Predicative Aspects of Order Theory in Univalent Foundations
PPredicative Aspects of Order Theory inUnivalent Foundations
Tom de Jong ! ˇ University of Birmingham, United Kingdom
Martín Hötzel Escardó ! ˇ University of Birmingham, United Kingdom
Abstract
We investigate predicative aspects of order theory in constructive univalent foundations. By predicat-ive and constructive, we respectively mean that we do not assume Voevodsky’s propositional resizingaxioms or excluded middle. Our work complements existing work on predicative mathematics byexploring what cannot be done predicatively in univalent foundations. Our first main result is thatnontrivial (directed or bounded) complete posets are necessarily large. That is, if such a nontrivialposet is small, then weak propositional resizing holds. It is possible to derive full propositionalresizing if we strengthen nontriviality to positivity. The distinction between nontriviality andpositivity is analogous to the distinction between nonemptiness and inhabitedness. We prove ourresults for a general class of posets, which includes directed complete posets, bounded completeposets and sup-lattices, using a technical notion of a δ V -complete poset. We also show that nontriviallocally small δ V -complete posets necessarily lack decidable equality. Specifically, we derive weakexcluded middle from assuming a nontrivial locally small δ V -complete poset with decidable equality.Moreover, if we assume positivity instead of nontriviality, then we can derive full excluded middle.Secondly, we show that each of Zorn’s lemma, Tarski’s greatest fixed point theorem and Pataraia’slemma implies propositional resizing. Hence, these principles are inherently impredicative and apredicative development of order theory must therefore do without them. Finally, we clarify, inour predicative setting, the relation between the traditional definition of sup-lattice that requiressuprema for all subsets and our definition that asks for suprema of all small families. Theory of computation → Constructive mathematics; Theory ofcomputation → Type theory
Keywords and phrases order theory, constructivity, predicativity, univalent foundations
We investigate predicative aspects of order theory in constructive univalent foundations.By predicative and constructive, we respectively mean that we do not assume Voevodsky’spropositional resizing axioms [26, 27] or excluded middle. Our work is situated in ourlarger programme of developing domain theory constructively and predicatively in univalentfoundations. In previous work [12], we showed how to give a constructive and predicativeaccount of many familiar constructions and notions in domain theory, such as Scott’s D ∞ model of untyped λ -calculus and the theory of continuous dcpos. The present workcomplements this and other existing work on predicative mathematics (e.g. [2, 21, 6]) byexploring what cannot be done predicatively, as in [7, 8, 9, 10, 11]. We do so by showingthat certain statements crucially rely on resizing axioms in the sense that they are equivalentto them. Such arguments are important in constructive mathematics. For example, theconstructive failure of trichotomy on the real numbers is shown [4] by reducing it to anonconstructive instance of excluded middle.Our first main result is that nontrivial (directed or bounded) complete posets are ne-cessarily large. In [12] we observed that all our examples of directed complete posets havelarge carriers. We show here that this is no coincidence, but rather a necessity, in thesense that if such a nontrivial poset is small, then weak propositional resizing holds. It is a r X i v : . [ m a t h . L O ] F e b Predicative Aspects of Order Theory in UF possible to derive full propositional resizing if we strengthen nontriviality to positivity inthe sense of [19]. The distinction between nontriviality and positivity is analogous to thedistinction between nonemptiness and inhabitedness. We prove our results for a general classof posets, which includes directed complete posets, bounded complete posets and sup-lattices,using a technical notion of a δ V -complete poset. We also show that nontrivial locally small δ V -complete posets necessarily lack decidable equality. Specifically, we can derive weakexcluded middle from assuming the existence of a nontrivial locally small δ V -complete posetwith decidable equality. Moreover, if we assume positivity instead of nontriviality, then wecan derive full excluded middle.Secondly, we prove that each of Zorn’s lemma, Tarski’s greatest fixed point theoremand Pataraia’s lemma implies propositional resizing. Hence, these principles are inherentlyimpredicative and a predicative development of order theory in univalent foundations mustthus forgo them.Finally, we clarify, in our predicative setting, the relation between the traditional definitionof sup-lattice that requires suprema for all subsets and our definition that asks for supremaof all small families. This is important in practice in order to obtain workable definitions ofdcpo, sup-lattice, etc. in the context of predicative univalent mathematics.Our foundational setup is the same as in [12], meaning that our work takes places inintensional Martin-Löf Type Theory and adopts the univalent point of view [24]. This meansthat we work with the stratification of types as singletons, propositions, sets, 1-groupoids,etc., and that we work with univalence. At present, higher inductive types other thanpropositional truncation are not needed. Often the only consequences of univalence neededhere are functional and propositional extensionality. An exception is Section 2.3. Full detailsof our univalent type theory are given at the start of Section 2. Related work
Curi investigated the limits of predicative mathematics in CZF [2] in a series of papers [7, 8,9, 10, 11]. In particular, Curi shows (see [7, Theorem 4.4 and Corollary 4.11], [8, Lemma 1.1]and [9, Theorem 2.5]) that CZF cannot prove that various nontrivial posets, includingsup-lattices, dcpos and frames, are small. This result is obtained by exploiting that CZF isconsistent with the anti-classical generalized uniformity principle GUP [25, Theorem 4.3.5].Our related Theorem 35 is of a different nature in two ways. Firstly, our theorem is in thespirit of reverse constructive mathematics [18]: Instead of showing that GUP implies thatthere are no non-trivial small dcpos, we show that the existence of a non-trivial small dcpois equivalent to weak propositional resizing, and that the existence of a positive small dcpois equivalent to full propositional resizing. Thus, if we wish to work with small dcpos, weare forced to assume resizing axioms. Secondly, we work in univalent foundations ratherthan CZF. This may seem a superficial difference, but a number of arguments in Curi’spapers [9, 10] crucially rely on set-theoretical notions and principles such as transitive set,set-induction, weak regular extension axiom wREA, which cannot even be formulated in theunderlying type theory of univalent foundations. Moreover, although Curi claims that thearguments of [7, 8] can be adapted to some version of Martin-Löf Type Theory, it is presentlynot known whether there is any model of univalent foundations which validates GUP.
Organization
Section 2 : Foundations and size matters, including impredicativity, relation to excludedmiddle, univalence and closure under embedded retracts.
Section 3 : Nontrivial and positive . de Jong and M. H. Escardó 3 δ V -complete posets and reductions to impredicativity and excluded middle. Section 4 :Predicative invalidity of Zorn’s lemma, Tarski’s fixed point theorem and Pataraia’s lemma.
Section 5 : Comparison of completeness w.r.t. families and w.r.t. subsets.
