Corrigendum & Addendum to "Categoricity-like Properties in the First Order Realm"

This paper serves as a corrigendum and addendum to the authors' 2024 work on categoricity-like properties in first-order logic, providing necessary corrections and supplementary insights to the original study.

The Gist

Imagine you and a friend wrote a very complex instruction manual for building a specific type of universe (a mathematical world). You called this manual "The Book of Sets." A few years later, you realized that while the conclusions in the book were correct, some of the steps you used to get there were shaky, and one of the tools you claimed was essential actually didn't work the way you thought.

This document is your Corrigendum and Addendum—a formal "Oops, we fixed it" note attached to your original book.

Here is the breakdown of what happened, explained simply:

1. The "Oops" Moment (The Corrigendum)

The authors, Ali and Mateusz, are fixing two specific problems in their original paper.

Problem A: The Broken Bridge (Theorem 39)

• The Original Idea: They tried to prove that certain mathematical rules (called solidity and tightness) don't hold for specific types of set theories. Think of "solidity" as asking: "If I build a house using these blueprints, is there only one possible house, or could I build a totally different one that still follows the rules?"
• The Mistake: In their original proof, they used a specific type of mathematical "glue" (a lemma) to connect their ideas. They realized the glue was too weak. They needed a stronger, more complex version of that glue to make the bridge hold.
• The Fix: They wrote a new, stronger theorem (Theorem 3) that acts as the heavy-duty glue. They also realized they mislabeled the complexity of the instructions (saying they were "Level 2" difficulty when they were actually "Level 3").
• The Result: The original conclusion stands! The rules still don't force a single unique universe. The only thing that changed is the path they took to prove it.

Problem B: The Broken Tool (Theorem 77)

• The Original Idea: They claimed that a specific set of rules (called Repl + Tarski) was so powerful that it forced everyone to build the exact same universe. They thought this was a "perfect" rule set.
• The Mistake: They relied on a helper rule (Lemma 79) that they thought was true. A reader pointed out that this helper rule is actually false.
• The Counter-Example: They built a "fake" universe using a mathematical trick called an "ultrapower" (imagine taking a photo of a city, zooming in infinitely, and creating a new city based on that zoom). In this new city, the helper rule fails.
• The Result: They have to withdraw their claim that Theorem 77 is true. They don't know yet if the original idea is true or false; they just know their proof was wrong. It's like realizing your map to the treasure is wrong, but you don't know if the treasure is actually there or not.

2. The "New Stuff" (The Addendum)

After fixing the mistakes, the authors look at what other mathematicians have been doing recently. It's like a "What's New in Math" newsletter:

• Simpler Proofs: Someone else found a simpler way to prove one of their earlier results, without needing to assume the existence of giant, mysterious "inaccessible" numbers.
• New Distinctions: Other researchers found new ways to tell different types of mathematical universes apart, showing that some rules are "solid" (unshakeable) while others are "loose."
• The Multiverse: There's new work on whether the idea of a "multiverse" (many different mathematical realities) can be categorized in a strict way.
• Foundational Questions: People are using these "solidity" concepts to ask big questions about the nature of reality and set theory.

The Big Picture Analogy

Think of the original paper as a recipe for a perfect cake.

1. The Fix: The authors realized they wrote the recipe using a measuring cup that was slightly the wrong size. They had to rewrite the steps to use the right cup, but the cake still tastes the same.
2. The Withdrawal: They also claimed that if you follow this specific recipe, you are guaranteed to get a cake that looks exactly like a photo. They realized their proof for that guarantee was wrong. They can't promise the photo-match anymore until they figure out a better proof.
3. The Update: They added a list of new recipes and techniques discovered by other bakers since they published the first book.

In short: The authors are being very honest scientists. They fixed their math, admitted where they were wrong, and shared the latest news from the field, ensuring that anyone trying to build these mathematical universes has the most accurate instructions possible.

Technical Summary

This document is a Corrigendum and Addendum to the authors' previous paper, "Categoricity-like Properties in the First Order Realm" (referenced as [6]). Authored by Ali Enayat and Mateusz Łełyk, the note addresses critical errors in the original proofs, provides a corrected framework, presents a counterexample to a specific lemma, and surveys recent developments in the field of model-theoretic categoricity and set theory.

Below is a detailed technical summary organized by problem, methodology, key contributions, results, and significance.

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1. Problem Statement

The original paper [6] investigated "categoricity-like" properties (such as solidity and tightness) within first-order theories, specifically focusing on fragments of Zermelo-Fraenkel set theory (ZF). The current note addresses two primary issues:

1. Proof Defects in Theorem 39: The original proof of Theorem 39 (regarding the non-solidity and non-tightness of $ZF_{\Pi_n}$) relied on a lemma (Lemma 40 of [6]) that was insufficiently strong and contained errors regarding the quantifier complexity of Kripke-Platek (KP) set theory axioms.
2. Invalidity of Theorem 77: The original claim that the scheme template $\tau_{Repl+Tarski}$ is both g-solid and strongly internally categorical was based on Lemma 79, which the authors have now proven to be false.

2. Methodology and Technical Corrections

A. Correction to Theorem 39 (Section 2)

The authors reconstruct the proof of Theorem 39 by introducing a stronger set-theoretical analogue of a result from arithmetic model theory.

