,there is an r ≤ q ′ with r (cid:13) ∗ ρ ∈ τ. Generically, there is such an r in f ( G ), soinductively q | = ρ ∈ τ. If c indexes a node, then by ii ) of the case hypothesis c | = σ = τ . Hence ⊥ | = σ = τ. Conversely, suppose there is no such p ∈ G. If p (cid:13) ∗ σ = τ because clause i )fails, then there is a witness q ≤ W p to that failure. We say that such a q is closeto p if dom ( q ) ⊆ dom ( p ). That means that q comes from p by changing some ∞ ’s to IN’s and not adding anything else. Observe that if i ) fails for p , then p can be extended to p ′ so that i ) fails for p ′ via a witness q close to p ′ . That’sbecause dom ( p ′ ) can be taken to be dom ( p ) ∪ dom ( q ), for b ∈ dom ( p ) p ′ ( b ) canbe taken to be p ( b ), and for b ∈ dom ( q ) \ dom ( p ) p ′ ( b ) can be taken to be q ( b ).Therefore D = { p | p (cid:13) ∗ σ = τ, or p violates i ) with a witness q close to p , or p violates ii ) } is dense.Suppose there were a p ∈ G violating i ) with a witness q close to p . Then q induces a legal weakening f ( G ) q of f ( G ), and so indexes a node. By the choiceof q, q | = σ i ∈ σ. If q | = σ i ∈ τ then inductively that would be ∗ -forced bysome r ≤ q . But by the choice of q there is no such r . Hence we would have q = σ = τ. If there is no such p then every p ∈ G violates ii ). Let p ∈ f ( G ) be suchthat p ⊇ G . Since ii ) fails for that p , then, with b from that failure, b indexes anode, b = σ = τ . In either case, ⊥ 6| = σ = τ.φ ∧ θ : Trivial. φ ∨ θ : Trivial. 15 → θ : Suppose p ∈ G and p (cid:13) ∗ φ → θ . If q | = φ then inductively, for some q ′ ∈ f ( G ) q , q ′ (cid:13) ∗ φ . Since we can take q ′ ≤ q ≤ W p , by i ) of the hypothesisthere is an r ≤ q such that r (cid:13) ∗ θ. By genericity, there is such an r in f ( G ) q .Hence q | = θ. If c | = φ then use ii ) of the hypothesis.Conversely, suppose there is no such p ∈ G. If p (cid:13) ∗ φ → θ because clause i ) fails, then there is a witness q ≤ W p to that failure, in which case p can beextended to p ′ so that i ) fails for p ′ via a witness q close to p ′ , where closenessis as defined above in the case for =, for the same reason as above. Therefore D = { p | p (cid:13) ∗ φ → θ, or p violates i ) with a witness q close to p , or p violates ii ) } is dense.Suppose there were a p ∈ G violating i ) with a witness q close to p . Then q induces a legal weakening f ( G ) q of f ( G ), and so indexes a node. By the choiceof q, q | = φ. If q | = θ then inductively that would be ∗ -forced by some r ≤ q .But by the choice of q there is no such r . Hence we would have q = φ → θ. If there is no such p then every p ∈ G violates ii ). Let p ∈ f ( G ) be suchthat p ⊇ G . Since ii ) fails for that p , then, with b from that failure, b indexes anode and b = φ → θ . In either case, ⊥ 6| = φ → θ. ∃ x φ ( x ): Trivial. ∀ x φ ( x ): As in the cases for = and → . Lemma 17. ⊥ | = IZF
Proof.
Just as in the last section, most of the axioms have soft proofs in thismodel. The only issue is with Separation. Given φ and σ , let Sep φ,σ be {h B, τ i | for some h B ′ , τ i ∈ σ with B ⊇ B ′ either i ) B = B p (cid:13) φ ( σ ), or ii ) ¬ b ′ ∈ B and ¬ b ′ (cid:13) ∗ φ } . By the previous lemma, this works.As in the previous section, this model does not satisfy FAN c , but does satisfy ¬¬ FAN c . For further discussion, see the questions at the end. c does not imply FAN Π Let G be P -generic exactly as in the last section. By convention, we say thatif G ( α ) = IN then G applied to any extension of α is also IN. Our goal is tohide G a bit better than before, so FAN c remains true, but not too well, so thatFAN Π is false.Let N be an ultrapower of M [ G ] using a non-principal ultrafilter on ω . TheKripke model has a bottom node ⊥ , and the successors of ⊥ are indexed bythe labels h n, α i , where n is a non-standard integer, and α ∈ ∗ either hasnon-standard length or G ( α ) = ∞ . Definition 14.
