On the Constructive Dedekind Reals

Robert S. Lubarsky

Department of Mathematical Sciences, Florida Atlantic University 777 Glades Road Boca Raton, FL 33431, USA

                                       rlubarsk@fau.edu

Michael Rathjen^{∗}

Department of Pure Mathematics, University of Leeds Leeds LS2 9JT, United Kingdom

                                 rathjen@math.ohio-state.edu

September 20, 2007

Abstract

In order to build the collection of Cauchy reals as a set in constructive set theory, the only Power Set-like principle needed is Exponentiation. In contrast, the proof that the Dedekind reals form a set has seemed to require more than that. The main purpose here is to show that Exponentiation alone does not suffice for the latter, by furnishing a Kripke model of constructive set theory, CZF with Subset Collection replaced by Exponentiation, in which the Cauchy reals form a set while the Dedekind reals constitute a proper class.

1 Introduction

In classical mathematics, one principal approach to defining the real numbers is to use equivalence classes of Cauchy sequences of rational numbers, and the other is the method of Dedekind cuts wherein reals appear as subsets of ℚ with special properties. Classically the two methods are equivalent in that the resulting field structures are easily shown to be isomorphic. As often happens in an intuitionistic setting, classically equivalent notions fork. Dedekind reals give rise to several demonstrably different collections of reals when only intuitionistic logic is assumed (cf. [18], Ch.5, Sect.5). Here we shall be concerned with the most common and fruitful notion of Dedekind real which crucially involves the (classically superfluous) condition of locatedness of cuts. These Dedekind reals are sometimes referred to as the constructive Dedekind reals but we shall simply address them as the Dedekind reals. Even in intuitionistic set theory, with a little bit of help from the countable axiom of choice (AC(ℕ, 2)^{1} suffices; see [4], 8.25), ℝ^{d} and ℝ^{c} are isomorphic (where ℝ^{d} and ℝ^{c} denote the collections of Dedekind reals and Cauchy reals, respectively). As ℝ^{c} is canonically embedded in ℝ^{d} we can view ℝ^{c} as a subset of ℝ^{d} so that the latter result can be stated as ℝ^{d} = ℝ^{c}. The countable axiom of choice is accepted in Bishop-style constructive mathematics but cannot be assumed in all intuitionistic contexts. Some choice is necessary for equating ℝ^{d} and ℝ^{c} as there are sheaf models of higher order intuitionistic logic in which ℝ^{d} is not isomorphic to ℝ^{c} (cf. [6]). This paper will show that the difference between ℝ^{d} and ℝ^{c} can be of a grander scale. When is the continuum a set? The standard, classical construction of ℝ as a set uses Power Set. Constructively, the weaker principle of Subset Collection (in the context of the axioms of Constructive Zermelo-Fraenkel Set Theory CZF) suffices, as does even the apparently even weaker principle of Binary Refinement [5]. In contrast, we shall demonstrate that there is a Kripke model of CZF with Exponentiation in lieu of Subset Collection in which the Cauchy reals form a set while the Dedekind reals constitute a proper class. This shows that Exponentiation and Subset Collection Axiom have markedly different consequences for the theory of Dedekind reals.

∗This material is based upon work supported by the National Science Foundation under Award No. DMS-0301162.

This paper proves the following theorems:

Theorem 1.1 (Fourman-Hyland [6]) IZF_{Ref} does not prove that the Dedekind reals equal the Cauchy reals.

Theorem 1.2 CZF_{Exp} (i.e. CZF with Subset Collection replaced by Exponentiation) does not prove that the Dedekind reals are a set.

Even though the proof of the first theorem given here could be converted easily to the original Fourman-Hyland proof of the same, it is still included because the conversion in the other direction, from the original sheaf proof to the current Kripke model, is not obvious (to us at least); one might well want to know what the Kripke model proof of this theorem is. Furthermore, it is helpful as background to understand the construction of the second proof. While the second proof could similarly be turned into a purely topological argument, albeit of a non-standard type, unlike Gauss, we do not wish to cover our tracks. The original intuition here was the Kripke model – indeed, we know of no other way to motivate the unusual topological semantics and term structure – and so it might be of practical utility to have that motivation present and up front. These benefits of presenting the Kripke constructions notwithstanding, this article is reader-friendly enough so that anyone who wanted to could simply skip the sections on constructing the models and go straight to the definitions of topological semantics (mod exchanging later on a few “true at node r”s with “forced by some neighborhood of r”s).

The paper is organized as follows. After a brief review of Constructive Zermelo-Fraenkel Set Theory and notions of real numbers, section 2 features a Kripke model of IZF_{Ref} in which ℝ^{d} ≠ ℝ^{c}. Here IZF_{Ref} denotes Intuitionistic Zermelo-Fraenkel Set Theory with the Reflection schema.^{2} In section 3 the model of section 2 undergoes refinements and pivotally techniques of [8] are put to use to engender a model of CZF_{Exp} in which ℝ^{d} is a proper class.

1∀ r ⊆ ℕ × 2[∀n ∈ ℕ ∃i ∈ {0, 1} ⟨n, i⟩ ∈ r → ∃f : ℕ → 2 ∀n ∈ ℕ ⟨n, f(n)⟩ ∈ r].

1.1 Constructive Zermelo-Fraenkel Set Theory

In this subsection we will summarize the language and axioms for CZF. The language of CZF is the same first order language as that of classical Zermelo-Fraenkel Set Theory, ZF whose only non-logical symbol is ∈. The logic of CZF is intuitionistic first order logic with equality. Among its non-logical axioms are Extensionality, Pairing and Union in their usual forms. CZF has additionally axiom schemata which we will now proceed to summarize.

Infinity: ∃x∀u[u ∈ x ↔ (∅ = u ∨ ∃v ∈ x u = v + 1)] where v + 1 = v ∪ {v}.

Set Induction: ∀x[∀y ∈ xϕ(y) → ϕ(x)] → ∀xϕ(x)

Bounded Separation: ∀a∃b∀x[x ∈ b ↔ x ∈ a ∧ ϕ(x)]

for all bounded formulae ϕ. A set-theoretic formula is bounded or restricted if it is constructed from prime formulae using ¬, ∧, ∨, →, ∀x ∈ y and ∃x ∈ y only.

Strong Collection: For all formulae ϕ,

∀a[∀x ∈ a∃yϕ(x, y) → ∃b [∀x ∈ a ∃y ∈ b ϕ(x, y) ∧ ∀y ∈ b ∃x ∈ a ϕ(x, y)]].

Subset Collection: For all formulae ψ,

∀a∀b∃c∀u [∀x ∈ a ∃y ∈ b ψ(x, y, u) →
∃d ∈ c [∀x ∈ a ∃y ∈ d ψ(x, y, u) ∧ ∀y ∈ d ∃x ∈ a ψ(x, y, u)]].

The Subset Collection schema easily qualifies as the most intricate axiom of CZF. To explain this axiom in different terms, we introduce the notion of fullness (cf. [1]).

Definition 1.3 As per usual, we use ⟨x, y⟩ to denote the ordered pair of x and y. We use Fun(g), dom(R), ran(R) to convey that g is a function and to denote the domain and range of any relation R, respectively.

For sets A, B let A × B be the cartesian product of A and B, that is the set of ordered pairs ⟨x, y⟩ with x ∈ A and y ∈ B. Let ^{A}B be the class of all functions with domain A and with range contained in B. Let mv(^{A}B) be the class of all sets R ⊆ A × B satisfying ∀u ∈ A ∃v ∈ B ⟨u, v⟩ ∈ R. A set C is said to be full in mv(^{A}B) if C ⊆ mv(^{A}B) and

2Reflection, Collection and Replacement are equivalent in classical set theory. Intuitionistically, Reflection implies Collection which in turn implies Replacement, however, these implications cannot be reversed (see [7] for the latter).

∀R ∈ mv(^{A}B) ∃S ∈ C S ⊆ R.

The expression mv(^{A}B) should be read as the collection of multi-valued functions from the set A to the set B.

Additional axioms we shall consider are:

Exponentiation: ∀x∀y∃z z = ^{x}y.

Fullness: ∀x∀y∃z z is full in mv(^{x}y).

The next result provides an equivalent rendering of Subset Collection.

Proposition 1.4 Let CZF^{−} be CZF without Subset Collection.

(i) CZF^{−} + Subset Collection = CZF^{−} + Fullness.

(ii) CZF ⊢ Exponentiation.

proof:[1], Proposition 2.2. (The equality in (i) is as theories: that is, they both prove the same theorems.)

1.2 The Cauchy and Dedekind reals

Definition 1.5 A fundamental sequence is a sequence (r_{n})_{n∈ℕ} of rationals, together with is a (Cauchy-)modulus f : ℕ → ℕ such that

∀k ∀m, n ≥ f(k) |r_{m} − r_{n}| < \frac{1}{2^{k}} ,

where all quantifiers range over ℕ.

Two fundamental sequences (r_{n})_{n∈ℕ}, (s_{n})_{n∈ℕ} are said to coincide (in symbols ≈) if

∀k∃n∀m ≥ n |r_{m} − s_{m}| < \frac{1}{2^{k}} .

≈ is indeed an equivalence relation on fundamental sequences. The set of Cauchy reals ℝ^{c} consists of the equivalence classes of fundamental sequences relative to ≈. For the equivalence class of (r_{n})_{n∈ℕ} we use the notation (r_{n})_{n∈ℕ}/ ≈.

The development of the theory of Cauchy reals in [18], Ch.5, Sect.2-4 can be carried out on the basis of CZF_{Exp}. Note that the axiom AC-NN! ^{3} is deducible in CZF_{Exp}.