Section 6 : Conclusionand future work.
We work with a subset of the type theory described in [24] and we mostly adopt theterminological and notational conventions of [24]. We include + (binary sum), Π (dependentproducts), Σ (dependent sum), Id (identity type), and inductive types, including (emptytype), (type with exactly one element ⋆ : ), N (natural numbers). We assume a universe U and two operations: for every universe U a successor universe U + with U : U + , andfor every two universes U and V another universe U ⊔ V such that for any universe U , wehave U ⊔ U ≡ U and U ⊔ U + ≡ U + . Moreover, ( − ) ⊔ ( − ) is idempotent, commutative,associative, and ( − ) + distributes over ( − ) ⊔ ( − ). We write U : ≡ U +0 , U : ≡ U +1 , . . . andso on. If X : U and Y : V , then X + Y : U ⊔ V and if X : U and Y : X → V , then thetypes Σ x : X Y ( x ) and Π x : X Y ( x ) live in the universe U ⊔ V ; finally, if X : U and x, y : X , thenId X ( x, y ) : U . The type of natural numbers N is assumed to be in U and we postulatethat we have copies U and U in every universe U . We assume function extensionality andpropositional extensionality tacitly, and univalence explicitly when needed. Finally, we use asingle higher inductive type: the propositional truncation of a type X is denoted by ∥ X ∥ and we write ∃ x : X Y ( x ) for ∥ P x : X Y ( x ) ∥ . We introduce the fundamental notion of a type having a certain size and specify theimpredicativity axioms under consideration (Section 2.2). We also note the relation toexcluded middle (Section 2.2) and univalence (Section 2.3). Finally in Section 2.4 we reviewembeddings and sections and establish our main technical result on size, namely that havinga certain size is closed under retracts whose sections are embeddings. ▶ Definition 1 (Size,
UF-Slice.html in [16]) . A type X in a universe U is said to have size V if it is equivalent to a type in the universe V . That is, X has-size V : ≡ P Y : V ( Y ≃ X ) . We consider various impredicativity axioms and their relation to (weak) excluded middle.The definitions and propositions below may be found in [15, Section 3.36], so proofs areomitted here. ▶ Definition 2 (Impredicativity axioms) .(i) By Propositional-Resizing U , V we mean the assertion that every proposition P in auniverse U has size V . (ii) The type of all propositions in a universe U is denoted by Ω U . Observe that Ω U : U + .We write Ω -Resizing U , V for the assertion that the type Ω U has size V . (iii) The type of all ¬¬ -stable propositions in a universe U is denoted by Ω ¬¬U , where aproposition P is ¬¬ -stable if ¬¬ P implies P . By Ω ¬¬ -Resizing U , V we mean theassertion that the type Ω ¬¬U has size V . (iv) For the particular case of a single universe, we write
Ω -Resizing U and Ω ¬¬ -Resizing U for the respective assertions that Ω U has size U and Ω ¬¬U has size U . Predicative Aspects of Order Theory in UF ▶ Proposition 3.(i)
The principle
Ω -Resizing U , V implies Propositional-Resizing U , V for every two universes U and V . (ii) The conjunction of
Propositional-Resizing U , V and Propositional-Resizing V , U implies Ω -Resizing U , V + for every two universes U and V . It is possible to define a weaker variation of propositional resizing for ¬¬ -stable propositionsonly (and derive similar connections), but we don’t have any use for it in this paper. ▶ Definition 4 ((Weak) excluded middle) .(i)
Excluded middle in a universe U asserts that for every proposition P in U either P or ¬ P holds. (ii) Weak excluded middle in a universe U asserts that for every proposition P in U either ¬ P or ¬¬ P holds. We note that weak excluded middle says precisely that ¬¬ -stable propositions are decidableand is equivalent to de Morgan’s Law. ▶ Proposition 5.
Excluded middle implies impredicativity. Specifically, (i)
Excluded middle in U implies Ω -Resizing U , U . (ii) Weak excluded middle in U implies Ω ¬¬ -Resizing U , U . Axioms should be subsingletons. For (weak) excluded middle this can be proved usingfunction extensionality. Assuming univalence we can prove that Propositional-Resizing U , V and Ω -Resizing U , V are subsingletons. More generally, univalence allows us to prove that thestatement that X has size V is a proposition. ▶ Proposition 6 (cf. has-size-is-subsingleton in [15]) . If V and U ⊔ V are univalentuniverses, then X has-size V is a proposition for every X : U . We can give a sort of converse to the above theorem. ▶ Proposition 7.
The type X has-size U is a proposition for every X : U if and only if U isa univalent universe. Proof.
Note that X has-size U is P Y : U Y ≃ X , so this can be found in [15, Section 3.14]. ◀ We show our main technical result on size here, namely that having a size is closed underretracts whose sections are embeddings. ▶ Definition 8 (Sections, retractions and embeddings) .(i) A section is a map s : X → Y together with a left inverse r : Y → X , i.e. the mapssatisfy r ◦ s ∼ id . We call r the retraction and say that X is a retract of Y . (ii) A function f : X → Y is an embedding if the map ap f : ( x = y ) → ( f ( x ) = f ( y )) is anequivalence for every x, y : X . (See [24, Definition 4.6.1(ii)].) (iii) A section-embedding is a section s : X → Y that moreover is an embedding. We alsosay that X is an embedded retract of Y . . de Jong and M. H. Escardó 5 We recall the following facts about embeddings and sections. ▶ Lemma 9.(i)
A function f : X → Y is an embedding if and only if all its fibres are subsingletons,i.e. Q y : Y is-subsingleton(fib f ( y )) . (See [24, Proof of Theorem 4.6.3].) (ii) If every section is an embedding, then every type is a set. (See [22, Remark 3.11(2)].) (iii)
Sections to sets are embeddings. (See [15, lc-maps-into-sets-are-embeddings ].)
In phrasing our results it is helpful to extend the notion of size from types to functions. ▶ Definition 10 (Size (for functions),
UF-Slice.html in [16]) . A function f : X → Y is saidto have size V if every fibre has size V . ▶ Lemma 11 (cf.
UF-Slice.html in [16]) .(i)
A type X has size V if and only if the unique map X → U has size V . (ii) If f : X → Y has size V and Y has size V , then so does X . (iii) If s : X → Y is a section-embedding and Y has size V , then s has size V too, regardlessof the size of X . Proof.
