Weakly Mahlo Cardinal

• Weakly Mahlo cardinals are infinite cardinals whose inaccessible subsets are stationary, ensuring that every club set includes an inaccessible cardinal.
• They form the foundational level of the Mahlo hierarchy, with higher degrees reflecting the stationarity of lower Mahlo cardinals in a recursive structure.
• Forcing techniques, such as club-shooting, allow researchers to modify Mahlo properties by preserving lower degrees while eliminating higher ones.

A weakly Mahlo cardinal, also known as a 0-Mahlo cardinal, is an infinite cardinal $\kappa$ such that the set of inaccessible cardinals below $\kappa$ is stationary in $\kappa$. In formal terms, let $$I = \{\alpha<\kappa \mid \alpha \text{ is inaccessible}\}$$; $\kappa$ is weakly Mahlo if for every club $C\subseteq\kappa$, $C\cap I\neq\emptyset$. This property forms the entry point to the broader Mahlo hierarchy, which captures an infinite increasing sequence of combinatorial large cardinal notions. The distinction between Mahlo and inaccessible cardinals, and the mechanisms for manipulating Mahlo properties via forcing, have considerable significance in modern set-theoretic research (Carmody, 2015).

1. Definition and Characterization

A cardinal $\kappa$ is weakly Mahlo (0-Mahlo) if $$I=\{\alpha<\kappa \mid \alpha\text{ is inaccessible}\}$$ is stationary in $\kappa$. This means that every closed unbounded subset (club) of $\kappa$ contains an inaccessible cardinal below $\kappa$. Equivalently:

$$\forall C\subseteq\kappa \text{ club},\; C\cap I\neq\emptyset.$$

This definition captures the idea that $\kappa$ reflects the property of being inaccessible throughout its initial segments in a stationary way. The stationary character of $I$ in $\kappa$ creates a combinatorial robustness that distinguishes weakly Mahlo from mere inaccessibility.

2. The Classical Mahlo Hierarchy

The Mahlo hierarchy extends the notion of weakly Mahlo to higher "degrees" indexed by ordinals. For any ordinal $\alpha$, a cardinal $\kappa$ is said to be $\alpha$-Mahlo if:

1. $\kappa$ is Mahlo (i.e., 0-Mahlo).
2. For every $\beta<\alpha$, the set of $\beta$-Mahlo cardinals below $\kappa$ is stationary in $\kappa$.

Formally: $$M_0(\kappa) = \{\gamma<\kappa \mid \gamma\text{ is Mahlo}\}$$

$$M_{\alpha+1}(\kappa) = \{\gamma<\kappa \mid \gamma\text{ is }\alpha\text{-Mahlo}\}$$

$$\kappa \text{ is }\alpha\text{-Mahlo} \Longleftrightarrow \forall \beta<\alpha,\; M_\beta(\kappa)\text{ is stationary in }\kappa$$

The higher Mahlo degrees recursively generalize the reflection of stationarity properties, resulting in an infinite ascending scale of largeness notions.

3. Relationship to $\alpha$-Inaccessible Cardinals

An uncountable cardinal $\kappa$ is defined to be $\alpha$-inaccessible if:

• $\alpha=0$: $\kappa$ is inaccessible.
• $\alpha=1$: $\kappa$ is inaccessible and a limit of inaccessibles.
• In general, $\kappa$ is inaccessible and for every $\beta<\alpha$, the set of $\beta$-inaccessible cardinals below $\kappa$ is unbounded in $\kappa$.

A central result (Theorem 3.5 of Chapter 3 in (Carmody, 2015)) establishes that if $\kappa$ is Mahlo, then $\kappa$ is $t$-inaccessible for every meta-ordinal term $t$ whose parameters are below $\kappa$; in particular, any $\alpha$-Mahlo $\kappa$ is at least $\alpha$-inaccessible. This suggests Mahlo cardinals sit strictly above all $\alpha$-inaccessibles for $\alpha<\kappa$.

4. Forcing Constructions and Modification of Mahlo Degrees

Forcing can be used to manipulate the Mahlo properties of a cardinal. Given $\kappa$ is $\alpha$-Mahlo in the ground model $V$, it is possible to "softly kill" the $(\alpha+1)$-Mahlo property but preserve all lower Mahlo degrees by using a club-shooting forcing.

Let

$$A = \{\gamma<\kappa \mid \gamma\text{ is }\alpha\text{-Mahlo in }V\},\quad \bar A = \kappa\setminus A$$

Define the forcing $\mathbb{P}$ where conditions are closed, bounded subsets $c\subseteq\kappa$ with $c\cap A=\varnothing$, ordered by end-extension. Lemma 4.2 states that if $A$ does not reflect into any set of $\beta$-Mahlo cardinals for $\beta<\alpha$, this forcing preserves every stationary subset of those $$B=\{\gamma<\kappa \mid \gamma\text{ is }\beta\text{-Mahlo}\}$$. Theorem 4.3 asserts that in $V[G]$, after forcing, $\kappa$ is no longer $(\alpha+1)$-Mahlo but remains $\alpha$-Mahlo (Carmody, 2015).

A similar argument applies at the base level: if $\kappa$ is Mahlo, one can shoot a club through $\kappa\setminus I$, making $\kappa$ every degree of inaccessible (i.e., $t$-inaccessible for all $t<\kappa$) but no longer Mahlo.

5. Summary of Interrelationships

The key structural relationships among weakly Mahlo, Mahlo, $\alpha$-Mahlo, and $\alpha$-inaccessible cardinals can be summarized as:

• $0$-Mahlo = Mahlo = "every club meets some inaccessible".
• $\alpha$-Mahlo $\implies$ $\alpha$-inaccessible, and indeed $t$-inaccessible for every meta-ordinal $t<\kappa$.
• Forcing by shooting a club through the complement of the $\alpha$-Mahlo cardinals is $\kappa$-strategically closed in all the weaker Mahlo degrees, so it preserves the $\beta$-Mahlo property for all $\beta<\alpha$, but ensures $\kappa$ is not $(\alpha+1)$-Mahlo.

Cardinal Property	Definition (Stationarity)	Inaccessibility Relation
0-Mahlo (Mahlo)	Inaccessibles below $\kappa$ stationary	Trivially inaccessible
$\alpha$-Mahlo	$\beta$-Mahlo ($\beta<\alpha$) stationary	At least $\alpha$-inaccessible
$\alpha$-inaccessible	$\beta$-inacc. unbounded $\forall \beta<\alpha$	Weaker than $\alpha$-Mahlo

6. Significance in Large Cardinal Theory

Weakly Mahlo cardinals provide the foundational level of the Mahlo hierarchy and a precise reflection property for inaccessibles. The fine structure between Mahlo and inaccessible degrees, illuminated through forcing constructions, demonstrates the delicate gradation in large cardinal strength and the sophistication of set-theoretic hierarchy. Forcing techniques developed for Mahlo degrees have broader applications for modifying other large cardinal features, such as the Mitchell rank for measurables and supercompactness, positioning the theory of Mahloness as a central component in ongoing research on the manipulation and characterization of large cardinal axioms (Carmody, 2015).