Definition 1.6 Let S ⊆ ℚ. S is called a left cut or Dedekind real if the following conditions are satisfied:

3(∀m ∈ ℕ ∃!n ∈ ℕ ϕ(m, n)) → (∃f : ℕ → ℕ ∀m ∈ ℕ ϕ(m, f(m)))

1. ∃r(r ∈ S) ∧ ∃r′(r′ ∉ S) (boundedness)

2. ∀r ∈ S ∃r′ ∈ S (r < r′) (openness)

3. ∀rs ∈ ℚ [r < s → r ∈ S ∨ s ∉ S] (locatedness)

For X ⊆ ℚ define X^{<} := {s ∈ ℚ : ∃r ∈ X s < r}. If S is a left cut it follows from openness and locatedness that S = S^{<}.

Lemma 1.7 Let r = (r_{n})_{n∈ℕ} and r′ = (r′_{n})_{n∈ℕ} be fundamental sequences of rationals. Define

X_{r} := \{s ∈ ℚ : ∃m s < (r_{f(m)} − \frac{1}{2^{m}})\}.

We then have

1. X_{r} is a Dedekind real.

2. X_{r} = X_{r′} if and only if (r_{n})_{n∈ℕ} ≈ (r′_{n})_{n∈ℕ}.

3. ℝ^{c} is a subfield of ℝ^{d} via the mapping (r_{n})_{n∈ℕ}/ ≈ ↦ X_{r}.

Proof: Exercise or see [4], Section 8.4.

2 ℝ^{d} ≠ ℝ^{c}

Theorem 2.1 (Fourman-Hyland [6]) IZF_{Ref} does not prove that the Dedekind reals equal the Cauchy reals.

2.1 Construction of the Model

Let M_{0} ≺ M_{1} ≺ ... be an ω-sequence of models of ZF set theory and of elementary embeddings among them, as indicated, such that the sequence from M_{n} on is definable in M_{n}, and such that each thinks that the next has nonstandard integers. Notice that this is easy to define (mod getting a model of ZF in the first place): an iterated ultrapower using any non-principal ultrafilter on ω will do. (If you’re concerned that this needs AC too, work in L of your starting model.) We will ambiguously use the symbol f to stand for any of the elementary embeddings inherent in the M_{n}-sequence.

Definition 2.2 The frame (underlying partial order) of the Kripke model M will be a (non-rooted) tree with ω-many levels. The nodes on level n will be the reals from M_{n}. r′ is an immediate successor of r iff r is a real from some M_{n}, r′ is a real from M_{n+1}, and r and r′ are infinitesimally close; that is, f(r) − r′, calculated in M_{n+1} of course, is infinitesimal, calculated in M_{n} of course. In other words, in M_{n}, r is that standard part of r′.

The Kripke structure will be defined like a forcing extension in classical set theory. That is, there will be a ground model, terms that live in the ground model, and an interpretation of those terms, which, after modding out by =, is the final model M. Since the current construction is mostly just a re-phrasing of the topological, i.e. Heyting-valued, model of [6], the similarity with forcing, i.e. Boolean-valued models, is not merely an analogy, but essentially the same material, and so it makes some sense to present it the way people are used to it.

Definition 2.3 The ground Kripke model has, at each node of level n, a copy of M_{n}. The transition functions (from a node to a following node) are the elementary embeddings given with the original sequence of models (and therefore will be notated by f again).

Note that by the elementarity of the extensions, this Kripke model is a model of classical ZF. More importantly, the model restricted to any node of level n is definable in M_{n}, because the original M-sequence was so definable.

Definition 2.4 The terms are defined at each node separately. For a node at level n, the terms are defined in M_{n}, inductively on the ordinals in M_{n}. At any stage α, a term of stage α is a set σ of the form {⟨σ_{i}, J_{i}⟩ | i ∈ I}, where I is some index set, each σ_{i} is a term of stage < α, and each J_{i} is an open subset of the real line.

(Often the terms at stage α are defined to be functions from the terms of all stages less than α, as opposed to the relations above, which may be non-total and multi-valued. This distinction makes absolutely no difference. Such a relation can be made total by sending all terms not yet in the domain to the empty set, and functional by taking unions of second components.)

Intuitively, each open set J is saying “the generic real is in me.” Also, each node r is saying “I am the generic, or at least somebody in my infinitesimal universe is.” So at node r, J should count at true iff r ∈ J. These intuitions will appear later as theorems. (Well, lemmas.)

The ground model can be embedded in this term structure: for x ∈ M_{n}, its canonical name x̂ is defined inductively as {⟨ŷ, ℝ⟩ | y ∈ x}. Terms of the form x̂ are called ground model terms.

Notice that the definition of the terms given above is uniform among the M_{n}’s, and so any term at a node gets sent by the transition function f to a corresponding term at any given later node. Hence we can use the same functions f yet again as the transition functions for this term model. (Their coherence on the terms follows directly from their coherence on the original M_{n}’s.)

At this point in the construction of the Kripke model, we have the frame, a universe (set of objects) at each node, and the transition functions. Now we need to define the primitive relations at each node. In the language of set theory, these are =_{M} and ∈_{M} (the subscript being used to prevent confusion with equality and membership of the ambient models M_{n}). This will be done via a forcing relation ⊩.

Definition 2.5 J ⊩ σ =_{M} τ and J ⊩ σ ∈_{M} τ are defined inductively on σ and τ, simultaneously for all open sets of reals J:

J ⊩ σ =_{M} τ iff for all ⟨σ_{i}, J_{i}⟩ ∈ σ J ∩ J_{i} ⊩ σ_{i} ∈_{M} τ and vice versa J ⊩ σ ∈_{M} τ iff for all r ∈ J there is a ⟨τ_{i}, J_{i}⟩ ∈ τ and J′ such that

r ∈ J′ ∩ J_{i} ⊩ σ =_{M} τ_{i}

(We will later extend this forcing relation to all formulas.)

Note that these definitions, for J, σ, τ ∈ M_{n}, can be evaluated in M_{n}, without reference to M_{n+1} or to future nodes or anything. Therefore J ⊩ ϕ (according to M_{n}) iff f(J) ⊩ f(ϕ) (according to M_{n+1}), by the elementarity of f. So we can afford to be vague about where various assertions are evaluated, since by this elementarity it doesn’t matter. (The same will be true when we extend forcing to all formulas.)

Definition 2.6 At node r, for any two terms σ and τ, σ =_{M} τ iff, for some J with r ∈ J, J ⊩ σ =_{M} τ.

Also, at r, σ ∈_{M} τ iff for some J with r ∈ J, J ⊩ σ ∈_{M} τ.

\underline{Notation} Satisfaction (in the sense of Kripke semantics) at node r will be notated with ⊧, as in “r ⊧ σ = τ”. This should not be confused with the forcing relation ⊩, even though the latter symbol is often used in the literature for Kripke satisfaction.

Thus we have a first-order structure at each node.

To have a Kripke model, the transition functions f must also respect this first-order structure, =_{M} and ∈_{M}; to wit:

Lemma 2.7 f is an =_{M} and ∈_{M}-homomorphism. That is, if σ =_{M} τ then f(σ) =_{M} f(τ), and similarly for ∈_{M}.

proof: If σ =_{M} τ then let J be such that r ∈ J and J ⊩ σ =_{M} τ. Then f(J) is open, r′ ∈ f(J) because r′ is infinitesimally close to r, and f(J) ⊩ f(σ) =_{M} f(τ) by elementarity. Hence f(σ) =_{M} f(τ). Similarly for ∈_{M}.

We can now conclude that we have a Kripke model.

Lemma 2.8 This Kripke model satisfies the equality axioms:

1. ∀x x = x

2. ∀x, y x = y → y = x

3. ∀x, y, z x = y ∧ y = z → x = z

4. ∀x, y, z x = y ∧ x ∈ z → y ∈ z

5. ∀x, y, z x = y ∧ z ∈ x → z ∈ y. proof: 1: It is easy to show with a simultaneous induction that, for all J and σ, J ⊩ σ =_{M} σ, and, for all ⟨σ_{i}, J_{i}⟩ ∈ σ, J ∩ J_{i} ⊩ σ_{i} ∈_{M} σ.

2: Trivial because the definition of J ⊩ σ =_{M} τ is itself symmetric.

3: For this and the subsequent parts, we need some lemmas.

Lemma 2.9 If J′ ⊆ J ⊩ σ =_{M} τ then J′ ⊩ σ =_{M} τ, and similarly for ∈_{M}.

proof: By induction on σ and τ.

Lemma 2.10 If J ⊩ ρ =_{M} σ and J ⊩ σ =_{M} τ then J ⊩ ρ =_{M} τ.

proof: Again, by induction on terms. Let ⟨ρ_{i}, J_{i}⟩ ∈ ρ. Then J ∩ J_{i} ⊩ ρ_{i} ∈_{M} σ, i.e. for all r ∈ J ∩ J_{i} there are ⟨σ_{j}, J_{j}⟩ ∈ σ and J′ ⊆ J ∩ J_{i} such that r ∈ J′ ∩ J_{j} ⊩ ρ_{i} =_{M} σ_{j}. Fix any r ∈ J ∩ J_{i}, and let ⟨σ_{j}, J_{j}⟩ ∈ σ and J′ be as given. By hypothesis, J ∩ J_{j} ⊩ σ_{j} ∈_{M} τ. So let ⟨τ_{k}, J_{k}⟩ ∈ τ and Ĵ ⊆ J ∩ J_{j} be such that r ∈ Ĵ∩ J_{k} ⊩ σ_{j} =_{M} τ_{k}. Let J̃ be J′ ∩ Ĵ∩ J_{j}. Note that J̃ ⊆ J ∩J_{i}, and that r ∈ J̃∩J_{k}. It remains only to show that J̃∩J_{k} ⊩ ρ_{i} =_{M} τ_{k}. Observing that J̃ ∩ J_{k} ⊆ J′ ∩ J_{j}, Ĵ ∩ J_{k}, it follows by the previous lemma that J̃∩ J_{k} ⊩ ρ_{i} =_{M} σ_{j}, σ_{j} =_{M} τ_{k}, from which the desired conclusion follows by the induction.