A truth value is a symbol of the form h n, α i , ¬h n, α i , or h∀ n, α i ,for n a natural number (in the first two cases) and α ∈ ∗ . Admittedly truthvalues of the first kind are also used to index nodes; whether truth values or nodesare intended in any particular case should be clear from the context. Terms aredefined inductively (through the ordinals) as sets of the form {h B i , σ i i | i ∈ I } ,where I is any index set, σ i a term, and B i a finite set of truth values. The sets at ⊥ will be the terms in M . The sets at any other node willbe analogous, that is, the terms in what N thinks is the ground model, i.e.16 κ ∈ ORD f ( M κ ). At ⊥ , h n, α i will always be true, ¬h n, α i always false, and h∀ n, α i true exactly when G ( α ) = IN. At node h m, β i , h n, α i is true exactlywhen h n, α i 6 = h m, β i , ¬h n, α i is true exactly when h n, α i = h m, β i , and h∀ n, α i true exactly when α = β. (Note that, perhaps perversely, the node h n, α i isexactly the node at which the truth value h n, α i is false . The reason behindthis choice is that the node h n, α i is where something special happens to thecorresponding truth value. If preferred, the reader can call that node ¬h n, α i instead.) This interpretation of the truth values induces an interpretation ofthe terms at all nodes.Let T n be the term {h{h n, α i} , ˆ α i | α ∈ ∗ } . Let C be a term namingthe function that on input n returns T n . T n at ⊥ looks like the full tree 2 ∗ ; T n at h n, α i looks like everything except α ; and T n at h m, α i , m = n, againlooks like 2 ∗ . The term for T n C ( n ) is given by {h{h∀ n, α i} , ˆ α i | α ∈ ∗ } , and isinterpreted as { α | G ( α ) = IN } at ⊥ and 2 ∗ \{ α } at h n, α i . Notice that T n C ( n )is not closed under extensions.The proof will be finished once we show that, at ⊥ , FAN c holds, IZF holds,and T n C ( n ) is a counter-example to FAN Π . Lemma 18. ⊥ 6| = FAN Π .Proof. It is clear that T n is decidable, and so T n C ( n ) is on the face of it Π . It is also clear that T n C ( n ) is not a uniform bar. So it suffices to show that ⊥ (cid:13) “ T n C ( n ) is a bar.”Let ⊥ | = “ Br is a branch through 2 ∗ . ” (Without loss of generality, it sufficesto start at ⊥ instead of at an arbitrary node.) Work beneath a conditionforcing that, so we can assume Br consists of sets of the form h B i , ˆ α i , forvarious α ∈ ∗ . If the standard part of Br , the part visible at ⊥ , is in theground model M , then, by the genericity of G, Br will hit G (i.e. for some α ∈ Br, G ( α ) = IN), which is how ⊥ interprets T n C ( n ). If the standardpart of Br were not in M , then contradictory facts about Br would be forcedby different forcing conditions. In particular, we would have p, q, and α with p (cid:13) “ ⊥ | = α ⌢ ∈ Br ” and q (cid:13) “ ⊥ | = α ⌢ ∈ Br. ” That means there are h B p , d α ⌢ i ∈ Br and h B q , d α ⌢ i ∈ Br, with B p and B q consisting only of truthvalues automatically true at ⊥ save for some of the form h∀ n, α i . But at somenode h n, α i with α non-standard, all of those latter truth values will be true.Hence h n, α i | = “ d α ⌢ , d α ⌢ ∈ Br, ” so ⊥ could not force Br to be a path.In order to prove the other facts, we will need to deal with truth at ⊥ . Definition 15.
For a forcing condition p , let | p | , the length of p , be the lengthof the longest α ∈ dom ( p ) . Let B p be {h n, α i | n, length ( α ) ≤| p |} ∪ {h∀ n, α i | length ( α ) ≤| p | and, for some initial segment β of α, p ( β ) = IN } . Definition 16.
