Morass (set theory)
Encyclopedia
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
.
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.
Shelah
and Stanley independently developed forcing axioms equivalent to the existence of morasses, to facilitate their use by non-experts. Going further, Velleman 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, whereby every subset of κ is built up through the branches of the morass, and mangroves, which are morasses stratified into levels (mangals) at which every branch must have a node.
Ronald Jensen
Ronald Björn Jensen is an American mathematician active in Europe, primarily known for his work in mathematical logic and set theory.-Career:...
in his proof that cardinal transfer theorems hold under the axiom of constructibility
Axiom of constructibility
The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. The axiom is usually written as V = L, where V and L denote the von Neumann universe and the constructible universe, respectively.- Implications :The axiom of...
.
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
Regular cardinal
In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. So, crudely speaking, a regular cardinal is one which cannot be broken into a smaller collection of smaller parts....
κ consists of a tree
Tree (set theory)
In set theory, a tree is a partially ordered set In set theory, a tree is a partially ordered set (poset) In set theory, a tree is a partially ordered set (poset) (T, In set theory, a tree is a partially ordered set (poset) (T, ...
of height κ + 1, with the top level having κ+-many nodes. The nodes are taken to be ordinals
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
, 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.
Variants and equivalents
Velleman andShelah
Saharon Shelah
Saharon Shelah is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey.-Biography:...
and Stanley independently developed forcing axioms equivalent to the existence of morasses, to facilitate their use by non-experts. Going further, Velleman 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
Kurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
constructible universe
Constructible universe
In mathematics, the constructible universe , denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis"...
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, whereby every subset of κ is built up through the branches of the morass, and mangroves, which are morasses stratified into levels (mangals) at which every branch must have a node.