The first two claims follow from the fact that for any map f : X → Y we have anequivalence X ≃ P y : Y fib f ( y ) (see [24, Lemma 4.8.2]). For the third claim, suppose that s : X → Y an embedding with retraction r : Y → X . By the second part of the proof ofTheorem 3.10 in [22], we have fib s ( y ) ≃ ∥ s ( r ( y )) = y ∥ , from which the claim follows. ◀▶ Lemma 12.(i) If X is an embedded retract of Y and Y has size V , then so does X . (ii) If X is a retract of a set Y and Y has size V , then so does X . Proof.
The first statement follows from (ii) and (iii) of Lemma 11. The second follows fromthe first and item (iii) of Lemma 9. ◀ We show that constructively and predicatively many structures from order theory (directedcomplete posets, bounded complete posets, sup-lattices) are necessarily large and necessarilylack decidable equality. We capture these structures by a technical notion of a δ V -completeposet in Section 3.1. In Section 3.2 we define when such structures are nontrivial and introducethe constructively stronger notion of positivity. Section 3.3 and Section 3.4 contain the twofundamental technical lemmas and the main theorems, respectively. Finally, Section 3.5considers alternative formulations of being nontrivial and positive that ensure that thesenotions are properties, as opposed to data and shows how the main theorems remain valid,assuming univalence. δ V -complete Posets We start by introducing a class of weakly complete posets that we call δ V -complete posets.The notion of a δ V -complete poset is a technical and auxiliary notion sufficient to make ourmain theorems go through. The important point is that many familiar structures (dcpos,bounded complete posets, sup-lattices) are δ V -complete posets (see Examples 15). Predicative Aspects of Order Theory in UF ▶ Definition 13 ( δ V -complete poset, δ x,y,P , W δ x,y,P ) . A poset ( X, ⊑ ) is δ V -complete for auniverse V if for every pair of elements x, y : X with x ⊑ y and every subsingleton P in V ,the family δ x,y,P : 1 + P → X inl( ⋆ ) x ;inr( p ) y ; has a supremum W δ x,y,P in X . N.B. In the above definition, the carrier of the poset and the values of the partial order canlive in arbitrary universes which may or may not be different from V . ▶ Remark 14 (Every poset is δ V -complete, classically). Consider a poset ( X, ⊑ ) and a pair ofelements x ⊑ y . If P : V is a decidable proposition, then we can define the supremum of δ x,y,P by case analysis on whether P holds or not. For if it holds, then the supremum is y ,and if it does not, then the supremum is x . Hence, if excluded middle holds in V , then thefamily δ x,y,P has a supremum for every P : V . Thus, if excluded middle holds in V , thenevery poset (in any universe) is δ V -complete.The above remark naturally leads us to ask whether the converse also holds, i.e. if everyposet is δ V -complete, does excluded middle in V hold? As far as we know, we can only getweak excluded middle in V , as we will later see in Proposition 18. This proposition alsoshows that in the absence of excluded middle, the notion of δ V -completeness isn’t trivial.For now, we focus on the fact that, also constructively and predicatively, there are manyexamples of δ V -complete posets. ▶ Examples 15.(i)
Every V -sup-lattices is δ V -complete. That is, if a poset X has suprema for all families I → X with I in the universe V , then X is δ V -complete. (ii) The V -sup-lattice Ω V is δ V -complete. The type Ω V of propositions in V is a V -sup-latticewith the order given by implication and suprema by existential quantification. Hence, Ω V is δ V -complete. Specifically, given propositions Q , R and P , the supremum of δ Q,R,P is given by Q ∨ ( R × P ) . (iii) The V -powerset P V ( X ) : ≡ X → Ω V of a type X is δ V -complete. Note that P V ( X ) isanother example of a V -sup-lattice (ordered by subset inclusion and with suprema givenby unions) and hence δ V -complete. (iv) Every V -bounded complete posets is δ V -complete. That is, if ( X, ⊑ ) is a poset withsuprema for all bounded families I → X with I in the universe V , then ( X, ⊑ ) is δ V -complete. A family α : I → X is bounded if there exists some x : X with α ( i ) ⊑ x for every i : I . For example, the family δ x,y,P is bounded by y . (v) Every V -directed complete poset (dcpo) is δ V -complete, since the family δ x,y,P is directed.We note that [12] provides a host of examples of V -dcpos. In Remark 14 we saw that if we can decide a proposition P , then we can define W δ x,y,P bycase analysis. What about the converse? That is, if δ x,y,P has a supremum and we knowthat it equals x or y , can we then decide P ? Of course, if x = y , then W δ x,y,P = x = y , sowe don’t learn anything about P . But what if add the assumption that x ̸ = y ? It turnsout that constructively we can only expect to derive decidability of ¬ P in that case. This . de Jong and M. H. Escardó 7 is due to the fact that x ̸ = y is a negated proposition, which is rather weak constructively,leading us to later define (see Definition 20) a constructively stronger notion for elements of δ V -complete posets. ▶ Definition 16 (Nontrivial) . A poset ( X, ⊑ ) is nontrivial if we have designated x, y : X with x ⊑ y and x ̸ = y . ▶ Lemma 17.
Let ( X, ⊑ , x, y ) be a nontrivial poset. We have the following implications forevery proposition P : V : (i) if the supremum of δ x,y,P exists and x = W δ x,y,P , then ¬ P is the case. (ii) if the supremum of δ x,y,P exists and y = W δ x,y,P , then ¬¬ P is the case. Proof.
Let P : V be an arbitrary proposition. (i) Suppose that x = W δ x,y,P and assume for a contradiction that we have p : P . Then y ≡ δ x,y,P (inr( p )) ⊑ W δ x,y,P = x, which is impossible by antisymmetry and ourassumptions that x ⊑ y and x ̸ = y . (ii) Suppose that y = W δ x,y,P and assume for a contradiction that ¬ P holds. Then x = W δ x,y,P = y , contradicting our assumption that x ̸ = y . ◀▶ Proposition 18 (cf. Section 4 of [12]) . Let be the poset with exactly two elements ⊑ .If is δ V -complete, then weak excluded middle in V holds. Proof.
Suppose that were δ V -complete and let P : V be an arbitrary subsingleton. Wemust show that ¬ P is decidable. Since has exactly two elements, the supremum W δ , ,P must be 0 or 1. But then we apply Lemma 17 to get decidability of ¬ P . ◀ That we can’t improve Lemma 17 to decidability of P is shown by the following observation. ▶ Proposition 19.
Recall Examples 15, which show that Ω V is δ V -complete. Suppose thatfor every two propositions Q and R with Q ⊑ R and Q ̸ = R we have that the equality R = W δ Q,R,P in Ω V implies P for every proposition P : V . Then excluded middle in V holds. Proof.