Returning to proving property 3, the hypothesis is that for some J and K containing r, J ⊩ ρ =_{M} σ and K ⊩ σ =_{M} τ. By the first lemma, J ∩K ⊩ ρ =_{M} σ, σ =_{M} τ, and so by the second, J ∩ K ⊩ ρ =_{M} τ, which suffices.

4: Let J ⊩ ρ =_{M} σ and K ⊩ ρ ∈_{M} τ. We will show that J ∩ K ⊩ σ ∈_{M} τ. Let r ∈ J ∩ K. By hypothesis, let ⟨τ_{i}, J_{i}⟩ ∈ τ, J′ ⊆ K be such that r ∈ J′ ∩ J_{i} ⊩ ρ =_{M} τ_{i}; without loss of generality J′ ⊆ J. By the first lemma, J′∩J_{i} ⊩ ρ =_{M} σ, and by the second, J′ ∩ J_{i} ⊩ σ =_{M} τ_{i}.

5: Similar, and left to the reader.

With this lemma in hand, we can now mod out by =_{M}, so that the symbol “=” is interpreted as actual set-theoretic equality. We will henceforth drop the subscript _{M} from = and ∈, although we will not distinguish notationally between a term σ and the model element it represents, σ’s equivalence class.

Note that, at any node r of level n, the whole structure M restricted to r and its successors is definable in M_{n}, satisfaction relation ⊧ and all. This will be useful when showing below that IZF holds. For instance, to show Separation, satisfaction r ⊧ ϕ(x) will have to be evaluated in order to define the right separation term in M_{n}, and so satisfaction must be definable in M_{n}.

2.2 The Forcing Relation

The primitive relations = and ∈ were defined in terms of open sets J. To put it somewhat informally, at r, σ = or ∈ τ if this is forced by a true set, and a set J is true at r if r ∈ J. In fact, this phenomenon propagates to non-primitive formulas. To show this, we extend the forcing relation J ⊩ ϕ from primitive to all (first-order, finitary) formulas. Then we prove as a lemma, the Truth Lemma, what was taken as a definition for the primitive formulas, that r ⊧ ϕ iff J ⊩ ϕ for some J containing r.

Definition 2.11 J ⊩ ϕ is defined inductively on ϕ:

J ⊩ σ = τ iff for all ⟨σ_{i}, J_{i}⟩ ∈ σ J ∩ J_{i} ⊩ σ_{i} ∈ τ and vice versa

J ⊩ σ ∈ τ iff for all r ∈ J there is a ⟨τ_{i}, J_{i}⟩ ∈ τ and J′ such that r ∈ J′ ∩ J_{i} ⊩ σ = τ_{i}

J ⊩ ϕ ∧ ψ iff J ⊩ ϕ and J ⊩ ψ

J ⊩ ϕ ∨ ψ iff for all r ∈ J there is a J′ containing r such that J ∩ J′ ⊩ ϕ or J ∩ J′ ⊩ ψ

J ⊩ ϕ → ψ iff for all J′ ⊆ J if J′ ⊩ ϕ then J′ ⊩ ψ

J ⊩ ∃x ϕ(x) iff for all r ∈ J there is a J′ containing r and a σ such that J ∩ J′ ⊩ ϕ(σ)

J ⊩ ∀x ϕ(x) iff for all r ∈ J and σ there is a J′ containing r such that J ∩ J′ ⊩ ϕ(σ)

Lemma 2.12 1. For all ϕ ∅ ⊩ ϕ.

2. If J′ ⊆ J ⊩ ϕ then J′ ⊩ ϕ.

3. If J_{i} ⊩ ϕ for all i then ⋃_{i} J_{i} ⊩ ϕ.

4. J ⊩ ϕ iff for all r ∈ J there is a J′ containing r such that J ∩ J′ ⊩ ϕ.

proof: 1. Trivial induction. The one observation to make regards negation, not mentioned above. As is standard, ¬ϕ is taken as an abbreviation for ϕ → ⊥, where ⊥ is any false formula. Letting ⊥ be “0=1”, observe that ∅ ⊩ ⊥.

2. Again, a trivial induction.

3. Easy induction. The one case to watch out for is →, where you need to invoke the previous part of this lemma.

4. Trivial, using 3.

Lemma 2.13 Truth Lemma: For any node r, r ⊧ ϕ iff J ⊩ ϕ for some J containing r.

proof: Again, by induction on ϕ, this time in detail for a change.

In all cases, the right-to-left direction (“forced implies true”) is pretty easy, by induction. (Note that only the → case needs the left-to-right direction in this induction.) Hence in the following we show only left-to-right (“if true at a node then forced”).

=: This is exactly the definition of =.

∈: This is exactly the definition of ∈.

∧: If r ⊧ ϕ ∧ ψ, then r ⊧ ϕ and r ⊧ ψ. Inductively let r ∈ J ⊩ ϕ and r ∈ J′ ⊩ ψ. J ∩ J′ suffices.

∨: If r ⊧ ϕ ∨ ψ, then without loss of generality r ⊧ ϕ . Inductively let r ∈ J ⊩ ϕ. J suffices.

→: Suppose to the contrary r ⊧ ϕ → ψ but no open set containing r forces such. Work in an infinitesimal neighborhood J around r. Since J ⊮ ϕ → ψ there is a J′ ⊆ J such that J′ ⊩ ϕ but J′ ⊮ ψ. By the previous part of this lemma, there is an r′ ∈ J′ such that no open set containing r′ forces ψ. At the node r′, by induction, r′ |̸= ψ, even though r′ ⊧ ϕ (since r′ ∈ J′ ⊩ ϕ). This contradicts the assumption on r (i.e. that r ⊧ ϕ → ψ), since r′ extends r (as nodes).

∃: If r ⊧ ∃x ϕ(x) then let σ be such that r ⊧ ϕ(σ). Inductively there is a J containing r such that J ⊩ ϕ(σ). J suffices.

∀: Suppose to the contrary r ⊧ ∀x ϕ(x) but no open set containing r forces such. Work in an infinitesimal neighborhood J around r. Since J ⊮ ∀x ϕ(x) there is an r′ ∈ J and σ such that for all J′ containing r′ J ∩ J′ ⊮ ϕ(σ). That is, no open set containing r′ forces ϕ(σ). Hence at the node r′, by induction, r′ |̸= ϕ(σ). This contradicts the assumption on r (i.e. that r ⊧ ∀x ϕ(x)).

2.3 The Final Proof

We now want to show that our model M satisfies certain global properties. If it had a bottom element ⊥, then we could express what we want by saying ⊥ ⊧ ϕ for certain ϕ. But it doesn’t. Hence we use the abbreviation M ⊧ ϕ for “for all nodes r, r ⊧ ϕ.”

Theorem 2.14 M ⊧ IZF_{Ref}.

proof: Note that, as a Kripke model, the axioms of intuitionistic logic are satisfied, by general theorems about Kripke models.

• Infinity: ω̂ will do. (Recall that the canonical name x̂ of any set x ∈ M_{n} is defined inductively as {⟨ŷ, ℝ⟩ | y ∈ x}.)

• Pairing: Given σ and τ, {⟨σ, ℝ⟩, ⟨τ, ℝ⟩} will do.

• Union: Given σ, {⟨τ, J ∩ J_{i}⟩ | for some σ_{i}, ⟨τ, J⟩ ∈ σ_{i} and ⟨σ_{i}, J_{i}⟩ ∈ σ} will do.

• Extensionality: We need to show that ∀x ∀y [∀z (z ∈ x ↔ z ∈ y) → x = y]. So let σ and τ be any terms at a node r such that r ⊧ “∀z (z ∈ σ ↔ z ∈ τ)”. We must show that r ⊧ “σ = τ”. By the Truth Lemma, let r ∈ J ⊩ “∀z (z ∈ σ ↔ z ∈ τ)”; i.e. for all r′ ∈ J, ρ there is a J′ containing r′ such that J ∩ J′ ⊩ ρ ∈ σ ↔ ρ ∈ τ. We claim that J ⊩ “σ = τ”, which again by the Truth Lemma suffices. To this end, let ⟨σ_{i}, J_{i}⟩ be in σ; we need to show that J ∩ J_{i} ⊩ σ_{i} ∈ τ. Let r′ be an arbitrary member of J ∩ J_{i} and ρ be σ_{i}. By the choice of J, let J′ containing r′ be such that J ∩ J′ ⊩ σ_{i} ∈ σ ↔ σ_{i} ∈ τ; in particular, J ∩ J′ ⊩ σ_{i} ∈ σ → σ_{i} ∈ τ. It has already been observed in 2.8, part 1, that J ∩ J′ ∩ J_{i} ⊩ σ_{i} ∈ σ, so J ∩ J′ ∩ J_{i} ⊩ σ_{i} ∈ τ. By going through each r′ in J ∩ J_{i} and using 2.12, part 3, we can conclude that J ∩ J_{i} ⊩ σ_{i} ∈ τ, as desired. The other direction (“τ ⊆ σ”) is analogous.

• Set Induction (Schema): Suppose r ⊧ “∀x ((∀y ∈ x ϕ(y)) → ϕ(x))”, where r ∈ M_{n} ; by the Truth Lemma, let J containing r force as much. We must show r ⊧ “∀x ϕ(x)”. Suppose not. Using the definition of satisfaction in Kripke models, there is an r′ ∈ M_{n′} extending (i.e. infinitesimally close to) r (hence in J in the sense of M_{n′}) and a σ such that r′ |̸= f(ϕ)(σ) (f the transition from from node r to r′). By elementarity, there is such an r′ in M_{n}. Let σ be such a term of minimal V -rank among all r′s ∈ J. Fix such an r′. By the Truth Lemma (and the choice of J), r′ ⊧ “(∀y ∈ σ ϕ(y)) → ϕ(σ)”. We claim that r′ ⊧ “∀y ∈ σ ϕ(y)”. If not, then for some r′′ extending r′ (hence in J) and τ, r′′ ⊧ τ ∈ f(σ) and r′′ |̸= f(ϕ)(τ). Unraveling the interpretation of ∈, this choice of τ can be substituted by a term τ of lower V -rank than σ. By elementarity, such a τ would exist in M_{n}, in violation of the choice of σ, which proves the claim. Hence r′ ⊧ ϕ(σ), again violating the choice of σ. This contradiction shows that r ⊧ “∀x ϕ(x)”.

