A Note on a Theorem of Apter

The paper establishes that the consistency of $\mathrm{ZF} + \mathrm{AD}_{\mathbb{R}}$ with $\Theta$ being measurable implies the consistency of $\mathrm{ZF}$ where $\Theta$ is simultaneously the least strongly regular and least measurable cardinal, while all uncountable cardinals below it have countable cofinality.

The Gist

Here is an explanation of the paper "A Note on a Theorem of Apter," translated into everyday language with creative analogies.

The Big Picture: Building a New Universe of Math

Imagine the universe of mathematics as a giant, infinite city called V. In this city, there are different types of "buildings" called cardinals (which represent sizes of infinity).

For a long time, mathematicians have been trying to understand the rules of this city. One of the most famous rules is the Axiom of Choice (AC). Think of AC as a "Universal Manager" who can instantly pick one item from any pile of socks, no matter how messy the pile is. This manager makes the city very orderly, but it also creates some weird, non-constructive patterns.

In recent decades, mathematicians have been exploring what happens if we fire the Universal Manager and replace them with a different rule called the Axiom of Determinacy (AD). Under AD, the city is built on the idea of perfect games. If you play a game of infinite length with another person, someone must have a winning strategy. This creates a very different, more "fair" city, but it breaks many of the old rules about how big things can be.

The Problem: Finding the "Smallest Giant"

In this new city (without the Universal Manager), mathematicians are looking for a specific type of building called a Measurable Cardinal.

• What is it? Think of a Measurable Cardinal as a "Super-Strong" building. It's so massive and stable that it can hold a special kind of "measure" (like a perfect scale) that can weigh any collection of smaller buildings without breaking.
• The Goal: Mathematicians want to find the smallest possible Super-Strong building in this city.

In the old city (with AC), the smallest Super-Strong building is usually huge. But in the new city (AD), we know that even small buildings (like the first infinite size, $\omega_1$) can be Super-Strong. However, there's a catch: we want this building to be Strongly Regular.

• Strongly Regular: Imagine a building that is so solid that no matter how many smaller pieces you try to stack on top of it, they never reach the top. It's "inaccessible" from below.

The big question the authors are asking is: Can we build a universe where the very first Super-Strong building is also the very first building that is Strongly Regular?

The Previous Attempts

Before this paper, other mathematicians (like Gitik, Hayut, and Karagila) showed that you could build such a universe, but it required a "magic wand" of immense power (a "Woodin limit of Woodin cardinals"). It was like saying, "To build this small house, you need a nuclear power plant."

Another mathematician, Apter, showed a way to do it using a specific tool called Prikry Forcing.

• The Tool (Prikry Forcing): Imagine you have a giant, solid tower (a Measurable Cardinal). You want to make it "wobbly" at the bottom (so it's no longer regular) without destroying its strength. Prikry Forcing is like a machine that drills a specific pattern of holes into the tower, making it wobble, while keeping the top intact.
• Apter's Trick: He used this machine to turn every Super-Strong building below a certain limit into a "wobbly" one, leaving only the very top one standing as both Super-Strong and Solid.

What This Paper Does

The authors of this paper (Rahman, Otto, and Sebastiano) took Apter's machine and refined it.

1. The Setup: They started with a universe that already has a "Super-Strong" limit called $\Theta$ (Theta). They assumed a specific condition called $\Theta_{meas}$, which basically means "There is a perfect scale for $\Theta$ that works even for infinite games."
2. The Construction: They used a "product" of Apter's machines. Imagine they had a row of towers (all the Super-Strong buildings below $\Theta$). They ran the "wobbly machine" on every single one of them at the same time.
3. The Result:
   • All the buildings below $\Theta$ became "wobbly" (they lost their regularity and their Super-Strong status).
   • The top building, $\Theta$, remained untouched.
   • Because everything below it is now wobbly, $\Theta$ becomes the smallest building that is both Super-Strong (Measurable) and Solid (Strongly Regular).

Why This Matters

The authors proved that you don't need the "nuclear power plant" (the massive Woodin cardinals) to build this specific city. You only need the "perfect scale" condition ($\Theta_{meas}$), which is a much weaker requirement.

The Analogy:
Imagine you want to build a fortress that is the only unbreakable wall in a kingdom.

• Old Way: You needed a dragon to burn down all other walls so yours was the only one left. (Requires huge power).
• New Way (This Paper): You just need a special paint that makes all other walls crumble into dust, leaving yours standing. It achieves the same result but uses much less energy.

