Lubarsky’s Kripke Models in Constructive Set Theory

• Lubarsky’s Kripke models are a semantic framework in constructive set theory that use inductive name hierarchies to analyze double complement axioms.
• The models vary the underlying frame and the class of admissible names to force different behaviors like DCom, NDCom, or ADCom.
• These constructions refine classical CZF techniques by showing, for example, that DCom implies Powerset and establishing equiconsistency results.

Searching arXiv for the specified paper and closely related work on Lubarsky's Kripke models in constructive set theory. Lubarsky's Kripke models, as developed and adapted in the study of the Axiom of Double Complement and its opposites, are internal set-theoretic models built over constructive ground theories such as $\mathsf{CZF}$ and $\mathsf{IZF}$ in order to analyze independence, compatibility, and relative consistency phenomena that are inaccessible by direct proof alone. In the treatment of "The Axiom of Double Complement and its opposites" (Jeon et al., 30 May 2026), these models are used to investigate $\mathsf{DCom}$ together with its opposites, $\mathsf{NDCom}$ and $\mathsf{ADCom}$, by varying the underlying frame $P$ and, in one case, restricting the class of admissible names. The resulting constructions recover and modify Lubarsky’s original $\mathsf{CZF}+\lnot\mathsf{Pow}$ techniques, and they yield metatheoretic results including $\mathsf{DCom}\rightarrow\mathsf{Pow}$ over $\mathsf{CZF}$, equiconsistency results for $\mathsf{NDCom}$ and $\mathsf{IZF}$0 with $\mathsf{IZF}$1, and persistence theorems under realizability (Jeon et al., 30 May 2026).

1. Motivating role in constructive set theory

In $\mathsf{IZF}$2 and $\mathsf{IZF}$3 one does not in general have full classical powersets, so internal semantic constructions become a primary tool for testing new axioms. The paper situates Lubarsky’s Kripke-model method in precisely this role: Kripke or sheaf models are built to establish independence of new principles, to determine compatibility or incompatibility with axioms such as Powerset, Separation, Collection, and Regular Extension (REA), and to calibrate relative consistency strength (Jeon et al., 30 May 2026).

Lubarsky’s original models exhibited $\mathsf{IZF}$4. The later development described in (Jeon et al., 30 May 2026) shows that these models are not tied to a single metatheoretic conclusion. By varying the frame $\mathsf{IZF}$5 and the allowed class of names, one can force $\mathsf{IZF}$6 or force its failure in distinct ways. This makes the method especially suitable for comparing $\mathsf{IZF}$7 with its weak and strong opposites, $\mathsf{IZF}$8 and $\mathsf{IZF}$9, inside constructive set theory.

The same framework also clarifies a central issue about $\mathsf{DCom}$0: its consistency, strength, and compatibility remain open problems in general, even though the paper establishes several sharp partial results. In particular, the paper states that $\mathsf{DCom}$1 proves Powerset over $\mathsf{DCom}$2 and is independent of $\mathsf{DCom}$3 (Jeon et al., 30 May 2026). This places Lubarsky-style Kripke semantics at the center of the subject rather than at its periphery.

2. General Kripke-model construction over a frame $\mathsf{DCom}$4

The basic setup begins with a classical model $\mathsf{DCom}$5 of $\mathsf{DCom}$6 or $\mathsf{DCom}$7 and a poset $\mathsf{DCom}$8 with least element $\mathsf{DCom}$9. For each ordinal $\mathsf{NDCom}$0 and node $\mathsf{NDCom}$1, one defines classes $\mathsf{NDCom}$2 of names of rank $\mathsf{NDCom}$3 at $\mathsf{NDCom}$4, together with transition maps $\mathsf{NDCom}$5 for $\mathsf{NDCom}$6 (Jeon et al., 30 May 2026).

The inductive clauses are as follows. Each $\mathsf{NDCom}$7 is a function

$\mathsf{NDCom}$8

These functions satisfy a monotonicity condition: whenever $\mathsf{NDCom}$9 and $\mathsf{ADCom}$0, then $\mathsf{ADCom}$1. They also satisfy a rank clause, namely

$\mathsf{ADCom}$2

The full collection of names at $\mathsf{ADCom}$3 is then

$\mathsf{ADCom}$4

and the restriction map is defined by $\mathsf{ADCom}$5 (Jeon et al., 30 May 2026).