Assume the hypothesis in the proposition. We are going to show that ¬¬ P → P forevery proposition P : V , from which excluded middle in V holds. Let P be a proposition in V and assume that ¬¬ P . This yields ̸ = P , so by assumption the equality P = W δ ,P,P implies P . But, recalling item (ii) of Examples 15, we have exactly this equality W δ ,P,P = P . ◀ We have seen that having a pair of elements x, y with x ⊑ y and x ̸ = y is very weak con-structively. As promised in the introduction of this section, we now introduce a constructivelystronger notion. ▶ Definition 20 (Strictly below, x ⊏ y ) . Let ( X, ⊑ ) be a δ V -complete poset and x, y : X .We say that x is strictly below y if x ⊑ y and, moreover, for every z ⊒ y and every proposition P : V , the equality z = W δ x,z,P implies P . Note that with excluded middle, x ⊏ y is equivalent to the conjunction of x ⊑ y and x ̸ = y .But constructively, the former is much stronger, as the following example and propositionillustrate. ▶ Example 21 (Strictly below in Ω V ) . Recall from Examples 15 that Ω V is δ V -complete. Let P : V be an arbitrary proposition. Observe that V ̸ = P precisely when ¬¬ P holds. However, V is strictly below P if and only if P holds. Predicative Aspects of Order Theory in UF ▶ Proposition 22.
For a δ V -complete poset ( X, ⊑ ) and x, y : X , we have that x ⊏ y impliesboth x ⊑ y and x ̸ = y . However, if the conjunction of x ⊑ y and x ̸ = y implies x ⊏ y forevery δ V -complete poset ( X, ⊑ ) and every x, y : X , then excluded middle in V holds. Proof.
Note that x ⊏ y implies x ⊑ y by definition. Now suppose that x ⊏ y and assume x = y for a contradiction. Since we assumed x ⊏ y , the equality y = W δ x,y, V implies that V holds. But this equality holds since x = y by our other assumption, so x ̸ = y , as desired.For P : Ω V we observed that V ̸ = P is equivalent to ¬¬ P and that V ⊏ P is equivalentto P , so if we had x ⊏ y → (( x ⊑ y ) × ( x ̸ = y )) in general, then we would have ¬¬ P → P for every proposition P in V , which is equivalent to excluded middle in V . ◀▶ Lemma 23.
Let ( X, ⊑ ) be a δ V -complete poset and x, y, z : X . The following hold: (i) If x ⊑ y ⊏ z , then x ⊏ z . (ii) If x ⊏ y ⊑ z , then x ⊏ z . Proof.
For (i), assume x ⊑ y ⊏ z , let P be an arbitrary proposition in V and suppose that z ⊑ w . We must show that w = W δ x,w,P implies P . But y ⊏ z , so we know that theequality w = W δ y,w,P implies P . Now observe that W δ x,w,P ⊑ W δ y,w,P , so if w = W δ x,w,P ,then w = W δ y,w,P , finishing the proof. For (ii), assume x ⊏ y ⊑ z , let P be an arbitraryproposition in V and suppose that z ⊑ w . We must show that w = W δ x,w,P implies P . But x ⊏ y and y ⊑ w , so this follows immediately. ◀▶ Proposition 24.
Let ( X, ⊑ ) be a V -sup-lattice and let y : X . The following are equivalent: (i) the least element of X is strictly below y ; (ii) for every family α : I → X with I : V and y ⊑ W α , there exists some element i : I . (iii) there exists some x : X with x ⊏ y . Proof.
Write ⊥ for the least element of X . By Lemma 23 we have: ⊥ ⊏ y ⇐⇒ ∃ x : X ( ⊥ ⊑ x ⊏ y ) ⇐⇒ ∃ x : X ( x ⊏ y ) , which proves the equivalence of (i) and (iii). It remains to prove that (i) and (ii) are equivalent.Suppose that ⊥ ⊏ y and let α : I → X with y ⊑ W α . Using ⊥ ⊏ y ⊑ W α and Lemma 23, wehave ⊥ ⊏ W α . Hence, we only need to prove W α ⊑ W δ ⊥ , W α, ∃ i : I , but α j ⊑ W δ ⊥ , W α, ∃ i : I forevery j : I , so this is true indeed. For the converse, assume that y satisfies (ii), suppose z ⊒ y and let P : V be a proposition such that z = W δ ⊥ ,z,P . We must show that P holds. Butnotice that y ⊑ z = W δ ⊥ ,z,P = W (( p : P ) z ), so P must be inhabited as y satisfies (ii). ◀ Item (ii) in Proposition 24 says exactly that y is a positive element in the sense of [19, p. 98].We note that item (iii) in Proposition 24 makes sense even when ( X, ⊑ ) is not a V -sup-lattice,but just a δ V -complete poset. Accordingly, we make the following definition. ▶ Definition 25 (Positive element) . An element of a δ V -complete poset is positive if it satisfiesitem (iii) in Proposition 24. ▶ Proposition 26.
Let D be a V -dcpo with a least element ⊥ . Then a compact element x : D is positive if and only if x ̸ = ⊥ . Proof.
One implication is taken care of by Proposition 22. For the converse, suppose that x ̸ = ⊥ . We show that ⊥ is strictly below x . For if x ⊑ y = W δ ⊥ ,y,P , then by compactness of x , there must exist i : + P such that x ⊑ δ ⊥ ,y,P ( i ) already. But i can’t be equal to inl( ⋆ ),since x is assumed to be different from ⊥ . Hence, i = inr( p ) and P must hold. ◀ . de Jong and M. H. Escardó 9 Looking to strengthen the notion of a nontrivial poset, we make the following definition,whose terminology is inspired by Definition 25. ▶ Definition 27 (Positive poset) . A δ V -complete poset X is positive if we have designated x, y : X with x strictly below y . ▶ Examples 28.(i)
The δ V -complete poset Ω V is positive with V ⊏ V . (ii) Let X : V be a set with a point x : X . The δ V -complete poset P V ( X ) (recall Examples 15)with points ∅ and { x } is positive. Here we employ the familiar set-theoretic notation { x } for what is formally the function ( y x = y ) : X → Ω V , which is well-defined,because X is assumed to be a set. Similarly, ∅ formally denotes the function y .By contrast, if we only know that X is nonempty (i.e. X ̸ = V ), then P V ( X ) withpoints ∅ and X (considered as a subset) is only nontrivial. We show that the type of propositions in V is a retract of any positive δ V -complete poset andthat the type of ¬¬ -stable propositions in V is a retract of any nontrivial δ V -complete poset. ▶ Definition 29 ( ∆ x,y : Ω V → X ) . Suppose that ( X, ⊑ , x, y ) is a nontrivial δ V -completeposet. We define ∆ x,y : Ω V → X by the assignment P W δ x,y,P . We will often omit the subscripts in ∆ x,y when it is clear from the context. ▶ Definition 30 (Locally small) . A δ V -complete poset ( X, ⊑ ) is locally small if its order hasvalues of size V , i.e. we have ⊑ V : X → X → V with ( x ⊑ y ) ≃ ( x ⊑ V y ) for every x, y : X . ▶ Examples 31.(i)
The V -sup-lattices Ω V and P V ( X ) (for X : V ) are locally small. (ii) All examples of V -dcpos in [12] are locally small. ▶ Lemma 32.
