In
descriptive set theory
In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to ot ...
, a branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Silver's dichotomy (also known as Silver's theorem)
[S. G. Simpson, "Subsystems of Z2 and Reverse Mathematics", p.442. Appearing in G. Takeuti, ''Proof Theory'' (1987), ISBN 0 444 87943 9.] is a statement about
equivalence relations
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relati ...
, named after
Jack Silver
Jack Howard Silver (23 April 1942 – 22 December 2016) was a set theorist and logician at the University of California, Berkeley.
Born in Montana, he earned his Ph.D. in Mathematics at Berkeley in 1966 under Robert Vaught before taking a posi ...
.
[L. Yanfang]
On Silver's Dichotomy
Ph.D thesis. Accessed 30 August 2022.[Sy D. Friedman]
Consistency of the Silver dichotomy in generalized Baire space
Fundamenta Mathematicae (2014). Accessed 30 August 2022.
Statement and history
A relation is said to be
coanalytic if its complement is an
analytic set. Silver's dichotomy is a statement about the equivalence classes of a coanalytic equivalence relation, stating any coanalytic equivalence relation either has
countably
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
many equivalence classes, or else there is a
perfect set
In general topology, a subset of a topological space is perfect if it is closed and has no isolated points. Equivalently: the set S is perfect if S=S', where S' denotes the set of all Limit point, limit points of S, also known as the derived se ...
of reals that are each incomparable to each other.
[A. Kechris]
New Directions in Descriptive Set Theory
(1999, p.165). Accessed 1 September 2022. In the latter case, there must be continuum many equivalence classes of the relation.
The first published proof of Silver's dichotomy was by Jack Silver, appearing in 1980 in order to answer a question posed by
Harvey Friedman
__NOTOC__
Harvey Friedman (born 23 September 1948)Handbook of Philosophical Logic, , p. 38 is an American mathematical logician at Ohio State University in Columbus, Ohio. He has worked on reverse mathematics, a project intended to derive the axi ...
.
[J. Silver]
Counting the number of equivalence classes of Borel and coanalytic equivalence relations
(Annals of Mathematical Logic, 1980, received 1977). Accessed 31 August 2022. One application of Silver's dichotomy appearing in recursive set theory is since equality restricted to a set
is coanalytic, there is no Borel equivalence relation
such that
, where
denotes
Borel-reducibility. Some later results motivated by Silver's dichotomy founded a new field known as invariant descriptive set theory, which studies definable equivalence relations. Silver's dichotomy also admits several weaker recursive versions, which have been compared in strength with subsystems of second-order arithmetic from
reverse mathematics
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in cont ...
,
while Silver's dichotomy itself is provably equivalent to
over
.
References
{{Reflist
Set theory