Forcing is defined recursively on formulas $\mathsf{ADCom}$6 with parameters in $\mathsf{ADCom}$7:

• $\mathsf{ADCom}$8 iff $\mathsf{ADCom}$9.
• $P$0 iff for all $P$1, $P$2.
• $P$3 iff $P$4 and $P$5.
• $P$6 iff $P$7 or $P$8.
• $P$9 iff for every $\mathsf{CZF}+\lnot\mathsf{Pow}$0, $\mathsf{CZF}+\lnot\mathsf{Pow}$1 implies $\mathsf{CZF}+\lnot\mathsf{Pow}$2.
• $\mathsf{CZF}+\lnot\mathsf{Pow}$3 iff for every $\mathsf{CZF}+\lnot\mathsf{Pow}$4 and every $\mathsf{CZF}+\lnot\mathsf{Pow}$5, $\mathsf{CZF}+\lnot\mathsf{Pow}$6.
• $\mathsf{CZF}+\lnot\mathsf{Pow}$7 iff there is $\mathsf{CZF}+\lnot\mathsf{Pow}$8 with $\mathsf{CZF}+\lnot\mathsf{Pow}$9.

By standard arguments together with the inductive-definition theorem, the paper shows that the $\mathsf{DCom}\rightarrow\mathsf{Pow}$0 axioms hold in $\mathsf{DCom}\rightarrow\mathsf{Pow}$1, and that full Separation or full Powerset lift to the Kripke model whenever they already hold in the ground model $\mathsf{DCom}\rightarrow\mathsf{Pow}$2 (Jeon et al., 30 May 2026).

3. Structural mechanisms: persistency, rank, slicing, and translation

A defining feature of these models is built-in persistency. Because the data of a name are given on all nodes above a base point and are constrained by the monotonicity clause, the semantics automatically tracks how membership information propagates upward through the frame (Jeon et al., 30 May 2026).

The paper isolates this structure further through a rank function

$\mathsf{DCom}\rightarrow\mathsf{Pow}$3

Using induction on the fully inductive $\mathsf{DCom}\rightarrow\mathsf{Pow}$4-closure, it is shown that each $\mathsf{DCom}\rightarrow\mathsf{Pow}$5. In the $\mathsf{DCom}\rightarrow\mathsf{Pow}$6 ground-model setting, a closure lemma states that if $\mathsf{DCom}\rightarrow\mathsf{Pow}$7 is forced, then the rank of $\mathsf{DCom}\rightarrow\mathsf{Pow}$8 is strictly below that of $\mathsf{DCom}\rightarrow\mathsf{Pow}$9 (Jeon et al., 30 May 2026). This rank stratification is essential in the later analysis of double-complement constructions.

Two additional devices play a recurrent role. First, for a fixed stage $\mathsf{CZF}$0, the slicing operation

$\mathsf{CZF}$1

converts membership at $\mathsf{CZF}$2 into a new name. Second, the translation maps

$\mathsf{CZF}$3

shift the domain upward by $\mathsf{CZF}$4. In eventual-constant models these maps become genuine isomorphisms (Jeon et al., 30 May 2026). The paper explicitly uses these translation isomorphisms in the proof that Separation holds in the eventually constant model $\mathsf{CZF}$5.

4. Linear frames and the forcing of $\mathsf{CZF}$6

For $\mathsf{CZF}$7, the decisive case is a linear frame $\mathsf{CZF}$8, such as $\mathsf{CZF}$9 or $\mathsf{NDCom}$0. In that setting the paper proves the lemma that if

$\mathsf{NDCom}$1

then $\mathsf{NDCom}$2 (Jeon et al., 30 May 2026). This rank drop is the key technical fact allowing the double-complement operation to be internalized as a set.

Given $\mathsf{NDCom}$3, the paper defines its double-complement name $\mathsf{NDCom}$4 by

$\mathsf{NDCom}$5

It is then verified that $\mathsf{NDCom}$6 and that

$\mathsf{NDCom}$7

Accordingly, in any linear $\mathsf{NDCom}$8-model of $\mathsf{NDCom}$9 one obtains $\mathsf{IZF}$00 (Jeon et al., 30 May 2026).

The significance of this construction is twofold. First, it shows that the semantic content of $\mathsf{IZF}$01 can be realized by a direct name-theoretic operation in linear Kripke frames. Second, when combined with the paper’s corollary that over $\mathsf{IZF}$02 one has $\mathsf{IZF}$03, it indicates that $\mathsf{IZF}$04 is not a minor closure principle but one with substantial set-existence consequences (Jeon et al., 30 May 2026).