1. For B a finite set of truth values, ¬h n, α i (cid:13) ∗ B iff h n, α i 6∈ B, h∀ n, α i 6∈ B, and the only truth value of the form ¬h m, β i in B is ¬h n, α i itself.2. σ ¬h n,α i = { σ ¬h n,α i i | for some h B i , σ i i ∈ σ, ¬h n, α i (cid:13) ∗ B i } .3. For φ ( σ , . . . , σ n ) in the language of the Kripke model, that is, with pa-rameters (displayed) terms, φ ¬h n,α i = φ ( σ ¬h n,α i , . . . , σ ¬h n,α i n ) . . ¬h n, α i (cid:13) ∗ φ , for φ in the language of the Kripke model, if φ ¬h n,α i is true(i.e. in V ). Note that φ ¬h n,α i is a formula with set parameters. Definition 17. p (cid:13) ∗ φ , for φ in the language of the Kripke model, i.e. when φ ’s parameters are terms, inductively on φ : • p (cid:13) ∗ σ ∈ τ if, for some h B i , τ i i ∈ τ , B i ⊆ B p and p (cid:13) ∗ σ = τ i . • p (cid:13) ∗ σ = τ if i ) for all h B i , σ i i ∈ σ and q ≤ p , if B i ⊆ B q then there is an r ≤ q suchthat r (cid:13) ∗ σ i ∈ τ , and symmetrically between σ and τ , and ii ) if n > | p | , and if either length( α ) > | p | or for no initial segment β of α do we have p ( β ) = IN, then ¬h n, α i (cid:13) ∗ σ = τ . • p (cid:13) ∗ φ ∧ θ if p (cid:13) ∗ φ and p (cid:13) ∗ θ . • p (cid:13) ∗ φ ∨ θ if p (cid:13) ∗ φ or p (cid:13) ∗ θ . • p (cid:13) ∗ φ → θ if i ) for all q ≤ p , if q (cid:13) ∗ φ then there is an r ≤ q such that r (cid:13) ∗ θ, and ii ) if n > | p | , and if either length( α ) > | p | or for no initial segment β of α do we have p ( β ) = IN, then ¬h n, α i (cid:13) ∗ φ → θ . • p (cid:13) ∗ ∃ x φ ( x ) if for some term σ p (cid:13) ∗ φ ( σ ) . • p (cid:13) ∗ ∀ x φ ( x ) if i ) for all terms σ and q ≤ p , there is an r ≤ q such that r (cid:13) ∗ φ ( σ ) , and ii ) if n > | p | , and if either length( α ) > | p | or for no initial segment β of α do we have p ( β ) = IN, then ¬h n, α i (cid:13) ∗ ∀ x φ ( x ) . Proposition 19. If p (cid:13) ∗ φ and q ≤ p then q (cid:13) ∗ φ. Proof.
Trivial induction on φ. Lemma 20. ⊥ | = φ iff p (cid:13) ∗ φ for some p ∈ G. Proof.
Inductively on φ.σ ∈ τ : ⊥ | = σ ∈ τ iff for some h B i , τ i i ∈ τ every member of B i is trueat ⊥ and ⊥ | = σ = τ i . The former clause holds iff B i contains nothing of theform ¬h n, α i , and if h∀ n, α i ∈ B i then G ( α ) = IN. Given such a B i , let p be asufficiently long initial segment of G forcing “ σ = τ i .” Such a p suffices. Theconverse direction is immediate. σ = τ : Suppose p ∈ G and p (cid:13) ∗ σ = τ. Then any member of σ at ⊥ is equalat ⊥ to some σ i , where h B i , σ i i ∈ σ and B i ⊆ B q for some q ∈ G . Then by thehypothesis and genericity there will be an extension r of q in G forcing σ i to bein τ . At any other node h n, α i , working in N , n is non-standard and so greaterthan | p | , and α also satisfies the conditions in ii ) (of the definition of ∗ -forcingequality), so “ σ = τ ” is true at these other nodes too.Conversely, suppose there is no such p ∈ G. With reference to the definitionof ∗ -forcing equality, observe that { p | p satisfies clause i ) } ∪ { p | for some h B i , σ i i ∈ σ, B i ⊆ B p , yet p has no extension ∗ -forcing σ i into τ } is dense.If G contains a member of the second set of that union, then the induced σ i witnesses that ⊥ 6| = σ = τ . If not, then G contains p satisfying i ). So no p ∈ G satisfies ii ). This also holds in N . In N , take p to be an initial segment of G of18on-standard length. The failure of ii ) for that p produces an n and α whichindex a node at which σ = τ , showing ⊥ 6| = σ = τ.φ ∧ θ : Trivial. φ ∨ θ : Trivial. φ → θ : Suppose p ∈ G and p (cid:13) ∗ φ → θ . Then it is direct that ⊥ | = φ → θ. Conversely, suppose there is no such p ∈ G. With reference to the definitionof ∗ -forcing implication, observe that { p | p satisfies clause i ) } ∪ { p | p (cid:13) ∗ φ yet p has no extension ∗ -forcing θ } is dense. If G contains a member of the secondset of that union, then inductively ⊥ | = φ and ⊥ 6| = θ , hence ⊥ 6| = φ → θ . Ifnot, then G contains p satisfying i ). So no p ∈ G satisfies ii ). This also holdsin N . In N , take p to be an initial segment of G of non-standard length. Thefailure of ii ) for that p produces an n and α which index a node at which φ → θ is false, showing ⊥ 6| = φ → θ. ∃ x φ ( x ): Trivial. ∀ x φ ( x ): As in the cases for = and → . Lemma 21. ⊥ | = FAN c Proof.