The Conclusion

The paper confirms that it is consistent (logically possible) to have a mathematical universe where:

1. The Axiom of Choice is false.
2. The "Super-Strong" and "Solid" properties are combined in the very first possible cardinal.
3. This doesn't require the strongest possible assumptions in mathematics; it can be done with a "lighter" set of rules.

The Open Questions (The "What's Next?")

The authors end by asking a few fun questions for the future:

• Can we make the "Super-Strong" building have a specific shape (like a specific type of wobble)?
• How "tall" can the hierarchy of these buildings get?
• Can we make the first "Solid" building be the 100th "Regular" building instead of the first?

In short, they successfully built a more efficient version of a mathematical universe, proving that you don't need the heaviest machinery to create a world where the smallest giant is also the strongest.

Technical Summary

Here is a detailed technical summary of the paper "A Note on a Theorem of Apter" by Rahman Mohammadpour, Otto Rajala, and Sebastiano Thei.

1. Problem Statement and Motivation

The paper addresses the problem of identifying the smallest possible measurable cardinal in the absence of the Axiom of Choice (AC).

• Context: Under ZFC, measurable cardinals are large cardinals (inaccessible, etc.). Under the Axiom of Determinacy (AD), the landscape changes drastically. For instance, under AD, $\omega_1$ can be measurable.
• Previous Work:
  • Jech (1960s): Showed it is consistent for $\omega_1$ to be measurable.
  • Steel: Under AD + $V=L(\mathbb{R})$, all regular cardinals below $\Theta$ are measurable.
  • Apter (2000s): Under AD + $V=L(\mathbb{R})$, constructed a model where the least measurable cardinal is the least regular limit cardinal. He used Prikry forcing to singularize measurable cardinals below $\Theta$ while preserving the measurability of others.
  • Gitik, Hayut, and Karagila: Showed that if $\kappa$ is a $\kappa^{++}$-supercompact, there is a model where $\kappa$ is the least strongly inaccessible and least measurable.
• The Gap: The authors aim to reduce the large cardinal strength required for the Gitik-Hayut-Karagila result. Specifically, they seek to construct a model where the least measurable cardinal coincides with the least "strongly regular" cardinal (a specific notion of inaccessibility) using weaker assumptions than a Woodin limit of Woodin cardinals.

2. Definitions and Notation

• $\Theta$: The least cardinal onto which the real numbers $\mathbb{R}$ cannot be mapped.
• Strongly Regular Cardinal: A cardinal $\kappa$ is strongly regular if it is uncountable and for every $\alpha < \kappa$ and every function $f: \mathcal{P}(\alpha) \to \kappa$, the range of $f$ is bounded in $\kappa$. (Note: This differs slightly from the standard definition of inaccessibility used by Gitik et al., though they are equivalent under AC).
• $\Theta_{\text{meas}}$: The conjunction of the Axiom of Determinacy for Reals (ADR) and the existence of an $\mathbb{R}$-complete measure on $\Theta$.
  • Note: A measure $\mu$ is $\mathbb{R}$-complete if the intersection of any sequence of sets in $\mu$ indexed by reals is also in $\mu$. This implies $\Theta$-completeness.
• ADR: Asserts that any $\omega$-length alternating two-player game with perfect information where players play reals is determined.

3. Methodology

The authors employ a symmetric extension technique combined with Prikry forcing, closely following the analysis of Apter [1].

A. The Forcing Setup

1. Assumption: Start with a model $V$ satisfying $\text{ZF} + \Theta_{\text{meas}}$.
2. Measurable Cardinals: Let $M = \{ \kappa < \Theta \mid \kappa \text{ is measurable} \}$. Under ADR, every regular cardinal below $\Theta$ is measurable.
3. Prikry Forcing: For each $\kappa \in M$, let $\mu_\kappa$ be the $\prec_\kappa$-least measure on $\kappa$ (well-ordering exists under AD). Define $\mathbb{P}_\kappa$ as the Prikry forcing associated with $\mu_\kappa$.
4. Product Forcing: Define $\mathbb{P}$ as the finite support product of $\{\mathbb{P}_\kappa \mid \kappa \in M\}$.
5. Symmetric Extension: Instead of taking the full generic extension $V[G]$, the authors construct a symmetric submodel $N$.
   • The model $N$ is defined inductively using a sublanguage of the forcing language that includes names for the generic filters $G_\kappa$ for $\kappa \in M$.
   • $N$ is constructed to preserve the structure of the ground model regarding cardinals while manipulating cofinalities.