• Separation (Schema): Let ϕ(x) be a formula and σ a term. Then {⟨σ_{i}, J ∩ J_{i}⟩ | ⟨σ_{i}, J_{i}⟩ ∈ σ and J ⊩ ϕ(σ_{i})} will do.

• Power Set: A term σ̄ is a normal form subset of σ if for all ⟨σ_{i}, J̄_{i}⟩ ∈ σ̄ there is a J_{i} ⊇ J̄_{i} such that ⟨σ_{i}, J_{i}⟩ ∈ σ. {⟨σ̄, ℝ⟩ | σ̄ is a normal form subset of σ} will do.

• Reflection (Schema): Recall that the statement of Reflection is that for every formula ϕ(x) (with free variable x and unmentioned parameters) and set z there is a transitive set Z containing z such that Z reflects the truth of ϕ(x) in V for all x ∈ Z. So to this end, let ϕ(x) be a formula and σ be a set at a node r of level n. Let k be such that the truth of ϕ(x) at node r and beyond is Σ_{k} definable in M_{n}. In M_{n}, let X be a set containing σ, r, and ϕ’s parameters such that X ≺_{k} M_{n}. Let τ be {⟨ρ, ℝ⟩ | ρ ∈ X is a term}. τ will do.

Just as in the case of regular, classical forcing, there is a generic element. In the case at hand, this generic can be identified with the term {⟨r̂, J⟩ | r is a rational, J is an open interval from the reals, and r < J}, where r < J if r is less than each element of J. We will call this term G. Note that at node r (of level n), every standard (in the sense of M_{n}) rational less than r gets into G, and no standard real greater than r will ever get into G. Of course, non-standard reals infinitesimally close to r are still up for grabs.

It is important in the following that, if r ⊧ σ ∈ ℚ, then there is a rational q in the sense of M_{n} (n r’s level) such that r ⊧ σ = q. That’s because rationals are (equivalence classes of) pairs of naturals, and the corresponding fact holds for naturals. And that last statement holds because M ⊧ “ℕ̂ is the set of natural numbers”, and the topological space on which the model is built is connected. Hence, at r, a Cauchy sequence of rationals is just what you’d think: a sequence with domain ℕ in the sense of M_{n}, range ℚ in the sense of M_{n}, with the right Cauchy condition on it, which gets extended to a larger domain at successors of r.

Proposition 2.15 M ⊧ “G is a constructive Dedekind real, i.e. a located left cut”.

proof: First off, r ⊧ “r − 1 ∈ G ∧ r + 1 ∉ G”. Secondly, if r ⊧ “s < t ∈ G”, then ⟨t, J⟩ ∈ G, where t < J and r ∈ J. Hence s < J, so ⟨s, J⟩ ∈ G, and r ⊧ s ∈ G. Finally, suppose r ⊧ “s, t ∈ ℚ ∧ s < t”. Either s < r or r < t. Since s and t are both standard (in the sense of M_{n}, n the level of r), either r ⊧ s ∈ G or r ⊧ t ∉ G respectively.

In order to complete the theorem, we need only prove the following:

Proposition 2.16 M ⊧ “The Dedekind real G is not a Cauchy real.”

proof: Recall that a Cauchy sequence is a function f : ℕ → ℚ such that for all k ∈ ℕ there is an m_{k} ∈ ℕ such that, for all i, j > m_{k}, f(i) and f(j) are within 2^{−k} of each other. Classically such a function k ↦ m_{k}, called a modulus of convergence, could be defined from f, but not constructively (see [10]). Often in a constructive setting a real number is therefore taken to be a pair of a Cauchy sequence and such a modulus (or an equivalence class thereof). We will prove the stronger assertion that G is not even the limit of a Cauchy sequence, even without a modulus. (A Dedekind real Y is the limit of the Cauchy sequence f exactly when r ∈ Y iff r < f(m_{k}) − 2^{−k} for some k, where m_{k} is an integer as above.)

Suppose r ⊧ “f is a Cauchy sequence”. By the Truth Lemma, there is an open set J containing r forcing the same. There are two cases.

CASE I: There is some open set J′ containing r forcing a value f(m) for each integer m in M_{n} (where r ∈ M_{n}). In this case, f is a ground model function; that is, in M_{n}, hence in each M_{k} with k ≥ n, g(m) can be defined as the unique l such that J′ ⊩ f(m) = l̂, and then J′ ⊩ f = ĝ. Since classical logic holds in M_{n}, either lim(f) is bounded away from r, say by a distance of 2^{−k}, or it’s not.

If it is, then r ⊧ G ≠ lim(f), as follows. Let J′′ be an interval around r of length less than 2^{−k}. J′′ ⊩ r̂− 2 ˆ−k ∈ G ∧ r̂+ 2 ˆ−k ∉ G, while f stays more than 2^{−k} away from r.

If on the other hand f is not bounded away from r, then the condition “s < f(m_{k}) − 2^{−k} for some k” becomes simply “s < r”. So then f would witness that s ∈ G iff s < r. But this is false: if r′ is less than r by an infinitesimal amount, then r′ ⊧ r̂′ < r̂ but r′ |̸= r̂′ ∈ G, and if r′ is greater than r by an infinitesimal amount, and s is between r and r′, then r′ ⊧ ŝ > r̂ but r′ ⊧ ŝ ∈ G.

CASE II: Not case I. That means that for any interval J′ around r, however small, there is some argument m to f such that J′ does not force any value f(m). By elementarity, in M_{n+1} pick J′ to be some infinitesimally small neighborhood around r, and m such an argument. Pick some value q that f(m) could have and the maximal (hence non-empty, proper, and open) subset of J′ forcing f(m̂) = q̂. Pick the maximal (hence non-empty, proper, and open) subset of J′ forcing f(m̂) ≠ q̂. These two subsets must be disjoint, lest the intersection force a contradiction. But an open interval cannot be covered by two disjoint, non-empty open sets. Hence there is an infinitesimal s in neither of those two subsets. Now consider the Kripke model at node s. f(m) is undefined at s. Otherwise, by the Truth Lemma, there would be some interval J containing s such that J ⊩ f(m̂) = p̂ for some particular rational p. Whether or not p = q would force s into one of the subsets or the other. Therefore, the node r cannot force that f is total, contradicting the hypothesis that r forced that f was a Cauchy sequence.

\underline{Comments and Questions} Those familiar with the proof via the (full) topological model, or sheaves, over ℝ, as in [6] for instance, will realize that it’s essentially the same as the one above. In fact, the topological/sheaf construction can be read off of the argument above. All of the proofs are based on constructing the right term and/or using an open set to force a statement. That is exactly what’s present in a topological model: the terms here are the standard terms for a topological model, and the forcing relation here is the standard topological semantics. So the Kripke superstructure is actually superfluous for this argument. Nonetheless, several questions arise.

What the Kripke structure has that the topological model doesn’t are the infinitesimals. Are they somehow hidden in the topological model? Are they dispensable in the Kripke model? Or are the models more than superficially different?

Also, is there some reason that the topology was necessary in the Kripke construction? The authors started this project with the idea of using a Kripke model, were led to infinitesimals, and did not suspect that any topological ideas would be necessary. (In some detail, suppose you’re looking at a Dedekind cut in a node of a Kripke model. By locatedness, if p < q then one of those two rationals gets put into either the lower or the upper cut; that is, we can remain undecided about the placement of at most one rational, which for simplicity we may as well take to be 0. Then why doesn’t the Cauchy sequence 1/n name this cut? That can happen only if, at some later node, the cut no longer looks to be around 0. But how can that happen if all other rationals are already decided? Only if at this later node there are new rationals that weren’t there at the old node. This leads directly to indexing nodes by infinitesimals, and having the cut look at any node as though it’s defining the infinitesimal at that node. Notice that there seems to be no reason to use topology here.) It was only after several attempts to define the terms, with their equality and membership relations, using just the partial order all failed that they were driven to the current, topological solution. Since this all happened before we became aware of the earlier Fourman-Hyland work, it is not possible that we were somehow pre-disposed toward turning to topology. Rather, it seems that topology is inherent in the problem. Is there some way to make that suggestion precise and to see why it’s true?

Indeed, this question becomes even more pressing in light of the next section. There topology is used in a similar way, but the terms and the semantics are like nothing we have seen before. Indeed, the construction following could not be in its essence a topological model of the kind considered so far in the literature, since the latter always model IZF, whereas the former will falsify Power Set (satisfying Exponentiation in its stead). So if there were some method to read off the topology from the problem in this section, it would be of great interest to see what that method would give us in the next problem.

There are other, soft reasons to have included the preceding construction, even though it adds little to the Fourman-Hyland argument. Conceivably, somebody could want to know what the paradigmatic Kripke model for the Cauchy and Dedekind reals differing is, and this is it. It also provides a nice warm-up for the more complicated work of the next section, to which we now turn.

3 The Dedekind Reals Are Not a Set

Theorem 3.1 CZF_{Exp} (i.e. CZF with Subset Collection replaced by Exponentiation) does not prove that the Dedekind reals form a set.