Suppose that at ⊥ we have a decidable set C ′ ⊆ ∗ inducing a c -bar C .We would like to show that at ⊥ the c -bar C is uniform, which means that, forsome k, C contains every sequence of length at least k ; in notation, C ⊇ ≥ k .This is equivalent with C ′ containing 2 ≥ k , which is what we will prove.Say that α ∈ ∗ is good if there is a natural number k such that, whenever n ≥ k and β ⊇ α has length at least k , ¬h n, β i (cid:13) ∗ C ′ ⊇ ≥ k . Observe thatif α ⌢ α ⌢ α (by taking k sufficiently large). So ifthe empty sequence hi is bad (i.e. not good) then there is a branch Br of badnodes. For each α ∈ Br , by the definition of badness, taking k to be the length | α | of α , we have some β ⊇ α and n ≥| α | such that ¬h n, β i 6 (cid:13) ∗ C ′ ⊇ ≥ k . Because ¬h n, β i (cid:13) ∗ φ is defined as the truth of φ ¬h n,β i in the classical universe V , we can reason classically and conclude that there is a γ ∈ ≥ k such that ¬h n, β i (cid:13) ∗ γ C ′ . By choosing α ’s of increasing length, we can get infinitelymany γ ’s of increasing length, in particular infinitely many distinct γ ’s. Hencethere is a branch Br such that each node in Br has infinitely many such γ ’sas extensions.That was all in M . Now with reference to N , if α ∈ Br N has standardlength, then the corresponding choice of γ is also standard, since it’s the same γ as in M . So if we choose a non-standard γ coming from the procedure above,that γ came from a non-standard α . Since N | = “ Br is infinite,” there is anon-standard node on Br N , with some such γ as an extension; since the nodechosen from Br N was non-standard, so is γ , and hence so is the α that γ camefrom. From α , we also have β ⊇ α and n ≥| α | with ¬h n, β i (cid:13) ∗ γ C ′ . Inparticular, h n, β i indexes a node in the model. But at ⊥ , C ′ induces a c -bar, so ⊥ | = “there is a node δ ∈ Br such that δ ∈ C ; that is, every extension of δ isin C ′ .” This contradicts the choice of γ .We conclude from this that hi is good. Fix k witnessing this goodness. Wewill show ⊥ | = C ′ ⊇ ≥ k .First, if δ ∈ ≥ k is standard, then, for any n and β non-standard, ¬h n, β i (cid:13) ∗ δ ∈ C ′ , so, with reference to the Kripke node h n, β i , h n, β i | = δ ∈ C ′ . Since C ′ is decidable, ⊥ | = δ ∈ C ′ .
19o finish the argument, we need only consider nodes h n, β i , and show h n, β i | = C ′ ⊇ ≥ k . If β has length at least k , this follows from the goodness of hi . Theonly other case is β of length less than k such that G ( β ) = ∞ . It sufficesto show that, for any such fixed β , in M there is a finite n such that, for all m ≥ n, ¬h m, β i (cid:13) ∗ C ′ ⊇ ≥ k .Toward that end, suppose not. Then for infinitely many m there is a γ oflength at least k such that ¬h m, β i (cid:13) ∗ γ C ′ . If those γ ’s are of boundedlength then one occurs infinitely often. For that fixed γ , by overspill there is anon-standard m such that ¬h m, β i (cid:13) ∗ γ C ′ . But h m, β i is a Kripke node, andwe already saw that, for δ ∈ ≥ k , ⊥ | = δ ∈ C ′ , which is a contradiction. Hencethere are infinitely many different γ ’s. That means there is a branch Br suchthat every node on Br has infinitely many different γ ’s as extensions. Pick anon-standard m such that the corresponding γ extends a non-standard node of Br . But again, ⊥ | = “ C is a c -bar,” so ⊥ | = “there is a node δ ∈ Br such that δ ∈ C ; i.e. every extension of δ is in C ′ .” This contradicts the choice of γ . Lemma 22. ⊥ | = IZF
Proof.
