© 1996, Topology Atlas

One common way to define an infinite structure is to construct it as a union of an increasing chain of substructures. For example, a structure of size ALEPH_1 can be constructed as a union of an increasing chain of countable structures. But the union of such a chain cannot have size ALEPH_2 or larger. A structure of size ALEPH_2 can be constructed as a union of countable substructures, but these substructures cannot form an increasing chain. One way to carry out such a construction is to use forcing, with the forcing conditions being countable substructures of the final structure. Another is to use a morass.

Morasses were invented by Ronald Jensen, who showed that their existence follows from the Axiom of Constructibility, V = L (see [1]). Morasses, as Jensen defined them, were closely tied to the structure of L. Later I defined simplified morasses, whose existence is equivalent to the existence of Jensen's morasses (see [7]). The definition of simplified morasses is based directly on the kinds of constructions for which morasses are usually used, and is not closely tied to the structure of L. A simplified morass of height omega_1, for example, can be thought of as a set of instructions for how to piece together countable pieces to make a structure of size ALEPH_2. More generally, a simplified morass of height kappa, for any regular infinite cardinal kappa, is used to construct a structure of size kappa^+ from pieces of size less than kappa. Although the smallest possible value for kappa in Jensen's morasses was omega_1, the definition of simplified morass makes sense in the case kappa = omega, and the existence of simplified morasses of height omega is provable in ZFC (see [4]).

Often if a structure of size kappa^+ can be constructed by forcing with conditions of size less than kappa, then the structure can also be constructed with a morass of height kappa. In fact, the existence of morasses is equivalent to a Martin's Axiom-type forcing axiom guaranteeing the existence of sufficiently generic sets for certain forcing notions. Sometimes if a construction cannot be carried out with a morass alone, additional structure can be added to the morass to make the construction possible. For example, there is a version of diamond for morasses. For an example of a topological application of this version of diamond, see [5]. There is also a function that can be defined easily from any simplified morass that is closely related to Todorcevic's function rho (see [2]).

The morasses discussed above are called gap-1 morasses, because there is a gap of one cardinal between the sizes of the pieces used in a construction and the size of the final structure constructed. Higher gap morasses and simplified morasses have also been defined, although so far they have found few applications. For more on higher gap morasses see [3] and [6].

[1] Devlin, Keith, Constructibility, Springer-Verlag, 1984

[2] Morgan, Charles, Morasses, Square, and Forcing Axioms, preprint.

[3] Morgan, Charles, The Equivalence of Morasses and Simplified Morasses in the Finite Gap Cases, Ph.D. dissertation.

[4] Velleman, Daniel, omega-Morasses, and a Weak Form of Martin's Axiom Provable in ZFC, Trans. Amer. Math. Soc. 285 (1984), 617-627.

[5] Velleman, Daniel, On a Topological Construction of Juhasz and Shelah, J. Symbolic Logic 57 (1992), 166-171.

[6] Velleman, Daniel, Simplified Gap-2 Morasses, Ann. Pure Appl. Logic 34 (1987), 171-208.

[7] Velleman, Daniel, Simplified Morasses, J. Symbolic Logic 49 (1984), 257- 271.