A locally small δ V -complete poset ( X, ⊑ ) with elements x ⊑ y is nontrivialif and only if the composite Ω ¬¬V , → Ω V ∆ x,y −−−→ X is a section. Proof.
Suppose first that ( X, ⊑ , x, y ) is nontrivial and locally small. We define r : X → Ω ¬¬V z z ̸⊑ V x. Note that negated propositions are ¬¬ -stable, so r is well-defined. Let P : V be anarbitrary ¬¬ -stable proposition. We want to show that r (∆ x,y ( P )) = P . By propositionalextensionality, establishing logical equivalence suffices. Suppose first that P holds. Then∆ x,y ( P ) ≡ W x,y,P = y , so r (∆ x,y ( P )) = r ( y ) ≡ ( y ̸⊑ V x ) holds by antisymmetry and ourassumptions that x ⊑ y and x ̸ = y . Conversely, assume that r (∆ x,y ( P )) holds, i.e. that wehave W δ x,y,P ̸⊑ V x . Since P is ¬¬ -stable, it suffices to derive a contradiction from ¬ P . Soassume ¬ P . Then x = W δ x,y,P , so r (∆ x,y ( P )) = r ( x ) ≡ x ̸⊑ V x , which is false by reflexivity.For the converse, assume that Ω ¬¬V , → Ω V ∆ x,y −−−→ X has a retraction r : Ω ¬¬V → X . Then V = r (∆ x,y ( V )) = r ( x ) and V = r (∆ x,y ( V )) = r ( y ), where we used that V and V are ¬¬ -stable. Since V ̸ = V , we get x ̸ = y , so ( X, ⊑ , x, y ) is nontrivial, as desired. ◀ The appearance of the double negation in the above lemma is due to the definition ofnontriviality. If we instead assume a positive poset X , then we can exhibit all of Ω V as aretract of X . ▶ Lemma 33.
A locally small δ V -complete poset ( X, ⊑ ) with elements x ⊑ y is positiveif and only if the map ∆ x,y : Ω V → X is a section. Proof.
Suppose first that ( X, ⊑ , x, y ) is positive and locally small. We define r : X Ω V z y ⊑ V z. Let P : V be arbitrary proposition. We want to show that r (∆ x,y ( P )) = P . Because ofpropositional extensionality, it suffices to establish a logical equivalence between P and r (∆( P )). Suppose first that P holds. Then ∆( P ) = y , so r (∆( P )) = r ( y ) ≡ ( y ⊑ V y ) holdsas well by reflexivity. Conversely, assume that r (∆( P )) holds, i.e. that we have y ⊑ V W δ x,y,P .Since W δ x,y,P ⊑ y always holds, we get y = W δ x,y,P by antisymmetry. But by assumption, x is strictly below y , so P must hold.For the converse, assume that ∆ x,y : Ω V → X has a retraction r : X → Ω V . We mustshow that the equality z = ∆ x,z ( P ) implies P for every z ⊒ y and proposition P : V .Assuming z = ∆ x,z ( P ), we have V = r (∆ x,z ( V )) = r ( z ) = r (∆ x,z ( P )) = P , so P must holdindeed. Hence, ( X, ⊑ , x, y ) is positive, as desired. ◀ We present our main theorems here, which show that, constructively and predicatively,nontrivial δ V -complete posets are necessarily large and necessarily lack decidable equality. ▶ Definition 34 (Small) . A δ V -complete poset is small if it is locally small and its carrierhas size V . ▶ Theorem 35.(i)
There is a nontrivial small δ V -complete poset if and only if Ω ¬¬ -Resizing V holds. (ii) There is a positive small δ V -complete poset if and only if Ω -Resizing V holds. Proof. (i) Suppose that ( X, ⊑ , x, y ) is a nontrivial δ V -complete poset. By Lemma 32, wecan exhibit Ω ¬¬V as a retract of X . But X has size V by assumption, so by Lemma 12 andthe fact that Ω ¬¬V is a set, the type Ω ¬¬V has size V as well. For the converse, note that(Ω ¬¬V , → , V , V ) is a nontrivial V -sup-lattice with W α given by ¬¬∃ i : I α i . And if we assumeΩ ¬¬ -Resizing V , then it is small.(ii) Suppose that ( X, ⊑ , x, y ) is a positive poset. By Lemma 33, we can exhibit Ω V as aretract of X . But X has size V by assumption, so by Lemma 12 and the fact that Ω V is aset, the type Ω V has size V as well. For the converse, note that (Ω V , → , V , V ) is a positive V -sup-lattice. And if we assume Ω -Resizing V , then it is small. ◀▶ Lemma 36 ( retract-is-discrete and subtype-is- ¬¬ -separated in [16]) .(i) Types with decidable equality are closed under retracts. (ii)
Types with ¬¬ -stable equality are closed under retracts. ▶ Theorem 37.
There is a nontrivial locally small δ V -complete poset with decidable equalityif and only if weak excluded middle in V holds. Proof.
Suppose that ( X, ⊑ , x, y ) is a nontrivial locally small δ V -complete poset with decidableequality. Then by Lemmas 32 and 36, the type Ω ¬¬V must have decidable equality too. Butnegated propositions are ¬¬ -stable, so this yields weak excluded middle in V . For theconverse, note that (Ω ¬¬V , → , V , V ) is a nontrivial V -sup-lattice that has decidable equalityif and only if weak excluded middle in V holds. ◀ . de Jong and M. H. Escardó 11 ▶ Theorem 38.
The following are equivalent: (i)
There is a positive locally small δ V -complete poset with ¬¬ -stable equality. (ii) There is a positive locally small δ V -complete poset with decidable equality. (iii) Excluded middle in V holds. Proof.