As before, all of the axioms have soft proofs, save for Separation. Given φ and σ , let Sep φ,σ be {h B i ∪ B p , τ i | h B i , τ i ∈ σ and p (cid:13) ∗ φ ( τ ) }∪{h B, τ i | for some ¬h n, α i ∈ B and some B i , h B i , τ i ∈ σ, ¬h n, α i (cid:13) ∗ B i , and ¬h n, α i (cid:13) ∗ φ ( τ ) } . By lemma 20, this works. Π does not imply FAN fullAs usual, let G be generic as above. In M [ G ], the Kripke model will have bottomnode ⊥ , and successor nodes labeled by those α ∈ ∗ such that G ( α ) = ∞ .As is standard, terms are defined inductively, and always subject to the usualrestrictions so as to have a Kripke model. That is, to define the full model [9]over any partial order h P, < i , at node p ∈ P a term σ is any function withdomain P ≥ p such that σ ( q ) is a set of terms at node q ; furthermore, withtransition function f qr for q < r , if τ ∈ σ ( q ) then f qr ( τ ) ∈ σ ( r ); finally, f pq isextended to σ by restriction: f pq ( σ ) = σ ↾ P ≥ q . That is called the full model,because everything possible is being thrown in. For the current construction, wewill take a sub-model of the full model by imposing one additional restriction:a term at any node α other than ⊥ must be in the ground model M .Let C be the term such that ⊥ | = “ ˆ β ∈ C ” ( β ∈ ∗ ) iff, for some initialsegment β ↾ n of β, G ( β ↾ n ) = IN, and, at node α = ⊥ , α | = “ ˆ β ∈ C ” iff β isnot an initial segment of α . Lemma 23. ⊥ | = FAN Π Proof. If ⊥ | = “ B ⊆ ∗ is decidable” then, for any β ∈ ∗ , ⊥ | = “ ˆ β ∈ B ” iff,for some node α = ⊥ , α | = “ ˆ β ∈ B ” iff the same holds for all α = ⊥ . Hence ⊥ | = “ B = ˆ B M ” for some set B M ∈ M . So if ⊥ | = “ B n is a sequence ofdecidable trees,” then that sequence is the image of a sequence of sets from M .Hence their intersection internally is the image of a set from M . So if T n B n is internally a bar, it is the image of a bar, and by the Fan Theorem in M isuniform. Lemma 24. ⊥ 6| = FAN full roof. At ⊥ , C is not uniform, so it suffices to show ⊥ | = “ C is a bar”. If ⊥ | = “ P is a path through 2 ∗ ” then ⊥ | = “ P is decidable”, and as above P is then the image of a ground model path. Generically, for some β along thatpath, G ( β ) = IN. For that β, ⊥ | = “ P goes through ˆ β and ˆ β ∈ C. ” Lemma 25. ⊥ | = IZF
Proof.