3.1 The Construction

Any model showing what is claimed must have certain properties. For one, the Dedekind reals cannot equal the Cauchy reals (since CZF_{Exp} proves that the Cauchy reals are a set). Hence the current model takes its inspiration from the previous one. Also, it must falsify Subset Collection (since CZF proves that the Dedekind reals are a set). Hence guidance is also taken from [8], where such a model is built.

The idea behind the latter is that a (classical, external) relation R on ℕ keeps on being introduced into the model via a term ρ but at a later node “disappears”; more accurately, the information ρ contains gets erased, because ρ grows into all of ℕ × ℕ, thereby melting away into the other sets present (to give a visual image). Since R is chosen so that it doesn’t help build any functions, ρ can be ignored when proving Exponentiation. On the other hand, while you’re free to include ρ in an alleged full set of relations, by the next node there is no longer any trace of R, so when R reappears later via a different term ρ′ your attempt at a full set no longer works.

In the present context, we will do something similar. The troublesome relation will be (essentially) the Dedekind real G from the previous construction. It will “disappear” in that, instead of continuing to change its mind about what it is at all future nodes, it will settle down to one fixed, standard real at all next nodes. But then some other real just like G will appear and pull the same stunt.

We now begin with the definition of the Kripke model, which ultimately is distributed among the next several definitions.

Definition 3.2 The underlying p.o. of the Kripke model is the same as above: a (non-rooted) tree with ω-many levels, the nodes on level n being the reals from M_{n}. r′ is an immediate successor of r iff r is a real from some M_{n}, r′ is a real from M_{n+1}, and r and r′ are infinitesimally close; that is, f(r) − r′, calculated in M_{n+1} of course, is infinitesimal, calculated in M_{n} of course. In other words, in M_{n}, r is that standard part of r′.

Definition 3.3 A term at a node of height n is a set of the form {⟨σ_{i}, J_{i}⟩ | i ∈ I} ∪ {⟨σ_{h}, r_{h}⟩ | h ∈ H}, where each σ is (inductively) a term, each J an open set of reals, each r a real, and H and I index sets, all in the sense of M_{n}.

The first part of each term is as in the previous section: at node r, J_{i} counts as true iff r ∈ J_{i}. The second part plays a role only when we decide to have the term settle down and stop changing. This settling down in described as follows.

Definition 3.4 For a term σ and real r ∈ M_{n}, σ^{r} is defined inductively in M_{n} on the terms as {⟨σ_{i}^{r}, ℝ⟩ | ⟨σ_{i}, J_{i}⟩ ∈ σ ∧ r ∈ J_{i}} ∪ {⟨σ_{h}^{r}, ℝ⟩ | ⟨σ_{h}, r⟩ ∈ σ}.

Note that σ^{r} is (the image of) a set from the ground model. It bears observation that (σ^{r})^{s} = σ^{r}.

What determines when a term settles down in this way is the transition function. In fact, from any node to an immediate successor, there will be two transition functions, one the embedding f as before and the other the settling down function. This fact of the current construction does not quite jive with the standard definition of a Kripke model, which has no room for alternate ways to go from one node to another. However, this move is standard (even tame) for categorical models, which allow for arbitrary arrows among objects. So while the standard categorical description of a partial order is a category where the objects are the elements of the order and there’s an arrow from p to q iff p ≤ q, the category we’re working in has two arrows from p to q (for immediate successors). If you’re still uncomfortable with this double arrow, or object to calling this object a Kripke model, then double not the arrows but the nodes. That is, replace each node s by two nodes s_{old} and s_{new}, and have the two arrows go to these two separate nodes. Now you have a very traditional Kripke model again. To save on subscripts, we will work instead with two arrows going from r to s.

Definition 3.5 If s is an immediate successor of r, then there are two transition functions from r to s, called f and g. f is the elementary embedding from M_{n} to M_{n+1} as applied to terms. g(σ) = f(σ)^{s}. Transition functions to non-immediate successors are arbitrary compositions of the immediate transition functions.

When considering g(σ), note that σ ∈ M_{n} and s ∈ M_{n+1}. However, for purposes other than the transition functions, we will have occasion to look at σ^{s} for both σ and s from M_{n}. In this case, please note that, since f is an elementary embedding, (f(σ))^{s} = f(σ^{s}).

It’s easy to see that for σ a (term for a) ground model set, f(σ) is also a ground model set, and for τ from the ground model (such as f(σ)) so is τ^{r}. Hence in this case f(σ) = g(σ).

We do not need to show that the transition functions are well-defined, since they are defined on terms and not on equivalence classes of terms. However, once we define =, we will show that = is an equivalence relation and that f and g respect =, so that we can mod out by = and still consider f and g as acting on these equivalence classes.

Speaking of defining =, we now do so, simultaneously with ∈ and inductively on the terms, like in the previous section. In an interplay with the settling down procedure, the definition is different from in the previous section.

Definition 3.6 J ⊩ σ =_{M} τ and J ⊩ σ ∈_{M} τ are defined inductively on σ and τ, simultaneously for all open sets of reals J:

J ⊩ σ =_{M} τ iff for all ⟨σ_{i}, J_{i}⟩ ∈ σ J∩J_{i} ⊩ σ_{i} ∈_{M} τ and for all r ∈ J σ^{r} = τ^{r}, and vice versa.

J ⊩ σ ∈_{M} τ iff for all r ∈ J there is a ⟨τ_{i}, J_{i}⟩ ∈ τ and J′ ⊆ J such that r ∈ J′ ∩ J_{i} ⊩ σ =_{M} τ_{i}, and for all r ∈ J ⟨σ^{r}, ℝ⟩ ∈ τ^{r}.

(We will later extend this forcing relation to all formulas.)

Definition 3.7 At a node r, for any two terms σ and τ, r ⊧ σ =_{M} τ iff, for some J with r ∈ J, J ⊩ σ =_{M} τ.

Also, r ⊧ σ ∈_{M} τ iff for some J with r ∈ J, J ⊩ σ ∈_{M} τ.

Thus we have a first-order structure at each node.

Corollary 3.8 The model just defined is a Kripke model. That is, the transition functions are =_{M} and ∈_{M}-homomorphisms. proof: Note that the coherence of the transition functions is not an issue for us. That is, normally one has to show that the composition of the transition functions from nodes p to q and from q to r is the transition function from p to r. However, in our case, the transition functions were given only for immediate successors, and arbitrary compositions are allowed. So there’s nothing about coherence to prove.

If r ⊧ σ =_{M} τ then let r ∈ J ⊩ σ =_{M} τ. For s an immediate successor of r, s is infinitesimally close to r, so s ∈ f(J). Also, by elementarity, f(J) ⊩ f(σ) =_{M} f(τ). Therefore, s ⊧ f(σ) =_{M} f(τ). Regarding g, by the definition of forcing equality, f(σ)^{s} = f(τ)^{s}, that is, g(σ) = g(τ). It is easy to see that for any term ρ ℝ ⊩ ρ =_{M} ρ, so s ∈ ℝ ⊩ g(σ) =_{M} g(τ), and s ⊧ g(σ) =_{M} g(τ).

Similarly for ∈_{M}.

Lemma 3.9 This Kripke model satisfies the equality axioms:

1. ∀x x = x

2. ∀x, y x = y → y = x

3. ∀x, y, z x = y ∧ y = z → x = z

4. ∀x, y, z x = y ∧ x ∈ z → y ∈ z

5. ∀x, y, z x = y ∧ z ∈ x → z ∈ y.

proof: Similar to the equality lemma from the previous section. For those who are concerned that the new forcing relation might make a difference and therefore want to see the details, here they come.

1: It is easy to show with a simultaneous induction that, for all J and σ, J ⊩ σ =_{M} σ, and, for all ⟨σ_{i}, J_{i}⟩ ∈ σ, J ∩ J_{i} ⊩ σ_{i} ∈_{M} σ. Those parts of the definition of =_{M} and ∈_{M} that are identical to those of the previous section follow by the same inductive argument of the previous section. The next clauses, in the current context, boil down to σ^{r} = σ^{r}, which is trivially true, and, for ⟨σ_{i}, J_{i}⟩ ∈ σ and r ∈ J_{i}, ⟨σ_{i}^{r}, ℝ⟩ ∈ τ^{r}, which follows immediately from the definition of τ^{r}.

2: Trivial because the definition of J ⊩ σ =_{M} τ is itself symmetric.

3: For this and the subsequent parts, we need some lemmas.

Lemma 3.10 If J′ ⊆ J ⊩ σ =_{M} τ then J′ ⊩ σ =_{M} τ, and similarly for ∈_{M}.

proof: By induction on σ and τ.

Lemma 3.11 If J ⊩ ρ =_{M} σ and J ⊩ σ =_{M} τ then J ⊩ ρ =_{M} τ. proof: The new part in the definition of J ⊩ ρ =_{M} τ is that for all r ∈ J ρ^{r} = τ^{r}. The corresponding new parts of the hypotheses are that, for such r, ρ^{r} = σ^{r} and σ^{r} = τ^{r}, from which the desired conclusion follows immediately.

The old part of the definition follows, as before, by induction on terms. Moreover, the proof is mostly identical. Starting with ⟨ρ_{i}, J_{i}⟩ ∈ ρ, we need to show that J ∩ J_{i} ⊩ ρ_{i} ∈ τ. The construction of τ_{k} remains as above. What’s new is the demand, by the additional clause in the definition of forcing ∈, that, for all r ∈ J ∩ J_{i}, ⟨ρ_{i}^{r}, ℝ⟩ ∈ τ^{r}. But that’s easy to see: ⟨ρ_{i}^{r}, ℝ⟩ ∈ ρ^{r}, by the definition of ρ^{r}, and, as we’ve already seen, ρ^{r} = τ^{r}.

Returning to proving property 3, the hypothesis is that for some J and K containing r, J ⊩ ρ =_{M} σ and K ⊩ σ =_{M} τ. By the first lemma, J ∩K ⊩ ρ =_{M} σ, σ =_{M} τ, and so by the second, J ∩ K ⊩ ρ =_{M} τ, which suffices.