B. Key Technical Lemmas

• Cardinal Preservation: The authors prove that $N$ preserves all cardinals of $V$. This relies on the fact that finite sub-products of Prikry forcing preserve cardinals (Apter [1]) and the symmetric nature of the extension.
• Cofinality Change: For every $\kappa \in M$, the forcing $\mathbb{P}_\kappa$ changes the cofinality of $\kappa$ to $\omega$. In the symmetric model $N$, all cardinals $\kappa < \Theta$ that were measurable in $V$ become singular with cofinality $\omega$.
• Lifting Measures: A crucial step is showing that the $\mathbb{R}$-complete measure on $\Theta$ in the ground model $V$ can be "lifted" to the symmetric model $N$.

4. Key Results and Proofs

Theorem 1.1 (Main Theorem)

Assuming $\text{ZF} + \Theta_{\text{meas}}$, there exists an extension $N$ satisfying $\text{ZF}$ such that:

1. $\Theta$ is a strongly regular cardinal.
2. $\Theta$ is the least uncountable regular cardinal.
3. $\Theta$ is the least measurable cardinal.
4. All uncountable cardinals below $\Theta$ have countable cofinality.

Proof Sketch of Main Properties in $N$:

• No Measurables Below $\Theta$:
  • In $V$, every regular $\kappa < \Theta$ is measurable.
  • The forcing $\mathbb{P}$ singularizes every $\kappa \in M$ (making $\text{cof}(\kappa) = \omega$).
  • Since measurable cardinals must be regular, no $\kappa < \Theta$ remains measurable in $N$.
• $\Theta$ is Strongly Regular:
  • The authors show that any function $f: \mathcal{P}(\alpha) \to \Theta$ in $N$ (for $\alpha < \Theta$) is bounded.
  • This is proven by coding the function into a ground model function $g: \mathcal{P}(\xi) \to \Theta$.
  • If $f$ were unbounded, $g$ would be unbounded. However, under AD, there is a surjection from $\mathbb{R}$ to $\mathcal{P}(\xi)$. An unbounded map from $\mathbb{R}$ to $\Theta$ would contradict the assumption that $\Theta$ carries an $\mathbb{R}$-complete measure (which implies $\Theta$ is not the image of a bounded map from $\mathbb{R}$ in a way that violates the measure's properties).
• $\Theta$ is Measurable:
  • The $\mathbb{R}$-complete measure $\mu$ on $\Theta$ in $V$ is lifted to $\mu^*$ in $N$.
  • The authors verify that $\mu^*$ is a normal, $\Theta$-complete ultrafilter in $N$.
  • The proof involves showing that for any set $x \subseteq \Theta$ in $N$, $x$ belongs to a finite sub-extension $V[G_c]$, where the measure can be lifted step-by-step using Lemma 2.4 (Apter).

5. Significance and Contributions

• Reduction of Consistency Strength: The paper demonstrates that the configuration where the least measurable cardinal is also the least strongly regular cardinal can be achieved from the assumption $\text{ZF} + \Theta_{\text{meas}}$.
  • This assumption is known to be consistent (Rachid and Sargsyan) and is weaker than the existence of a Woodin limit of Woodin cardinals, which was the strength required for the similar result by Gitik, Hayut, and Karagila.
• Refinement of Inaccessibility: The authors introduce and utilize the notion of "strongly regular" cardinals, distinguishing it from other definitions of inaccessibility in the context of ZF without AC, while showing they coincide in the specific model constructed.
• Methodological Advancement: The paper successfully adapts Apter's Prikry forcing machinery to a symmetric extension framework to simultaneously singularize all cardinals below $\Theta$ while preserving the measurability of $\Theta$ itself.

6. Open Questions

The paper concludes with several open questions regarding the consistency strength and structural properties of such models:

1. Exact Consistency Strength: What is the exact consistency strength of "$\Theta$ is the least inaccessible and the least measurable"? (The authors conjecture it is weaker than a Woodin limit of Woodin cardinals).
2. Measure Concentration: Can the least measurable cardinal carry a normal measure concentrating on points of cofinality $\omega_1$ or on regular cardinals?
3. Order of Measurability ($o(\Theta)$): How large can the Mitchell order $o(\Theta)$ be in such a model?
4. Position in the Hierarchy: Is it consistent that $\Theta$ is the least measurable and strongly regular, but the $\xi$-th uncountable regular cardinal for some $\xi < \Theta$?

Conclusion

This paper provides a significant step forward in understanding the structure of large cardinals under the Axiom of Determinacy. By lowering the consistency strength required to make the least measurable cardinal the least strongly regular cardinal, the authors bridge the gap between the results of Apter and Gitik et al., suggesting that the "gap" between measurability and inaccessibility in ZF is smaller than previously thought, provided one works within the framework of $\Theta_{\text{meas}}$.