Not only are most of the axioms trivial to verify, in this case even Sep-aration is too. Given a formula φ , term σ , and node nd , let Sep φ,σ ( nd ) be { τ | τ ∈ σ and nd | = φ ( τ ) } . The reason that at node α this is in M is that, at α , φ ’s parameters can also be interpreted in M , and so truth at α is definablein M . • We have seen that FAN full holds in every topological model, and thatFAN Π holds in the model over any connected Heyting algebra. Are thereany other sufficient or necessary properties for any of the various fan the-orems we have been considering to hold or fail in a Heyting-valued model? • As a particular instance of the previous question, if a Heyting algebrasatisfies FAN ∆ (resp. FAN c ), does it automatically satisfy FAN c (resp.FAN Π )? • Although we were not able to make use of any Heyting algebras otherthan Ω, some seem worthwhile to investigate, as possibly separating someof these fan theorems, or perhaps having some other interesting properties.We would include among these K ( T ) for various natural spaces T , suchas 2 N . We would also include other ways of killing points, such as over ameasure space τ with measure λ modding out by sets of measure 0: U ∼ V ⇐⇒ λ ( U ) = λ ( V ) = λ ( V ∩ U )(two opens are equivalent if their symmetric difference is of measure zero).The space τ / ∼ should be a Heyting algebra, which we will denote byanalogy with K as L ( τ ). Of particlar interest seem to be L ( I ) and L ( I × I ). • In the models presented here, the principles in question were not true.That’s different from their being false (meaning their negations beingtrue). We expect this could be done by iterating the constructions pre-sented here. That is, to each terminal node of the model append anothermodel of the same kind, starting with the ambient universe of that termi-nal node as the new ground model. By iterating this procedure infinitelyoften, one is left with a Kripke model with no terminal nodes. In orderstill to have a model of IZF, to get the Power Set Axiom for instance,terms for all of these bars from the iteration might have to be present at ⊥ , or perhaps some other fix would work. So this suggestion would atleast take some work to implement, and might even demand some newideas.It would be even better, or at least different, if we had a model withone fixed counter-example. Maybe the models presented here could be so21weaked. For instance, for FAN ∆ , could we just throw away the terminalnodes? For FAN c , it might work not to stop a node just because ¬ b ′ istrue, but rather to continue extending the node to allow finitely many ¬ b ′ sto be true. Or maybe a more radical idea is needed. • One of the referees asked about the role of Choice here. It is not thathard to see that Dependent Choice fails in most (or all) of these models.Are there some nice choice principles that are true here? Are there othermodels in which DC or other choice principles of interest hold? Are theresignificant fragments of Choice that are incompatible with these separa-tions? • Within reverse classical mathematics, many weakenings of Weak K¨onig’sLemma (classically equivalent to the Fan Theorem) have been identified.Of interest to us here is Weak Weak K¨onig’s Lemma. Whereas WKLstates that any bar (closed under extension, for simplicity) contains anentire level of 2 ∗ , WWKL states that any bar contains half of a level.(WWKL has been shown to be connected to the development of measuretheory.) In our context, any of the principles we have been consideringcould be so weakened, yielding Weak FAN ∆ , Weak FAN c , Weak FAN Π ,and Weak FAN full . Clearly any principle implies its weak correlate (e.g.FAN Π implies Weak FAN Π ), and any weak principle implies the weakprinciples lower down (e.g. Weak FAN Π implies Weak FAN c ), forminga bit of a square. Are there any implications along the diagonal (e.g.between FAN c and Weak FAN Π )? Are these weak principles even naturalor interesting, by being equivalent with interesting theorems? • Are there any other interesting principles to be found here, for instanceΠ n -FAN for n >
1, or adaptations of reverse math principles beneathWKL other than WWKL?
References [1] Michael Beeson,
Foundations of Constructive Mathematics , Springer-Verlag, 1985.[2] Josef Berger,
The logical strength of the uniform continuity theorem , Logical Approachesto Computational Barriers, Proceedings of CiE 2006, LNCS 3988, 2006, pp. 35–39.[3] ,
A separation result for varieties of Brouwer’s Fan Theorem , Proceedings ofthe 10th Asian Logic Conference (ALC 10), Kobe University in Kobe, Hyogo, Japan,September 1-6, 2008, 2010, pp. 85–92.[4] Errett Bishop and Douglas Bridges,
Constructive Analysis , Springer-Verlag, 1985.[5] Hannes Diener,
Compactness under constructive scrutiny , Ph.D. Thesis, 2008.[6] Hannes Diener and Iris Loeb,
Sequences of real functions on [0, 1] in constructive reversemathematics , Annals of Pure and Applied Logic (2009), pp. 50-61.[7] M. Fourman and J. Hyland,
Sheaf models for analysis , Applications of Sheaves (MichaelFourman, Christopher Mulvey, and Dana Scott, eds.), Lecture Notes in Mathematics,vol. 753, Springer Berlin / Heidelberg, 1979, pp. 280-301.[8] William Julian and Fred Richman,
A uniformly continuous function on [0,1] that iseverywhere different from its infimum , Pacific Journal of Mathematics (1984),pp. 333-340.[9] Robert S. Lubarsky,
Independence Results around Constructive ZF , Annals of Pure andApplied Logic (2005), pp. 209-225.[10] Stephen Simpson,
Subsystems of Second Order Arithmetic , ASL/Cambridge UniversityPress, 2009., ASL/Cambridge UniversityPress, 2009.