4: Let J ⊩ ρ =_{M} σ and K ⊩ ρ ∈_{M} τ. We will show that J ∩ K ⊩ σ ∈_{M} τ. Let r ∈ J ∩ K. By hypothesis, let ⟨τ_{i}, J_{i}⟩ ∈ τ, J′ ⊆ K be such that r ∈ J′ ∩ J_{i} ⊩ ρ =_{M} τ_{i}; without loss of generality J′ ⊆ J. By the first lemma, J′∩J_{i} ⊩ ρ =_{M} σ, and by the second, J′ ∩ J_{i} ⊩ σ =_{M} τ_{i}. Furthermore, ρ^{r} = σ^{r} and ⟨ρ^{r}, ℝ⟩ ∈ τ^{r}, hence ⟨σ^{r}, ℝ⟩ ∈ τ^{r}.

5: Similar, and left to the reader.

With this lemma in hand, we can now mod out of =_{M} at each node, and have a model in which equality is actually =.

3.2 The Forcing Relation

As above, we define a forcing relation J ⊩ ϕ, with J an open set of reals and ϕ a formula. The definition should be read as pertaining to all formulas with parameters from a fixed M_{n}, and is to be interpreted in said M_{n}.

Definition 3.12 For ϕ = ϕ(σ_{0}, ..., σ_{i}) a formula with parameters σ_{0}, ..., σ_{i}, ϕ^{r} is ϕ(σ_{0}^{r}, ..., σ_{i}^{r}).

Definition 3.13 J ⊩ ϕ is defined inductively on ϕ:

J ⊩ σ = τ iff for all ⟨σ_{i}, J_{i}⟩ ∈ σ J ∩ J_{i} ⊩ σ_{i} ∈ τ and for all r ∈ J σ^{r} = τ^{r}, and vice versa

J ⊩ σ ∈ τ iff for all r ∈ J there is a ⟨τ_{i}, J_{i}⟩ ∈ τ and J′ ⊆ J such that r ∈ J′ ∩ J_{i} ⊩ σ = τ_{i} and for all r ∈ J ⟨σ^{r}, ℝ⟩ ∈ τ^{r}

J ⊩ ϕ ∧ ψ iff J ⊩ ϕ and J ⊩ ψ

J ⊩ ϕ∨ψ iff for all r ∈ J there is a J′ ⊆ J containing r such that J ∩J′ ⊩ ϕ or J ∩ J′ ⊩ ψ

J ⊩ ϕ → ψ iff for all J′ ⊆ J if J′ ⊩ ϕ then J′ ⊩ ψ, and, for all r ∈ J, if ℝ ⊩ ϕ^{r} then ℝ ⊩ ψ^{r}

J ⊩ ∃x ϕ(x) iff for all r ∈ J there is a J′ containing r and a σ such that J ∩ J′ ⊩ ϕ(σ)

J ⊩ ∀x ϕ(x) iff for all r ∈ J and σ there is a J′ containing r such that J ∩ J′ ⊩ ϕ(σ), and for all r ∈ J and σ there is a J′ containing r such that J′ ⊩ ϕ^{r}(σ).

(Notice that in the last clause, σ is not interpreted as σ^{r}.)

Lemma 3.14 1. For all ϕ ∅ ⊩ ϕ.

2. If J’ ⊆ J ⊩ ϕ then J’ ⊩ ϕ.

3. If J_{i} ⊩ ϕ for all i then ⋃_{i} J_{i} ⊩ ϕ.

4. J ⊩ ϕ iff for all r ∈ J there is a J’ containing r such that J ∩ J’ ⊩ ϕ.

5. For all ϕ, J if J ⊩ ϕ then for all r ∈ J ℝ ⊩ ϕ^{r}.

6. If ϕ contains only ground model terms, then either ℝ ⊩ ϕ or ℝ ⊩ ¬ϕ.

proof:

1. Trivial induction, as before.

2. Again, a trivial induction.

3. By induction. As in the previous section, for the case of →, you need to invoke the previous part of this lemma. All other cases are straightforward.

4. Trivial, using 3.

5. By induction on ϕ.

Base cases: = and ∈: Trivial from the definitions of forcing = and ∈.

∨ and ∧: Trivial induction.

→: Suppose J ⊩ ϕ → ψ and r ∈ J. We must show that ℝ ⊩ ϕ^{r} → ψ^{r}. For the first clause, suppose K ⊆ ℝ and K ⊩ ϕ^{r}. If K = ∅ then K ⊩ ψ^{r}. Else let s ∈ K. Inductively, since s ∈ K ⊩ ϕ^{r}, ℝ ⊩ (ϕ^{r})^{s}. But (ϕ^{r})^{s} = ϕ^{r}, so ℝ ⊩ ϕ^{r}. Using the hypothesis on J, ℝ ⊩ ψ^{r}, and so by part 2 above, K ⊩ ψ^{r}. For the second clause, let s ∈ ℝ. If ℝ ⊩ (ϕ^{r})^{s} then ℝ ⊩ ϕ^{r}. By the hypothesis on J, ℝ ⊩ ψ^{r}, and ψ^{r} = (ψ^{r})^{s}.

∃: If J ⊩ ∃x ϕ(x) and r ∈ J, let J′ and σ be such that r ∈ J ∩ J′ ⊩ ϕ(σ). By induction, ℝ ⊩ ϕ^{r}(σ^{r}). σ^{r} witnesses that ℝ ⊩ ∃x ϕ^{r}(x).

∀: Let J ⊩ ∀x ϕ(x) and r ∈ J. We need to show that ℝ ⊩ ∀x ϕ^{r}(x). For the first clause, we will show that for any σ, ℝ ⊩ ϕ^{r}(σ). By part 4 above, it suffices to let s ∈ ℝ be arbitrary, and find a J′ containing s such that J′ ⊩ ϕ^{r}(σ). By the hypothesis on J, for every τ there is a J′′ containing r such that J′′ ⊩ ϕ^{r}(τ). Introducing new notation here, let τ be shift_{r−s}σ, which is σ with all the intervals shifted by r − s hereditarily. So we have r ∈ J′′ ⊩ ϕ^{r}(shift_{r−s}σ). Now shift by s − r. Letting J′ be the image of J′′, note that s ∈ J′, the image of shift_{r−s}σ is just σ, and the image of ϕ^{r} is just ϕ^{r}. Since the forcing relation is unaffected by this shift, we have s ∈ J′ ⊩ ϕ^{r}(σ), as desired.

The second clause follows by the same argument. Given any s ∈ ℝ and σ, we need to show that there is a J′ containing s such that J′ ⊩ (ϕ^{r})^{s}(σ). But (ϕ^{r})^{s} = ϕ^{r}, and we have already shown that for all σ ℝ ⊩ ϕ^{r}(σ).

6. If ℝ ⊮ ϕ, then we have to show that ℝ ⊩ ¬ϕ, that is ℝ ⊩ ϕ → ⊥. Since ϕ^{r} = ϕ, the second clause in forcing an implication is exactly the case hypothesis. The first clause is that for all non-empty J ⊆ ℝ J ⊮ ϕ. If, to the contrary, J ⊩ ϕ for some non-empty J, then, letting r ∈ J, by the previous part of this lemma, ℝ ⊩ ϕ^{r}; since ϕ^{r} = ϕ, this contradicts the case hypothesis.

Lemma 3.15 Truth Lemma: For any node r, r ⊧ ϕ iff J ⊩ ϕ for some J containing r.

proof: By induction on ϕ, in detail.

=: Trivial, by the definition of =_{M}.

∈: Trivial, by the definition of ∈_{M}.

∧: Trivial.

∨: Trivial.

→: First we do the left-to-right direction (“if true at a node then forced”). Suppose that for node r of tree height n, r ⊧ ϕ → ψ. Note that for s ∈ M_{n+1}

infinitesimally close to r, if ℝ ⊩ f(ϕ)^{s} then (inductively) f(ϕ)^{s} holds at any successor node to r, in particular the one labeled s. Since r ⊧ ϕ → ψ, and at node s g(ϕ) = f(ϕ)^{s} and g(ψ) = f(ψ)^{s}, f(ψ)^{s} would also hold at s. Inductively f(ψ)^{s} would be forced by some (non-empty) open set. Choosing any t from that open set, by part 5 of this lemma, ℝ ⊩ (f(ψ)^{s})^{t}. Also (f(ψ)^{s})^{t} = f(ψ)^{s}. So ℝ ⊩ f(ϕ)^{s} implies ℝ ⊩ f(ψ)^{s} for all s infinitely close to r. Hence by overspill the same must hold for all s in some finite interval J containing r, and the corresponding assertion in M_{n}: for all s ∈ J if ℝ ⊩ ϕ^{s} then ℝ ⊩ ψ^{s}. Note that the same holds also for all subsets of J.

Suppose for a contradiction that no subset of J containing r forces ϕ → ψ. In M_{n+1} let J′ be an infinitesimal neighborhood around r. So J′ ⊮ f(ϕ) → f(ψ). Since J′ ⊆ J, the second clause in the definition of J′ ⊩ f(ϕ) → f(ψ) is satisfied. Hence the first clause is violated. Let J′′ ⊆ J′ be such that J′′ ⊩ f(ϕ), but J′′ ⊮ f(ψ). By part 4 of this lemma and the inductive hypothesis, let s ∈ J′′ be such that s |̸= f(ψ). But s ⊧ f(ϕ). So s |̸= f(ϕ) → f(ψ). This contradicts r ⊧ ϕ → ψ.

