Weak Propositional Resizing Principle

• Weak Propositional Resizing Principle is an axiom in type theory that equates each mere proposition in a higher universe with an equivalent in a lower universe.
• It underpins the construction of small order-theoretic structures like nontrivial complete posets, bridging predicativity with impredicative methods.
• The principle enables reverse mathematics by linking the existence of small dcpo and posets to key fixed-point theorems and classical maximality principles.

The weak propositional resizing principle is a foundational axiom in type theory and univalent foundations concerning the relationship between propositions in higher universes and their representations in lower universes. Specifically, it posits that every mere proposition in a universe $U_i$ can be equivalently represented by a proposition in a smaller universe $U_j$, without necessarily collapsing all propositions into a single universe as full propositional resizing would. This principle has deep connections to the existence and predicative construction of certain order-theoretic structures, with notable implications for the formulability and applicability of key impredicative results in mathematics.

1. Formal Definition and Basic Properties

Let $\mathsf{Prop}_i$ denote the type of mere propositions ($(-1)$-types) living in universe $U_i$. A proposition $P:\mathsf{Prop}_i$ is $U_j$-small if there exists $Q:\mathsf{Prop}_j$ such that $Q \simeq P$. The weak propositional resizing principle asserts: $$\forall\, (P : \mathsf{Prop}_i)\, \exists\, (Q : \mathsf{Prop}_j) .\, Q \simeq P$$ This is denoted as $\mathsf{WPropRes}_{i,j}$. If $j<i$, the resizing is termed "weak"; if $j \leq i$, full propositional resizing obtains, often denoted $\mathsf{PropRes}_{i,i}$. The weak form avoids collapsing all proposition universes but allows for equivalence witnesses across universe boundaries (Jong et al., 2021, Jong et al., 2021).

A refined version focuses on $\neg\neg$-stable propositions. If $\Omega_{U}$ is the type of all mere propositions in $U$, the collection of $\neg\neg$-stable propositions is

$$\Omega^{\neg\neg}_U \coloneqq \sum_{P: \Omega_U} (\neg\neg P \to P)$$

Weak propositional resizing then amounts to $\Omega^{\neg\neg}_U$ having the size of a lower universe $V$, i.e., there exists $Y: V$ such that $Y \simeq \Omega^{\neg\neg}_U$.

2. Connections to Order-Theoretic Structures

The principal mathematical content of weak propositional resizing arises from its equivalence with the existence of nontrivial small order-theoretic structures. Specifically:

• A poset $(X, \le)$ in $U_\ell$ is nontrivial if there exist $x,y : X$ with $x\le y$ and $x \neq y$.
• It is directed complete (dcpo) if every directed family indexed by $I:U_\ell$ has a supremum in $X$.
• It is small if $X : U_\ell$ and $x \le y : \mathsf{Prop}_\ell$ for all $x, y : X$, with the identity relation factoring through $\mathsf{Prop}_\ell$—implying $X$ is a set in $U_\ell$.

The existence of a nontrivial small dcpo or bounded-complete poset in $U_\ell$ is equivalent to the weak propositional resizing $\mathsf{WPropRes}_{i,\ell}$. In constructive settings, nontriviality suffices for weak resizing, while strengthening to "positivity" (existence of $x \ll y$ in the domain-theoretic sense) yields full resizing. The distinction between nontrivial and positive is constructively significant and analogous to nonemptiness versus inhabitedness (Jong et al., 2021, Jong et al., 2021).

3. Reverse Mathematics: Resizing from Small Complete Posets

Central results in constructive univalent foundations have shown that:

• Reverse implication: If there exists a nontrivial small dcpo (or more generally, a small nontrivial $\delta$-complete poset), then weak propositional resizing holds. The construction uses retractions from $\mathsf{Prop}_i$ into the poset, showing that the smallness of the poset entails the smallness of the universe of mere propositions.
• Conversely, if weak propositional resizing holds, then $\Omega^{\neg\neg}_U$ is itself a small nontrivial $\delta$-complete poset.

