Martin's maximum (MM) is a strong forcing axiom in set theory, formulated as a maximal strengthening of Martin's axiom (MA). It asserts that for every partially ordered set $P$ that preserves stationary subsets of $\omega_{1}$ (i.e., $P$ is stationary set preserving), and for every collection $\mathcal{D}$ of at most $\aleph_{1}$ dense subsets of $P$, there exists a filter $G\subseteq P$ meeting every dense set in $\mathcal{D}$. Formally,
$$ \text{If } P \text{ is stationary‑set‑preserving and } |\mathcal{D}|\le\aleph_{1}, \text{ then } \exists,G\subseteq P;(\forall D\in\mathcal{D})(G\cap D eq\varnothing). $$
The axiom is named after Donald A. Martin, who introduced Martin's axiom, and the term “maximum” reflects the fact that MM is, in a precise sense, the strongest possible forcing axiom that can hold for the class of stationary‑set‑preserving posets while still being consistent with ZFC (assuming suitable large‑cardinal hypotheses).
Historical context and consistency
- Origin. Martin’s maximum was first introduced by Foreman, Magidor, and Shelah in the early 1980s as a natural maximal extension of MA for the class of stationary‑set‑preserving forcings.
- Consistency strength. The consistency of MM relative to ZFC requires the existence of a supercompact cardinal. More precisely, if there is a supercompact cardinal, then there is a forcing extension in which MM holds. The converse is not known; however, MM implies the existence of many large‑cardinal‑like combinatorial properties at $\omega_{1}$ and $\omega_{2}$.
Major consequences
MM has a wide range of combinatorial and structural implications, several of which are unattainable from weaker axioms such as MA or the Proper Forcing Axiom (PFA). Notable consequences include:
| Area | Consequence under MM |
|---|---|
| Continuum size | The cardinality of the continuum is $\mathfrak{c}= \aleph_{2}$. |
| Stationary reflection | Every stationary subset of $\omega_{1}$ reflects to an uncountable regular cardinal below $\omega_{2}$. |
| Saturation of ideals | The non‑stationary ideal $\mathrm{NS}{\omega{1}}$ on $\omega_{1}$ is $\aleph_{2}$-saturated. |
| Clubs and guessing | Strong club‑guessing principles fail; in particular, there are no $\diamondsuit$-type sequences on $\omega_{1}$. |
| Structure of the reals | Many regularity properties (e.g., Lebesgue measurability, the Baire property) hold for all sets of reals definable from real and ordinal parameters. |
| Tree properties | The tree property holds at $\aleph_{2}$; i.e., every $\aleph_{2}$-tree has a cofinal branch. |
| Uniformization | Certain uniformization problems for projective sets are resolved positively. |
These outcomes make MM a central principle in the study of the interaction between forcing, large cardinals, and the combinatorics of small uncountable cardinals.
Variants and strengthenings
- MM$^+$ – An enhancement that additionally asserts the existence of filters meeting $\aleph_{2}$ many dense sets for a broader class of forcings (those preserving stationary subsets of $\omega_{2}$).
- MM$^{++}$ – Introduced by Foreman, Magidor, and Shelah, this version asserts the forcing axiom for all $\omega_{1}$-preserving posets with $\le \aleph_{1}$ many dense sets and simultaneously yields the saturation of the non‑stationary ideal on $\omega_{1}$ together with additional structural consequences.
- MM$^{+++}$ – A further strengthening studied in the context of $\Omega$-logic and categoricity results, whose consistency strength exceeds that of a supercompact cardinal and approaches huge cardinals.
Relationship to other forcing axioms
- Martin's axiom (MA). MA applies to c.c.c. (countable chain condition) posets and asserts the existence of filters meeting $\le\mathfrak{c}$ many dense sets. MM extends MA to a far larger class of posets (those preserving stationary subsets of $\omega_{1}$) while limiting the number of dense sets to $\aleph_{1}$.
- Proper Forcing Axiom (PFA). PFA is the forcing axiom for all proper posets. Since every stationary‑set‑preserving poset is proper, MM implies PFA, but MM is strictly stronger; many statements provable from MM are independent of PFA.
- Bounded Martin's maximum (BMM). A weaker, “bounded” version that restricts the axiom to statements of a certain syntactic form; BMM is equiconsistent with the existence of a Woodin cardinal.
Open problems and research directions
- Exact consistency strength. While a supercompact cardinal suffices for MM, it is unknown whether a weaker large‑cardinal assumption (e.g., a strongly compact cardinal) is also sufficient.
- Interaction with inner model theory. The compatibility of MM with canonical inner models containing large cardinals remains a subject of ongoing investigation.
- Determinacy connections. The extent to which MM influences projective determinacy and related regularity properties for sets of reals is not fully settled.
References (selected)
- Foreman, M., Magidor, M., & Shelah, S. (1988). Martin’s Maximum, saturated ideals, and nonregular ultrafilters. Annals of Mathematics, 127(1), 1–47.
- Jech, T. (2003). Set Theory (3rd ed.). Springer.
- Todorčević, S. (2007). Trees and linearly ordered sets. Handbook of Set-Theoretic Topology, 235–285.
- Cummings, J. (2003). Iterated forcing and elementary embeddings. Handbook of Set Theory, 765–834.
This entry presents a concise, factual overview of Martin’s maximum as it is understood in contemporary set‑theoretic literature.