Chang's conjecture
Encyclopedia
In model theory
, a branch of mathematical logic
, Chang's conjecture, attributed to Chen Chung Chang
by , states that every model of type (ω2,ω1) for a countable language has an elementary submodel of type (ω1, ω). A model is of type (α,β) if it is of cardinality α and a unary relation is represented by a subset of cardinality β. The usual notation is
.
The axiom of constructibility
implies that Chang's conjecture fails. Silver
proved the consistency of Chang's conjecture from the consistency of an ω1-Erdős cardinal
. Hans-Dieter Donder showed the reverse implication: if CC holds, then ω2 is ω1-Erdős in K
.
More generally, Chang's conjecture for two pairs (α,β), (γ,δ) of cardinals is the claim
that every model of type (α,β) for a countable language has an elementary submodel of type (γ,δ).
The consistency of
was shown by Laver
from the consistency of a huge cardinal.
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
, a branch of mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, Chang's conjecture, attributed to Chen Chung Chang
Chen Chung Chang
Chen Chung Chang is a mathematician who works in model theory. He obtained his PhD from Berkeley in 1955 on "Cardinal and Ordinal Factorization of Relation Types" under Alfred Tarski. He wrote the standard text on model theory. Chang's conjecture is named after him...
by , states that every model of type (ω2,ω1) for a countable language has an elementary submodel of type (ω1, ω). A model is of type (α,β) if it is of cardinality α and a unary relation is represented by a subset of cardinality β. The usual notation is
.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...
implies that Chang's conjecture fails. Silver
Jack Silver
Jack Howard Silver is a set theorist and logician at the University of California, Berkeley. He has made several deep contributions to set theory...
proved the consistency of Chang's conjecture from the consistency of an ω1-Erdős cardinal
Erdos cardinal
In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by .The Erdős cardinal κ is defined to be the least cardinal such that for every function...
. Hans-Dieter Donder showed the reverse implication: if CC holds, then ω2 is ω1-Erdős in K
Core model
In set theory, the core model is a definable inner model of the universe of all sets. Even though set theorists refer to "the core model", it is not a uniquely identified mathematical object. Rather, it is a class of inner models that under the right set theoretic assumptions have very special...
.
More generally, Chang's conjecture for two pairs (α,β), (γ,δ) of cardinals is the claim
that every model of type (α,β) for a countable language has an elementary submodel of type (γ,δ).
The consistency of
was shown by LaverRichard Laver
Richard Laver is an American mathematician, working in set theory. He is a professor emeritus at the Department of Mathematics of the University of Colorado at Boulder.-His main results:Among Laver's notable achievements some are the following....
from the consistency of a huge cardinal.