Morass (set theory)

In axiomatic set theory, a mathematical discipline, a morass is an infinite combinatorial structure, used to create "large" structures from a "small" number of "small" approximations. They were invented by Ronald Jensen in his proof that cardinal transfer theorems hold under the axiom of constructibility.

Overview

Whilst it is possible to define so-called gap-n morasses for n > 1, they are so complex that focus is usually restricted to the gap-1 case, except for specific applications. The "gap" is essentially the cardinal difference between the size of the "small approximations" used and the size of the ultimate structure.

A (gap-1) morass on an uncountable regular cardinal κ consists of a tree of height κ + 1, with the top level having κ⁺-many nodes. The nodes are taken to be ordinals, and functions π between these ordinals are associated to the edges in the tree order. It is required that the ordinal structure of the top level nodes be "built up" as the direct limit of the ordinals in the branch to that node by the maps π, so the lower level nodes can be thought of as approximations to the (larger) top level node. A long list of further axioms is imposed to have this happen in a particularly "nice" way.^[1][2]

Variants and equivalents

Velleman^[2] and Shelah and Stanley^[3] independently developed forcing axioms equivalent to the existence of morasses, to facilitate their use by non-experts. Going further, Velleman^[4] showed that the existence of morasses is equivalent to simplified morasses, which are vastly simpler structures. However, the only known construction of a simplified morass in Gödel's constructible universe is by means of morasses, so the original notion retains interest.

Other variants on morasses, generally with added structure, have also appeared over the years. These include universal morasses,^[5] whereby every subset of κ is built up through the branches of the morass, and mangroves,^[6] which are morasses stratified into levels (mangals) at which every branch must have a node.

References

1. K. Devlin. Constructibility. Springer, Berlin, 1984.
2. Velleman, Daniel J. (1982). "Morasses, diamond, and forcing". Ann. Math. Logic 23: 199–281. doi:10.1016/0003-4843(82)90005-5. Zbl 0521.03034.
3. S. Shelah and L. Stanley. S-forcing, I: A "black box" theorem for morasses, with applications: Super-Souslin trees and generalizing Martin's axiom, Israel Journal of Mathematics, 43 (1982), pp 185–224.
4. Velleman, Dan (1984). "Simplified morasses". Journal of Symbolic Logic 49 (1): 257–271. doi:10.2307/2274108. Zbl 0575.03035.
5. K. Devlin. Aspects of Constructibility, Lecture Notes in Mathematics 354, Springer, Berlin, 1973.
6. Brooke-Taylor, A.; Friedman, S. (2009). "Large cardinals and gap-1 morasses". Annals of Pure and Applied Logic 159 (1-2): 71–99. arXiv:0801.1912v1. doi:10.1016/j.apal.2008.10.007. Zbl 1165.03033.