For the right-to-left direction (“if forced then true”), suppose r ∈ J ⊩ ϕ → ψ. If r ⊧ ϕ, then inductively let r ∈ J′ ⊩ ϕ. So r ⊧ ψ, which persists at all future nodes. Hence r ⊧ ϕ → ψ. The same argument applies unchanged to any extension of r reached via a composition of only the f-style transition functions. The other cases are compositions which include at least one g; without loss of generality we can assume we’re using g to go from r to an immediate extension s. If s ⊧ g(ϕ), i.e. s ⊧ f(ϕ)^{s}, then by induction and by part 5 ℝ ⊩ f(ϕ)^{s}. Also, by elementarity s ∈ J ⊩ f(ϕ) → f(ψ). Hence, by the definition of forcing →, ℝ ⊩ f(ψ)^{s}, so s ⊧ f(ψ)^{s}, i.e. s ⊧ g(ψ).

∃: If r ⊧ ∃x ϕ(x), then let σ be such that r ⊧ ϕ(σ). Inductively there is a J containing r such that J ⊩ ϕ(σ). J suffices. In the other direction, if r ∈ J ⊩ ∃x ϕ(x), let J′ and σ be such that r ∈ J ∩ J′ ⊩ ϕ(σ). Inductively r ⊧ ϕ(σ), so r ⊧ ∃x ϕ(x).

∀: For the left-to-right direction, suppose at node r that r ⊧ ∀x ϕ(x). If there were no interval J forcing the first clause in the ∀-forcing definition, then let J be an infinitesimal neighborhood around r. Let s ∈ J and σ be such that there is no J′ containing s such that J′ ⊩ f(ϕ)(σ). Inductively s |̸= f(ϕ)(σ), which is a contradiction.

If there were no interval J forcing the second clause in the ∀-forcing definition, then let J be an infinitesimal neighborhood around r. Let s ∈ J and σ be such that there is no J′ containing s such that J′ ⊩ f(ϕ)^{s}(σ). Inductively s |̸= f(ϕ)^{s}(σ), i.e. s |̸= g(ϕ)(σ), which is a contradiction.

The right-to-left direction is trivial.

3.3 The Final Proof

It remains to show only

Theorem 3.16 M ⊧ CZF_{Exp}

and

Theorem 3.17 M ⊧ The Dedekind reals do not form a set.

proof: The only axioms below, the proofs of which are essentially different from the corresponding proofs in section 2, are Set Induction, Strong Collection, Separation, and, of course, Exponentiation.

• Infinity: As in the previous section, ω̂ will do. (Recall that the canonical name x̂ of any set x ∈ M_{n} is defined inductively as {⟨ŷ, ℝ⟩ | y ∈ x}.)

• Pairing: As in the previous section, given σ and τ, {⟨σ, ℝ⟩, ⟨τ, ℝ⟩} will do.

• Union: Given σ, {⟨τ, J ∩ J_{i}⟩ | for some σ_{i}, ⟨τ, J⟩ ∈ σ_{i} and ⟨σ_{i}, J_{i}⟩ ∈ σ} ∪ {⟨τ, r⟩ | for some σ_{i}, ⟨τ, r⟩ ∈ σ_{i} and ⟨σ_{i}, r⟩ ∈ σ} will do.

• Extensionality: We need to show that ∀x ∀y [∀z (z ∈ x ↔ z ∈ y) → x = y]. So let σ and τ be any terms at a node r such that r ⊧ “∀z (z ∈ σ ↔ z ∈ τ)”. We must show that r ⊧ “σ = τ”. By the Truth Lemma, let r ∈ J ⊩ “∀z (z ∈ σ ↔ z ∈ τ)”; i.e. for all r′ ∈ J, ρ there is a J′ containing r′ such that J ∩ J′ ⊩ ρ ∈ σ ↔ ρ ∈ τ, and a J′′ containing r′ such that J′′ ⊩ ρ ∈ σ^{r′} ↔ ρ ∈ τ^{r′}. We claim that J ⊩ “σ = τ”, which again by the Truth Lemma suffices.

To this end, let ⟨σ_{i}, J_{i}⟩ be in σ; we need to show that J ∩ J_{i} ⊩ σ_{i} ∈ τ. Let r′ be an arbitrary member of J ∩ J_{i} and ρ be σ_{i}. By the choice of J, let J′ containing r′ be such that J ∩ J′ ⊩ σ_{i} ∈ σ ↔ σ_{i} ∈ τ; in particular, J ∩ J′ ⊩ σ_{i} ∈ σ → σ_{i} ∈ τ. It has already been observed in 3.9, part 1, that J ∩J′ ∩J_{i} ⊩ σ_{i} ∈ σ, so J ∩J′ ∩J_{i} ⊩ σ_{i} ∈ τ. By going through each r′ in J ∩ J_{i} and using 3.14, part 3, we can conclude that J ∩ J_{i} ⊩ σ_{i} ∈ τ, as desired. The other direction (“τ ⊆ σ”) is analogous. Thus the first clause in J ⊩ “σ = τ” is satisfied.

The second clause is that, for each r ∈ J, σ^{r} = τ^{r}. That holds because σ^{r} and τ^{r} are ground model terms: σ^{r} = x̂ and τ^{r} = ŷ for some x, y ∈ M_{n}. If x ≠ y, then let z be in their symmetric difference. Then no set would force ẑ ∈ x̂ ↔ ẑ ∈ ŷ, contrary to the assumption on J.

• Set Induction (Schema): Suppose r ⊧ “∀x ((∀y ∈ x ϕ(y)) → ϕ(x))”, where r ∈ M_{n}; by the Truth Lemma, let J containing r force as much. We must show r ⊧ “∀x ϕ(x)”; again by the Truth Lemma, it suffices to show that J forces the same.

If not, then there is a σ and an r ∈ J such that either there is no J′ containing r forcing ϕ(σ) or there is no J′ containing r forcing ϕ^{r}(σ). In M_{n}, pick such a σ of minimal V -rank. We will show that neither option above is possible.

By the choice of J and 3.14 part 3, J ⊩ “(∀y ∈ σ ϕ(y)) → ϕ(σ)”. If we show that J ⊩ “∀y ∈ σ ϕ(y)”, then we can conclude that J ⊩ ϕ(σ), eliminating the first option above.

Toward the first clause in forcing a universal, let τ be a term. We claim that J ⊩ “τ ∈ σ → ϕ(τ)”, which suffices. Regarding the first clause in forcing an implication, suppose K ⊆ J and K ⊩ τ ∈ σ. Unraveling the definition of forcing ∈, for each s ∈ K, there is an L containing s forcing τ to equal some term ρ of lower V -rank than σ. By the minimality of σ, some neighborhood of s forces ϕ(ρ), hence also ϕ(τ) (perhaps by restricting to L). By 3.14 part 3, K also forces ϕ(τ). Thus the first clause in J ⊩ “τ ∈ σ → ϕ(τ)” is satisfied. The second clause in forcing an implication is that, for all r ∈ J, if ℝ ⊩ τ^{r} ∈ σ^{r} then ℝ ⊩ ϕ^{r}(τ^{r}). If ℝ ⊩ τ^{r} ∈ σ^{r}, then τ^{r} is forced to be equal to some ground model term x̂ of lower V -rank than σ. By the minimality of σ, ℝ ⊩ ϕ^{r}(x̂), i.e. ℝ ⊩ ϕ^{r}(τ^{r}). Thus the first clause in J ⊩ “∀y ∈ σ ϕ(y)” is satisfied.

Toward the second clause in that universal, given τ and r ∈ J, we must find a J′ containing r with J′ ⊩ τ ∈ σ^{r} → ϕ^{r}(τ). We claim that J suffices; that is, (i) for all K ⊆ J, if K ⊩ τ ∈ σ^{r} then K ⊩ ϕ^{r}(τ), and (ii) for all s ∈ J, if ℝ ⊩ τ^{s} ∈ σ^{r} then ℝ ⊩ ϕ^{r}(τ^{s}) (using here that (σ^{r})^{s} = σ^{r} and (ϕ^{r})^{s} = ϕ^{r}). Regarding (i), if K ⊆ J forces τ ∈ σ^{r}, then for each t ∈ K there is a neighborhood L of t forcing τ = x̂, for some x ∈ M_{n}. If L did not force ϕ^{r}(x̂), then, by 3.14 part 6, ℝ ⊩ ¬ϕ^{r}(x̂), where x̂ has lower rank than σ, contradicting the choice of σ. So L ⊩ ϕ^{r}(x̂), and L ⊩ ϕ^{r}(τ). Since each t ∈ K has such a neighborhood, K ⊩ ϕ^{r}(τ). Similarly for (ii): If ℝ ⊩ τ^{s} ∈ σ^{r}, then τ^{s} has lower rank than σ; by the minimality of σ, it cannot be the case that ℝ ⊩ ¬ϕ^{r}(τ^{s}), hence, by 3.14 part 6, ℝ ⊩ ϕ^{r}(τ^{s}).

We have just finished proving that J ⊩ “∀y ∈ σ ϕ(y)”, and so J ⊩ ϕ(σ). Hence the first option provided by the minimality of σ is not possible. The other option is that for some r ∈ J there is no J′ containing r forcing ϕ^{r}(σ). We will show this also is not possible.

Fix r ∈ J. By the choice of J (using the second clause in the definition of forcing ∀), there is a J′ containing r such that J′ ⊩ “(∀y ∈ σ ϕ^{r}(y)) → ϕ^{r}(σ)”. If we show that J′ ⊩ “∀y ∈ σ ϕ^{r}(y), then we can conclude J′ ⊩ ϕ^{r}(σ), for our contradiction.