This equivalence is established through the construction of explicit maps:

• $\Delta: \mathsf{Prop}_i \to X$, defined using supremum constructions in the poset.
• $r: X \to \mathsf{Prop}_i$, defined as $r(z) = \neg(x \leq z)$.

Their composition yields a retraction $r \circ \Delta = \mathsf{id}_{\mathsf{Prop}_i}$. Univalence ensures closure of smallness under retracts, thus propagating smallness from $X$ to $\mathsf{Prop}_i$ (Jong et al., 2021, Jong et al., 2021).

Strengthening to positivity, namely, the existence of a strictly-positive pair $x \ll y$, upgrades the consequence to full propositional resizing. Here, further structural properties of the poset facilitate a section for every $z \geq y$, embedding the entire proposition universe inside itself up to equivalence.

4. Consequences for Order Theory and Impredicativity

A direct consequence is the predicative inaccessibility of certain classical order-theoretic fixed-point theorems within univalent foundations. If predicativity is taken seriously (i.e., without resizing or excluded middle), then:

• Nontrivial small complete posets cannot exist without weak propositional resizing.
• Standard classical results such as Tarski's fixed-point theorem, Zorn's lemma, and Pataraia’s lemma imply full propositional resizing, collapsing these principles to strong impredicative requirements.

For example, in predicative settings, there are no small nontrivial sup-lattices or complete lattices obeying the classical maximal or fixed-point properties unless resizing holds. Similarly, even constructing certain categories of ordinals illustrates the necessity of large universe sizes for genuinely complete posets (Jong et al., 2021, Jong et al., 2021).

5. Model-Theoretic Independence and Failure

The independence of weak (and full) propositional resizing from standard constructive type-theoretic and homotopical foundations is sharply exhibited by model constructions:

• In the cubical assembly model, which supports a univalent, impredicative universe, weak propositional resizing fails. Specifically, there exist homotopy propositions in a cubical fibrant universe $U^F$ such that the canonical resizing map $\eta_A: A \to A^*$ is not an equivalence.
• This demonstrates that even full impredicativity and univalence do not imply resizing; resizing must be posited as a separate axiom if needed (Uemura, 2018).

Such counterexamples highlight the necessity of treating resizing principles distinctly and emphasize the limitations of relying on impredicative universes alone for foundational purposes.

6. Implications for Foundational Methodologies

The interplay between weak propositional resizing and predicative foundations clarifies the boundaries for constructive developments in order theory, domain theory, and related areas:

• The principle explicitly parametrizes what can and cannot be done predicatively in the presence of universes.
• It identifies precisely the impredicative content underlying the existence of certain “small” structures and classical arguments.
• The failure of resizing in univalent impredicative models suggests that any predicative development must avoid the assumption of small nontrivial complete posets, or else concede resizing and its associated impredicativity.
• In practice, for the preservation of predicativity, development must work with fixed large domains or refrain from classical maximality principles unless universe-level resizing is explicitly postulated (Jong et al., 2021, Jong et al., 2021, Uemura, 2018).

7. Summary Table: Logical Relationships

Structure/Principle	Implies	Size/Resizing Needed
Nontrivial small dcpo/$\delta$-complete poset	Weak resizing	$\mathsf{WPropRes}_{i,j}$
Small positive dcpo/$\delta$-complete poset	Full resizing	$\mathsf{PropRes}_{i,i}$
Zorn's lemma (for small posets)	Full resizing	$\mathsf{PropRes}_{i,i}$
Tarski's fixed-point theorem (small lattices)	Full resizing	$\mathsf{PropRes}_{i,i}$
Cubical assembly model with univalent universe	– (resizing fails)	– (independent/failed)

A plausible implication is that formal predicative mathematics within univalent foundations necessitates the careful avoidance or explicit management of resizing requirements in any theory dealing with complete posets, sup-lattices, or DCPOs. The weak propositional resizing principle thus stands at the boundary of constructive, predicative methods and classical, impredicative arguments (Jong et al., 2021, Jong et al., 2021, Uemura, 2018).