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 for his proof that cardinal transfer theorems hold under the axiom of constructibility. A far less complex but equivalent variant known as a simplified morass was introduced by Velleman, and the term morass is now often used to mean these simpler structures.
Overview
editWhilst 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 (also called a (κ,1)-morass) 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
editVelleman[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, mangroves,[6] which are morasses stratified into levels (mangals) at which every branch must have a node, and quagmires.[7]
Simplified morass
editVelleman [8] defined gap-1 simplified morasses which are much simpler than gap-1 morasses, and showed that the existence of gap-1 morasses is equivalent to the existence of gap-1 simplified morasses.
Roughly speaking: a (κ,1)-simplified morass M = < φ→, F⇒ > contains a sequence φ→ = < φβ : β ≤ κ > of ordinals such that φβ < κ for β < κ and φκ = κ+, and a double sequence F⇒ = < Fα,β : α < β ≤ κ > where Fα,β are collections of monotone mappings from φα to φβ for α < β ≤ κ with specific (easy but important) conditions.
Velleman's clear definition can be found in,[9] where he also constructed (ω0,1) simplified morasses in ZFC. In [10] he gave similar simple definitions for gap-2 simplified morasses, and in [11] he constructed (ω0,2) simplified morasses in ZFC.
Higher gap simplified morasses for any n ≥ 1 were defined by Morgan [12] and Szalkai.[13][14]
Roughly speaking: a (κ,n + 1)-simplified morass (of Szalkai) M = < M→, F⇒ > contains a sequence M→ = < Mβ : β ≤ κ > of (< κ,n)-simplified morass-like structures for β < κ and Mκ a (κ+,n) -simplified morass, and a double sequence F⇒ = < Fα,β : α < β ≤ κ > where Fα,β are collections of mappings from Mα to Mβ for α < β ≤ κ with specific conditions.
References
edit- ^ K. Devlin. Constructibility. Springer, Berlin, 1984.
- ^ a b 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.
- ^ Shelah, S.; Stanley, L. (1982). "S-forcing, I: A "black box" theorem for morasses, with applications: Super-Souslin trees and generalizing Martin's axiom". Israel Journal of Mathematics. 43: 185–224. doi:10.1007/BF02761942.
- ^ Velleman, Dan (1984). "Simplified morasses". Journal of Symbolic Logic. 49 (1): 257–271. doi:10.2307/2274108. Zbl 0575.03035.
- ^ K. Devlin. Aspects of Constructibility, Lecture Notes in Mathematics 354, Springer, Berlin, 1973.
- ^ Brooke-Taylor, A.; Friedman, S. (2009). "Large cardinals and gap-1 morasses". Annals of Pure and Applied Logic. 159 (1–2): 71–99. arXiv:0801.1912. doi:10.1016/j.apal.2008.10.007. Zbl 1165.03033.
- ^ Kanamori, Akihiro (1983). "Morasses in combinatorial set theory". In Mathias, A.R.D. (ed.). Surveys in set theory. London Mathematical Society Lecture Note Series. Vol. 87. Cambridge: Cambridge University Press. pp. 167–196. ISBN 0-521-27733-7. Zbl 0525.03036.
- ^ D. Velleman. Simplified Morasses, Journal of Symbolic Logic 49, No. 1 (1984), pp 257–271.
- ^ D. Velleman. Simplified Morasses, Journal of Symbolic Logic 49, No. 1 (1984), pp 257–271.
- ^ D. Velleman. Simplified Gap-2 Morasses, Annals of Pure and Applied Logic 34, (1987), pp 171–208.
- ^ D. Velleman. Gap-2 Morasses of Height ω0, Journal of Symbolic Logic 52, (1987), pp 928–938.
- ^ Ch. Morgan. The Equivalence of Morasses and Simplified Morasses in the Finite Gap Case, PhD.Thesis, Merton College, UK, 1989.
- ^ I. Szalkai. Higher Gap Simplified Morasses and Combinatorial Applications, PhD-Thesis (in Hungarian), ELTE, Budapest, 1991. English abstract: http://math.uni-pannon.hu/~szalkai/Szalkai-1991d-MorassAbst-.pdf
- ^ I. Szalkai. An Inductive Definition of Higher Gap Simplified Morasses, Publicationes Mathematicae Debrecen 58 (2001), pp 605–634. http://math.uni-pannon.hu/~szalkai/Szalkai-2001a-IndMorass.pdf