5. Tree frames, eventual constancy, and the opposites $\mathsf{IZF}$05 and $\mathsf{IZF}$06

The same semantic template yields opposite behaviors once the frame or the name class is altered. The paper presents two canonical variants (Jeon et al., 30 May 2026).

Frame	Name class	Result
Linear $\mathsf{IZF}$07 such as $\mathsf{IZF}$08 or $\mathsf{IZF}$09	Full $\mathsf{IZF}$10	Forces $\mathsf{IZF}$11
$\mathsf{IZF}$12	Full $\mathsf{IZF}$13	Forces $\mathsf{IZF}$14
$\mathsf{IZF}$15	Eventually constant names $\mathsf{IZF}$16	Forces $\mathsf{IZF}$17

For $\mathsf{IZF}$18, the frame is the full binary tree $\mathsf{IZF}$19. The paper shows that there is a proper class of names $\mathsf{IZF}$20 all forced at all nodes to lie in $\mathsf{IZF}$21. Because no single Kripke-set can collect them as the double complement of $\mathsf{IZF}$22, one obtains $\mathsf{IZF}$23 (Jeon et al., 30 May 2026). The argument turns on branching: unlike the linear case, binary splitting obstructs the formation of a unique collecting set for the relevant double-complement information.

For $\mathsf{IZF}$24, the construction restricts attention to names that become constant after some finite stage. Lubarsky’s “first model” is recast as follows. For each $\mathsf{IZF}$25,

$\mathsf{IZF}$26

One then sets

$\mathsf{IZF}$27

and the quantifiers in the forcing relation are taken over $\mathsf{IZF}$28 alone (Jeon et al., 30 May 2026).

Within this model, $\mathsf{IZF}$29 still holds; the proof uses the translation isomorphisms $\mathsf{IZF}$30 to establish Separation. However, any forced “double complement of $\mathsf{IZF}$31” would have to contain names encoding inhabited subsets of $\mathsf{IZF}$32 at all stages, and this is impossible in the eventually constant setting. Hence $\mathsf{IZF}$33 holds in $\mathsf{IZF}$34 (Jeon et al., 30 May 2026). A plausible implication is that eventual constancy converts a semantic limitation—stabilization after finitely many stages—into a precise obstruction to double-complement closure.

6. Metatheoretic consequences and persistence under realizability

The model constructions support several principal theorems. The paper states the corollary

$\mathsf{IZF}$35

over $\mathsf{IZF}$36. The proof proceeds by obtaining $\mathsf{IZF}$37 and then using exponentiation and subset-collection to derive full Powerset (Jeon et al., 30 May 2026). The same work also states that $\mathsf{IZF}$38 is independent of $\mathsf{IZF}$39 (Jeon et al., 30 May 2026).

For the opposites, the paper proves equiconsistency results. $\mathsf{IZF}$40 is equiconsistent with $\mathsf{IZF}$41, with the $\mathsf{IZF}$42 model built over $\mathsf{IZF}$43 starting from $\mathsf{IZF}$44. Likewise, $\mathsf{IZF}$45 is equiconsistent with $\mathsf{IZF}$46, since the eventually constant model $\mathsf{IZF}$47 is built over a $\mathsf{IZF}$48 ground (Jeon et al., 30 May 2026). These results isolate a sharp asymmetry: while $\mathsf{IZF}$49 does not add consistency strength over $\mathsf{IZF}$50, $\mathsf{IZF}$51 already implies Powerset and therefore has much stronger consequences over the same base theory.

A further theorem concerns realizability persistence. Under mild extra Separation or REA assumptions, McCarty-style realizability preserves $\mathsf{IZF}$52, $\mathsf{IZF}$53, and $\mathsf{IZF}$54 (Jeon et al., 30 May 2026). The paper explicitly notes that one may therefore combine $\mathsf{IZF}$55 with standard constructive principles such as Choice, Church’s Thesis, Markov’s Principle, and Uniformity in a realizability model. It also reports that §§6–7 exhibit functional-realizability failures of Powerset, hence satisfaction of $\mathsf{IZF}$56 (Jeon et al., 30 May 2026).

Taken together, these results establish Lubarsky’s Kripke models as a flexible semantic apparatus for constructive set theory. Their methodological core is stable—the inductive name hierarchy over a frame $\mathsf{IZF}$57—but their mathematical output depends delicately on linearity versus branching, and on whether the model permits unrestricted names or only eventually constant ones. The open problems surrounding the consistency, strength, and compatibility of $\mathsf{IZF}$58 remain, but the paper shows that Lubarsky-style models already organize a substantial portion of the known landscape (Jeon et al., 30 May 2026).