Toward the first clause in forcing a universal, let τ be a term. We claim that J′ ⊩ “τ ∈ σ → ϕ^{r}(τ)”, which suffices. Regarding the first clause in forcing an implication, suppose K ⊆ J′ and K ⊩ τ ∈ σ. We need to show K ⊩ ϕ^{r}(τ). It suffices to find a neighborhood of each s ∈ K forcing ϕ^{r}(τ). So let s ∈ K. Unraveling the definition of forcing ∈, there is a K′ containing s forcing τ to equal some term ρ of lower V -rank than σ. Shift (as in the proof of 3.14 part 5) by r − s. Since the parameters in ϕ^{r} are all ground model terms, shift_{r−s}(ϕ^{r}) = ϕ^{r}. Also, rk(shift_{r−s}ρ) = rk(ρ) < rk(σ). By the minimality of σ, there is some neighborhood L of r forcing ϕ^{r}(shift_{r−s}ρ). Shifting back, we get shift_{s−r}(L) containing s and forcing ϕ^{r}(ρ). Then s ∈ K′ ∩ shift_{s−r}(L) ⊩ ϕ^{r}(τ), as desired.

The second clause in forcing an implication is that, for all s ∈ J′, if ℝ ⊩ τ^{s} ∈ σ^{s} then ℝ ⊩ ϕ^{r}(τ^{s}). If ℝ ⊩ τ^{s} ∈ σ^{s}, then τ^{s} is forced to be equal to some ground model term x̂ of lower V -rank than σ. By the minimality of σ, it cannot be the case that ℝ ⊩ ¬ϕ^{r}(x̂), so, by 3.14 part 6, ℝ ⊩ ϕ^{r}(x̂), i.e. ℝ ⊩ ϕ^{r}(τ^{s}). Thus the first clause in J ⊩ “∀y ∈ σ ϕ(y)” is satisfied.

Toward the second clause in that universal, given τ and s ∈ J′, we must find a neighborhood of s forcing τ ∈ σ^{s} → ϕ^{r}(τ). We claim that J′ suffices; that is, (i) for all K ⊆ J′, if K ⊩ τ ∈ σ^{s} then K ⊩ ϕ^{r}(τ), and (ii) for all t ∈ J′, if ℝ ⊩ τ^{t} ∈ σ^{s} then ℝ ⊩ ϕ^{r}(τ^{t}). For (i), if K ⊆ J′ forces τ ∈ σ^{s}, then for each t ∈ K there is a neighborhood L of t forcing τ = x̂, for some x ∈ M_{n}. If L did not force ϕ^{r}(x̂), then, by 3.14 part 6, ℝ ⊩ ¬ϕ^{r}(x̂), where x̂ has lower rank than σ, contradicting the choice of σ. So L ⊩ ϕ^{r}(x̂), and L ⊩ ϕ^{r}(τ). Since each t ∈ K has such a neighborhood, K ⊩ ϕ^{r}(τ). Similarly for (ii): If ℝ ⊩ τ^{t} ∈ σ^{s}, then τ^{t} has lower rank than σ; by the minimality of σ, it cannot be the case that ℝ ⊩ ¬ϕ^{r}(τ^{t}), hence, by 3.14 part 6, ℝ ⊩ ϕ^{r}(τ^{t}).

• Exponentiation: Let σ and τ be terms at node r. Let C be {⟨ρ, J⟩ | rk(ρ) < max(rk(σ), rk(τ))+ω, and J ⊩ ρ : σ → τ is a function}∪{⟨ĥ, s⟩ | h : σ^{s} → τ^{s} is a function}. (The restriction on ρ is so that C is set-sized.) We claim that C suffices.

Let s be any immediate extension of r. (The case of non-immediate extensions follows directly from this case.) If s ⊧ “ρ : f(σ) → f(τ) is a function”, then s ⊧ ρ ∈ f(C) by the first clause in the definition of C. If s ⊧ “ρ : g(σ) → g(τ) is a function” and ρ is a ground model term, then s ⊧ ρ ∈ g(C) by the second clause. What we must show is that for any node r and sets X and Y, if r ⊧ “ρ : X̂ → Ŷ is a function”, then some neighborhood of r forces ρ equal to a ground model function.

By the Truth Lemma, let r ∈ J ⊩ “ρ : X̂ → Ŷ is a function”. We claim that for all x ∈ X there is a y ∈ Y such that for each s ∈ J s ⊧ ρ(x̂) = ŷ. If not, let x be otherwise. Let y be such that r ⊧ ρ(x̂) = ŷ. For each immediate successor s of r, s ⊧ f(ρ)(f(x̂)) = f(ŷ). By overspill the same holds for some neighborhood around r (sans the f’s). If this does not hold for all s ∈ J, let s be an endpoint in J of the largest interval around r for which this does hold. Repeating the same argument around s, there is a y′ such that, for all t in some neighborhood of s, t ⊧ ρ(x̂) = ŷ′. This neighborhood of s must overlap that of r, though. So y = y′, contradicting the choice of s. So the value ρ(x̂) is fixed on the whole interval J, and ρ is forced by J to equal a particular ground model function.

• Separation: Although CZF contains only Δ_{0} Separation, full Separation holds here. Let ϕ(x) be a formula and σ a term. Then {⟨σ_{i}, J ∩ J_{i}⟩ | ⟨σ_{i}, J_{i}⟩ ∈ σ and J ⊩ ϕ(σ_{i})} ∪ {⟨x, s⟩ | ⟨x, ℝ⟩ ∈ σ^{s} and ℝ ⊩ ϕ^{s}(x)} will do.

• Strong Collection: If r ⊧ ∀x ∈ σ ∃y ϕ(x, y), let r ∈ J force as much. For each ⟨σ_{i}, J_{i}⟩ ∈ σ and s ∈ J ∩ J_{i}, let τ_{i,s} and J_{i,s} be such that s ∈ J_{i,s} ⊩ ϕ(σ_{i}, τ_{i,s}). Also, for each s ∈ J and ⟨x, ℝ ∈ σ^{s}, let τ_{x,s} be such that some neighborhood of s forces ϕ^{s}(x, τ_{x,s}). (Notice that, by 3.14 part 5, ℝ ⊩ ϕ^{s}(x, τ_{x,s}^{s}).) Then {⟨τ_{i,s}, J_{i,s}⟩ | i ∈ I, s ∈ J} ∪ {⟨τ_{x,s}, s⟩ | s ∈ J, x ∈ σ^{s}} suffices.

proof: First note that the same generic term from the last section, G := {⟨r̂, J⟩ |

r is a rational, J is an open interval from the reals, and r < J}, still defines a Dedekind cut. In fact, half of the proof of such is just the argument from the last section itself. That’s because most of the properties involved with being a Dedekind real are local. For instance, if s < t are rationals at any given node, then it must be checked at that node whether s ∈ G or t ∉ G. For this, the earlier arguments work unchanged. The same applies to images of G at later nodes, as long as such image satisfies the same definition, i.e. is of the form f(G). We must check what happens when G settles down. Cranking through the definition, G^{s} = {⟨r̂, ℝ⟩ | r < s}, which is the Dedekind cut standing for s, and which satisfies all the right properties.

Furthermore, although G can become a ground model real, it isn’t one itself: there are no J and r such that J ⊩ G = r̂. That’s because there is a K ⊆ J with either r < s < K or K < s < r (some s). In the former case, K ⊩ ŝ ∈ G, i.e. K ⊩ r̂ < G; in the latter, K ⊩ ¬ŝ ∈ G, i.e. K ⊩ r̂ > G.

Finally, to see that the Dedekind reals do not form a set, let σ be a term at any node. g(σ) is a ground model term. So if any J ⊩ G ∈ g(σ), then for some K ⊆ J, K ⊩ G = r̂ for some real r, which we just saw cannot happen. So σ cannot name the set of Dedekind reals.

Not infrequently, when some weaker axioms are shown to hold, what interests people is not why the weaker ones are true but why the stronger ones aren’t. The failure of Subset Collection, and hence of Power Set too, is exactly what the previous paragraph is about, but perhaps the failure of Power Set is more clearly seen on the simpler set 1 = {0} = {∅}. After applying a settling down transition function g to a set σ, the only subsets of 1 in g(σ) are 0 and 1. But in M 1 has more subsets than that. For instance, at node r, {⟨∅, J⟩ | min J > r} is 0 at all future nodes s < f(r) and 1 at all future nodes s > f(r). So σ could never have been the power set of 1 to begin with, because g(σ) is missing some such subsets.

This same example also shows that Reflection fails: that the power set of 1 does not exist is true in M, as above, but not true within any set, as once that set settles down, {0, 1} is indeed the internal power set of 1.

\underline{Comments and Questions} The settling down process, as explained when introduced, was motivated by the construction in [8]. The change in the terms (adding members based not on open sets but on individual real numbers, to be used only when settling down) was quickly seen to be necessary to satisfy Separation. But where does the unusual topological semantics come from? The topology is the same: the space is still ℝ, the only things that force statements are the same open sets as before; it’s just a change in the meaning of the forcing relation ⊩, the semantics. It is no surprise that there would have to be some change, in the base cases (= and ∈) if nowhere else. But why exactly those changes as presented in the inductive cases? The authors found them through a bothersome process of trial and error, and have no explanation for them.

Would this new semantics have interesting applications elsewhere? As an example of a possible kind of application, consider the topology of this article. Under the standard semantics, there is a Dedekind cut which is not a Cauchy sequence. With the new semantics, the collections of Dedekind and Cauchy reals differ (the latter being a set and the former not), but not for that reason. In fact, any Dedekind cut is not not equal to a Cauchy real: just apply a settling down transition. These collections differ because the Dedekind reals include some things that are just not yet Cauchy reals. So this semantics might be useful for gently separating concepts, getting sets (or classes) of things to be unequal without producing any instance of one which is not the other.

Finally, which axioms does this new semantics verify? For instance, in [10] a topological model for a generic Cauchy sequence is given. Analogously with the current construction, wherein a generic Dedekind cut in a model with settling down implies that there is no set of Dedekind cuts, in a model with settling down and a generic Cauchy sequence the Cauchy sequences are not a set. That would mean that not even Exponentiation holds. So what does hold generally under this semantics?

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