Note that (ii) ⇒ (i), so we are left to show that (iii) ⇒ (ii) and that (i) ⇒ (iii). Forthe first implication, note that (Ω V , → , V , V ) is a positive V -sup-lattice that has decidableequality if and only if excluded middle in V holds. To see that (i) implies (iii), supposethat ( X, ⊑ , x, y ) is a positive locally small δ V -complete poset with ¬¬ -stable equality. Thenby Lemmas 33 and 36 the type Ω V must have ¬¬ -stable equality. But this implies that ¬¬ P → P for every proposition P in V which is equivalent to excluded middle in V . ◀ Lattices, bounded complete posets and dcpos are necessarily large and necessarily lackdecidable equality in our predicative constructive setting. More precisely, ▶ Corollary 39.(i)
There is a nontrivial small V -sup-lattice (or V -bounded complete poset or V -dcpo)if and only if Ω ¬¬ -Resizing V holds. (ii) There is a positive small V -sup-lattice (or V -bounded complete poset or V -dcpo)if and only if Ω -Resizing V holds. (iii) There is a nontrivial locally small V -sup-lattice (or V -bounded complete poset or V -dcpo)with decidable equality if and only if weak excluded middle in V holds. (iv) There is a positive locally small V -sup-lattice (or V -bounded complete poset or V -dcpo)with decidable equality if and only if excluded middle in V holds. The above notions of non-triviality and positivity are data rather than property. Indeed, anontrivial poset ( X, ⊑ ) is (by definition) equipped with two designated points x, y : X suchthat x ⊑ y and x ̸ = y . It is natural to wonder if the propositionally truncated versions ofthese two notions yield the same conclusions. In this section we show that this is indeed thecase if we assume univalence. The need for the univalence assumption comes from the factthat the notion of having a given size is property precisely if univalence holds, as shown inPropositions 6 and 7. ▶ Definition 40 (Nontrivial/positive in an unspecified way) . A poset ( X, ⊑ ) is nontrivialin an unspecified way if there exist some elements x, y : X such that x ⊑ y and x ̸ = y ,i.e. ∃ x : X ∃ y : X (( x ⊑ y ) × ( x ̸ = y )) . Similarly, we can define when a poset is positive in anunspecified way by truncating the notion of positivity. ▶ Theorem 41.
Suppose that the universes V and V + are univalent. (i) There is a small δ V -complete poset that is nontrivial in an unspecified way if and onlyif Ω ¬¬ -Resizing V holds. (ii) There is a small δ V -complete poset that is positive in an unspecified way if and only if Ω -Resizing V holds. Proof. (i) Suppose that ( X, ⊑ ) is a δ V -complete poset that is nontrivial in an unspecifiedway. By Proposition 6 and univalence of V and V + , type Ω ¬¬V has-size V is a proposition.By the universal property of the propositional truncation, in proving that Ω ¬¬V has-size V wecan therefore assume that are given points x, y : X with x ⊑ y and x ̸ = y . The result thenfollows from Theorem 35. (ii) By reduction to item (ii) of Theorem 35. ◀ Similarly, we can prove the following theorems by reduction to Theorems 37 and 38. ▶ Theorem 42.(i)
There is a locally small δ V -complete poset with decidable equality that is nontrivial inan unspecified way if and only if weak excluded middle in V holds. (ii) There is a locally small δ V -complete poset with decidable equality that is positive in anunspecified way if and only if excluded middle in V holds. In this section we construct a particular example of a V -sup-lattice that will prove very usefulin studying the predicative validity of some well-known principles in order theory. ▶ Definition 43 (Lifting, cf. [14]) . Fix a proposition P U in a universe U . Lifting P U withrespect to a universe V is defined by L V ( P U ) : ≡ X Q :Ω V ( Q → P U ) . This is a subtype of Ω V and it is closed under V -suprema (in particular, it contains theleast element). ▶ Examples 44.(i) If P U : ≡ U , then L V ( P U ) ≃ (cid:16)P Q :Ω V ¬ Q (cid:17) ≃ (cid:16)P Q :Ω V Q = V (cid:17) ≃ . (ii) If P U : ≡ U , then L V ( P U ) ≡ (cid:16)P Q :Ω V ( Q → U ) (cid:17) ≃ Ω V . What makes L V ( P U ) useful is the following observation. ▶ Lemma 45.
Suppose that the poset L V ( P U ) has a maximal element Q : Ω V . Then P U is equivalent to Q , which is the greatest element of L V ( P U ) . In particular, P U has size V .Conversely, if P U is equivalent to a proposition Q : Ω V , then Q is the greatest elementof L V ( P U ) . Proof.
Suppose that L V ( P U ) has a maximal element Q : Ω V . We wish to show that Q ≃ P U .By definition of L V ( P U ), we already have that Q → P U . So only the converse remains.Therefore suppose that P U holds. Then, V is an element of L V ( P U ). Obviously Q → V ,but Q is maximal, so actually Q = 1 V , that is, Q holds, as desired. Thus, Q ≃ P U .It is then straightforward to see that Q is actually the greatest element of L V ( P U ), since L V ( P U ) ≃ P Q ′ :Ω V ( Q ′ → Q ). For the converse, assume that P U is equivalent to a proposition Q : Ω V . Then, as before, L V ( P U ) ≃ P Q ′ :Ω V ( Q ′ → Q ), which shows that Q is indeed thegreatest element of L V ( P U ). ◀▶ Corollary 46.
Let P U be a proposition in U . The V -sup-lattice L V ( P U ) has all V -infima ifand only if P U has size V . Proof.
Suppose first that L V ( P U ) has all V -infima. Then it must have a infimum for theempty family V → L V ( P U ). But this infimum must be the greatest element of L V ( P U ). Soby Lemma 45 the proposition P U must have size V .Conversely, suppose that P U is equivalent to a proposition Q : V . Then the infimum of afamily α : I → L V ( P U ) with I : V is given by ( Q × Π i : I α i ) : V . ◀▶ Definition 47 ( Zorn’s-Lemma V , U , T ) . Let U , V and T be universes. Zorn’s-Lemma V , U , T asserts that every pointed V -dcpo with carrier in U and order taking values in T (cf. [12])has a maximal element. . de Jong and M. H. Escardó 13 It important to note that Zorn’s lemma does not imply the Axiom of Choice in the absenceof excluded middle [3]. If it did, then the following would be useless, since the Axiom ofChoice implies excluded middle, which in turn implies propositional resizing. ▶ Theorem 48.
Zorn’s-Lemma V , V + ⊔U , V implies Propositional-Resizing U , V . In particular, Zorn’s-Lemma V , V + , V implies Propositional-Resizing V + , V . Proof.