• The Core Issue: The original proof failed because it did not account for the correct quantifier complexity of KP axioms (specifically, the Collection scheme for $\Delta_0$ formulas requires $\Pi_3$ complexity, not $\Pi_2$) and lacked a robust characterization of definable submodels.
• New Tool: Theorem 3: The authors introduce Theorem 3, which analyzes the submodel $K_n(M)$ (elements definable by $\Sigma_n$ formulas) within a model $M \models ZF^- + V=L$.
  • Key Result (Theorem 3b): $K_n(K_n(M)) = K_n(M)$, meaning every element in the $\Sigma_n$-definable submodel is itself $\Sigma_n$-definable within that submodel.
  • Key Result (Theorem 3c): $K_n(M)$ satisfies $ZF_{\Pi_{n+1}}$ but fails the Collection scheme for $\Sigma_{n+1}$ formulas ($\neg Coll(\Sigma_{n+1})$).
• Bi-interpretability: The authors establish Theorem 2, proving that for $n \geq 3$, the standard model of arithmetic $\mathbb{N}$ is bi-interpretable with $K_n(M_0)$ (where $M_0$ is a specific non-standard model of $ZF + V=L + \neg Con_{ZF}$).
• Resolution: By leveraging the bi-interpretability between $\mathbb{N}$ and these set-theoretic models, the authors demonstrate that $ZF_{\Pi_n}$ is neither solid nor tight. The proof relies on the fact that $Coll(\Sigma_n)$ has a complexity of at most $\Pi_{n+3}$, allowing for the construction of models that distinguish between different levels of the hierarchy.

B. Counterexample to Theorem 77 (Section 3)

The authors withdraw the claim regarding the categoricity of $\tau_{Repl+Tarski}$ by disproving Claim 6 (formerly Lemma 79 of [6]).

• The False Claim: Claim 6 asserted that if $K$ is a submodel of $M$ (both models of ZF) and $K$ is "parametrically definable" in $M$ in a specific way, then $K$ must be isomorphic to an initial segment of $M$, isomorphic to $M$, or isomorphic to a rank-initial segment.
• The Counterexample Construction:
  1. Let $K$ be an $\omega$-standard model of ZFC containing a non-principal ultrafilter $U$ over $\mathcal{P}(\omega)$.
  2. Let $M$ be the internal ultrapower of $K$ modulo $U$.
  3. Properties:
     • $M$ is an elementary extension of $K$.
     • $K$ is a conservative elementary extension of $M$ (satisfying the hypothesis of Claim 6).
     • However, $K$ is $\omega$-standard, while $M$ is $\omega$-nonstandard (due to the ultrapower construction).
• Conclusion: Since $K$ and $M$ differ in their standardness of $\omega$, no isomorphism or rank-initial segment embedding exists between them as required by the three conditions of Claim 6. Thus, the lemma is false, and the proof of Theorem 77 collapses. The authors note that while the theorem might still be true, no alternative proof or counterexample to the theorem itself is currently available.

3. Key Contributions and Results

1. Refined Characterization of Definable Submodels: The paper establishes that for $n \geq 3$, the submodel of $\Sigma_n$-definable elements in a $V=L$ model behaves in a highly structured way, allowing for bi-interpretability with arithmetic. This bridges the gap between arithmetic model theory and set theory.
2. Correction of Axiomatic Complexity: The note clarifies that the axioms of KP (Kripke-Platek set theory) are $\Pi_3$-axiomatizable (specifically regarding the Collection scheme), correcting a previous $\Pi_2$ assumption.
3. Disproof of a Structural Lemma: The paper provides a concrete counterexample using internal ultrapowers to show that parametric definability does not force the rigid structural relationships (isomorphism or initial segment) previously hypothesized.
4. Survey of Recent Advances (Section 4): The authors summarize significant progress in the field since the publication of [6]:
   • Meadows & Chen [4]: Provided a simpler proof that $Z_2 + \Pi^1_\infty$-AC and $ZF^- + \forall x |x| \leq \aleph_0$ are not definitionally equivalent, avoiding the need for inaccessible cardinals.
   • Gruza, Kołodziejczyk, & Łełyk [8]: Constructed solid subtheories of Peano Arithmetic (PA) extending $I\Sigma_n$ where PA is not interpretable, answering Question B of [6] positively.
   • Gruza & Łełyk [9]: Formalized "categoricity up to an ordinal height" and showed that metatheoretic assumptions (like GB class theory) are crucial for establishing categoricity results.
   • Elliot Glazer [7]: Investigated the solidity of $ZCR$ (Zermelo set theory + Choice + Axiom of Ranks), leaving the solidity of $ZR$ as an open problem.

4. Significance

• Foundational Rigor: The paper is essential for researchers working on the model theory of set theory and the hierarchy of categoricity. It prevents the propagation of incorrect proofs regarding the structural rigidity of set-theoretic models.
• Clarification of "Solidity" and "Tightness": By correcting the proof of Theorem 39, the authors solidify the understanding that first-order fragments of ZF ($ZF_{\Pi_n}$) lack the strong categoricity properties (solidity/tightness) that might be expected, reinforcing the distinction between first-order and second-order set theories.
• Methodological Impact: The use of internal ultrapowers to generate counterexamples to structural lemmas provides a new tool for testing the limits of definability and interpretability in set theory.
• Future Directions: The addendum highlights that the field is moving toward a more nuanced understanding of how metatheoretic assumptions (e.g., the presence of class theories like GB) influence categoricity results, suggesting that "internal categoricity" is highly sensitive to the logical framework in which it is evaluated.

In summary, this note serves as a critical correction to the literature, replacing flawed arguments with rigorous proofs based on definable submodels and bi-interpretability, while simultaneously mapping the current landscape of research into categoricity-like properties in first-order logic.