Suppose that Zorn’s-Lemma V , V + ⊔U , V were true. Then L V ( P ) : V + ⊔U has a maximalelement for every P : Ω U . Hence, by Lemma 45, every P : Ω U has size V . ◀ We can also use Lemma 45 to show that the following version of Tarski’s fixed pointtheorem [23] is not available predicatively. ▶ Definition 49 ( Tarski’s-Theorem V , U , T ) . The assertion
Tarski’s-Theorem V , U , T says thatevery monotone endofunction on a V -sup-lattice with carrier in a universe U and order takingvalues in a universe T has a greatest fixed point. ▶ Theorem 50.
Tarski’s-Theorem V , V + ⊔U , V implies Propositional-Resizing U , V . In particular, Tarski’s-Theorem V , V + , V implies Propositional-Resizing V + , V . Proof.
Suppose that Tarski’s-Theorem V , V + ⊔U , V were true and let P : Ω U be arbitrary.Consider the V -sup-lattice L V ( P ) : V + ⊔ U . By assumption, the identity map on this posethas a greatest fixed point, but this must be the greatest element of L V ( P ), which impliesthat P has size V by Lemma 45. ◀ Another famous fixed point theorem, for dcpos this time, is due to Pataraia [20, 13]which says that every monotone endofunction on a pointed dcpo has a least fixed point.(A dcpo is called pointed if it has a least element.) A crucial step in proving Pataraia’stheorem is the observation that every dcpo has a greatest monotone inflationary endofunction.(An endomap f : X → Y is inflationary when x ⊑ f ( x ) for every x : X .) We refer to thisintermediate result as Pataraia’s lemma. ▶ Definition 51 ( Pataraia’s-Lemma V , U , T , Pataraia’s-Theorem V , U , T ) .(i) Pataraia’s-Theorem V , U , T says that every monotone endofunction on a pointed V -dcpowith carrier in a universe U and order taking values in a universe T has a leastfixed point. (ii) Pataraia’s-Lemma V , U , T says that every V -dcpo with carrier in a universe U and ordertaking values in a universe T has a greatest monotone inflationary endofunction. A careful analysis of the proof in [13, Section 2] shows that in our predicative setting wecan still prove that Pataraia’s-Lemma V , U⊔T , U⊔T implies Pataraia’s-Theorem V , U , T . However,Pataraia’s lemma is not available predicatively. ▶ Theorem 52.
Pataraia’s-Lemma V , V + ⊔U , V implies Propositional-Resizing U , V . In particular, Pataraia’s-Lemma V , V + , V implies Propositional-Resizing V + , V . Proof.
Suppose that Pataraia’s-Lemma V , V + ⊔U , V were true and let P : Ω U be arbitrary.Consider the V -dcpo L V ( P ) : V + ⊔ U . By assumption, it has a greatest monotone inflationaryendomap g : L V ( P ) → L V ( P ). We claim that g ( V ) is a maximal element of L V ( P ), whichwould finish the proof by Lemma 45. So suppose that we have Q : L V ( P ) with g ( V ) ⊑ Q .Then we must show that Q ⊑ g ( V ). Define f Q : L V ( P ) → L V ( P ) by Q ′ Q ′ ∨ Q . Notethat f Q is monotone and inflationary, so that g ⊑ f Q . Hence, g ( V ) ⊑ f Q ( V ) = Q , asdesired. ◀ ▶ Remark 53.
For a single universe U , the usual proofs (see resp. [23] and [13, Section 2])of Tarski’s-Theorem U , U , U , Pataraia’s-Lemma U , U , U and (hence) Pataraia’s-Theorem U , U , U arealso valid in our predicative setting. However, in light of Theorem 35 and Theorem 35, thesestatements are not useful predicatively, because one would never be able to find interestingexamples of posets to apply the statements to.Finally, we note that Zorn’s lemma implies Pataraia’s lemma with the following universeparameters. Together with Theorem 52 this yields another proof that Zorn’s-Lemma V , V + , V implies Propositional-Resizing V + , V . ▶ Lemma 54.
Zorn’s-Lemma V , U⊔T , U⊔T implies
Pataraia’s-Lemma V , U , T . Proof.
Assume Zorn’s-Lemma V , U⊔T , U⊔T and let D : U be V -dcpo with order taking valuesin T . Consider the type MI D of monotone and inflationary endomaps on D . We can orderthese maps pointwise to get a V -dcpo with carrier and order taking values in U ⊔ T . Finally,MI D has a least element: the identity map. Hence, by our assumption, it has a maximalelement g : D → D . It remains to show that g is in fact the greatest element. To this end,let f : D → D be an arbitrary monotone inflationary endomap on D . We must show that f ⊑ g . Since f is inflationary, we have g ⊑ f ◦ g . So by maximality of g , we get g = f ◦ g .But f is monotone and g is inflationary, so f ⊑ f ◦ g = g , finishing the proof. ◀ The answer to the question whether Pataraia’s theorem (or similarly, a least fixed pointtheorem version of Tarki’s theorem) is inherently impredicative or (by contrast) does admita predicative proof has eluded us thus far.
In traditional impredicative foundations, completeness of posets is usually formulated usingsubsets. For instance, dcpos are defined as posets D such that every directed subset D hasa supremum in D . Examples 15 are all formulated using small families instead of subsets.While subsets are primitive in set theory, families are primitive in type theory, so this couldbe an argument for using families above. However, that still leaves the natural question ofhow the family-based definitions compare to the usual subset-based definitions, especiallyin our predicative setting, unanswered. This section aims to answer this question. We firststudy the relation between subsets and families predicatively and then clarify our definitionsin the presence of impredicativity. In our answers we will consider sup-lattices, but similararguments could be made for posets with other sorts of completeness, such as dcpos. All Subsets
We first show that simply asking for completeness w.r.t. all subsets is not satisfactory from apredicative viewpoint. In fact, we will now see that even asking for all subsets X → Ω T forsome fixed universe T is problematic from a predicative standpoint. ▶ Theorem 55.
Let U and V be universes and fix a proposition P U : U . Recall L V ( P U ) from Definition 43, which has V -suprema. Let T be any type universe. If L V ( P U ) hassuprema for all subsets L V ( P U ) → Ω T , then P U has size V independently of T . Proof.
Let T be a type universe and consider the subset S of L V ( P U ) given by Q T .Note that S has a supremum in L V ( P U ) if and only if L V ( P U ) has a greatest element, butby Lemma 45, the latter is equivalent to P U having size V . ◀ . de Jong and M. H. Escardó 15All Subsets Whose Total Spaces Have Size V The proof above illustrates that if we have a subset S : X → Ω T , then there is no reason whythe total space P x : X x ∈ S : ≡ P x : X ( S ( x ) holds) should have size T . In fact, for S ( x ) : ≡ T as above, the latter is equivalent to asking that X has size T . ▶ Definition 56 (Total space of a subset, T ) . Let T be a universe, X a type and S : X → Ω T a subset of X . The total space of S is defined as T ( S ) : ≡ P x : X x ∈ S. A naive attempt to solve the problem described in Theorem 55 would be to stipulatethat a V -sup-lattice X should have suprema for all subsets S : X → Ω V for which T ( S )has size V . Somewhat less naively, we might be more liberal and ask for suprema of subsets S : X → Ω U⊔V for which T ( S ) has size V . Here the carrier of X is in a universe U . Perhapssurprisingly, even this more liberal definition is too weak to be useful as the following exampleshows. ▶ Example 57 (Naturally occurring subsets whose total spaces are not necessarily small) . Let X be a poset with carrier in U and suppose that it has suprema for all (directed) subsets S : X → Ω U⊔V for which T ( S ) has size V . Now let f : X → X be a Scott continuousendofunction on X . We would want to construct the least fixed point of f as the supremumof the directed subset S : ≡ {⊥ , f ( ⊥ ) , f ( ⊥ ) , . . . } . Now, how do we show that its total space T ( S ) ≡ P x : X ( ∃ n : N x = f n ( ⊥ )) has size V ? A first guess might be that N ≃ T ( S ), whichwould do the job. However, it’s possible that f m ( ⊥ ) = f m +1 ( ⊥ ) for some natural number m ,which would mean that T ( S ) ≃ Fin( m ) for the least such m . The problem is that in theabsence of decidable equality on X we might not be able to decide which is the case. But X seldom has decidable equality, as we saw in Theorems 37 and 38. ▶ Remark 58.
The example above also makes clear that it is undesirable to impose aninjectivity condition on families, as the family N → X, n f n ( ⊥ ) is not necessarily injective.In fact, for every type X : U there is an equivalence between embeddings I , → X with I : V and subsets of X whose total spaces have size V , cf. [16, Slice.html ]. All V -covered Subsets The point of Example 57 is analogous to the difference between Bishop finiteness andKuratowski finiteness. Inspired by this, we make the following definition. ▶ Definition 59 ( V -covered subset) . Let X be a type, T a universe and S : X → Ω T a subsetof X . We say that S is V -covered for a universe V if we have a type I : V with a surjection e : I ↠ T ( S ) . In the example above, the subset S : ≡ {⊥ , f ( ⊥ ) , f ( ⊥ ) , . . . } is U -covered, because N ↠ T ( S ). ▶ Theorem 60.
For X : U and any universe V we have an equivalence between V -coveredsubsets X → Ω U⊔V and families I → X with I : V . Proof.
The forward map φ is given by ( S, I, e ) ( I, pr ◦ e ). In the other direction, wedefine ψ by mapping ( I, α ) to the triple (
S, I, e ) where S is the subset of X given by S ( x ) : ≡ ∃ i : I x = α ( i ) and e : I ↠ T ( S ) is defined as e ( i ) : ≡ ( α ( i ) , | ( i, refl) | ). The composite φ ◦ ψ is easily seen to be equal to the identity. To show that ψ ◦ φ equals the identity, weneed the following intermediate result, which is proved using function extensionality andpath induction. ▷ Claim.
Let
S, S ′ : X → Ω U⊔V , e : I → T ( S ) and e ′ : I → T ( S ′ ). If S = S ′ andpr ◦ e ∼ pr ◦ e ′ , then ( S, e ) = ( S ′ , e ′ ).The result then follows from the claim using function extensionality and propositionalextensionality. ◀▶ Corollary 61.
Let X be a poset with carrier in U and let V be any universe. Then X hassuprema for all V -covered subsets X → Ω U⊔V if and only if X has suprema for all families I → X with I : V . Families and Subsets in the Presence of Impredicativity
Finally, we compare our family-based approach to the subset-based approach in the presenceof impredicativity. ▶ Theorem 62.
Assume
Ω -Resizing T , U for every universe T . Then the following areequivalent for a poset X in a universe U : (i) X has suprema for all subsets; (ii) X has suprema for all U -covered subsets; (iii) X has suprema for all subsets whose total spaces have size U ; (iv) X has suprema for all families I → X with I : U . Proof.
Clearly (i) ⇒ (ii) ⇒ (iii). We show that (iii) implies (i), which proves the equivalenceof (i)–(iii). Assume that X has suprema for all subsets whose total spaces have size U andlet S : X → Ω T be any subset of X . Using Ω -Resizing T , U , the total space T ( S ) has size U .So X has a supremum for S by assumption, as desired. Finally, (ii) and (iv) are equivalentby Corollary 61. ◀ Notice that (iv) in Theorem 62 implies that X has suprema for all families I → X with I : V and V such that V ⊔ U ≡ U . Typically, in the examples of [12] for instance, U : ≡ U and V : ≡ U , so that V ⊔ U ≡ U holds. Thus, our V -families-based approach generalizes thetraditional subset-based approach. Firstly, we have shown, constructively and predicatively, that nontrivial dcpos, boundedcomplete posets and sup-lattices are all necessarily large and necessarily lack decidableequality. We did so by deriving a weak impredicativity axiom or weak excluded middlefrom the assumption that such nontrivial structures are small or have decidable equality,respectively. Strengthening nontriviality to the (classically equivalent) positivity condition,we derived a strong impredicativity axiom and full excluded middle.Secondly, we proved that Zorn’s lemma, Tarski’s greatest fixed point theorem andPataraia’s lemma all imply impredicativity axioms. Hence, these principles are inherentlyimpredicative and a predicative development of order theory (in univalent foundations) mustthus do without them.Thirdly, we clarified, in our predicative setting, the relation between the traditionaldefinition of a lattice that requires completeness with respect to subsets and our definitionthat asks for completeness with respect to small families.In future work, we wish to study the predicative validity of Pataraia’s theorem and Tarski’s least fixed point theorem. Curi [9, 10] develops predicative versions of Tarki’s fixed pointtheorem in some extensions of CZF. It is not clear whether these arguments could be adapted . de Jong and M. H. Escardó 17 to univalent foundations, because they rely on the set-theoretical principles discussed in theintroduction. The availability of such fixed-point theorems would be especially useful forapplication to inductive sets [1], which we might otherwise introduce in univalent foundationsusing higher inductive types [24]. In another direction, we have developed a notion ofapartness [5] for continuous dcpos [12] that is related to the notion of being strictly belowintroduced in this paper. Namely, if x ⊑ y are elements of a continuous dcpo, then x isstrictly below y if x is apart from y . In upcoming work, we give a constructive analysis ofthe Scott topology [17] using this notion of apartness. References Peter Aczel. An introduction to inductive definitions. In Jon Barwise, editor,
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