In
mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
, indiscernibles are objects that cannot be distinguished by any
property
Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of the property, an owner of property may have the right to consume, alter, share, r ...
or
relation defined by a
formula. Usually only
first-order formulas are considered.
Examples
If ''a'', ''b'', and ''c'' are
distinct and is a
set of indiscernibles, then, for example, for each binary formula
, we must have
:
Historically, the
identity of indiscernibles
The identity of indiscernibles is an ontological principle that states that there cannot be separate objects or entities that have all their properties in common. That is, entities ''x'' and ''y'' are identical if every predicate possessed by ''x'' ...
was one of the
laws of thought
The laws of thought are fundamental axiomatic rules upon which rational discourse itself is often considered to be based. The formulation and clarification of such rules have a long tradition in the history of philosophy and logic. Generally th ...
of
Gottfried Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...
.
Generalizations
In some contexts one considers the more general notion of order-indiscernibles, and the term sequence of indiscernibles often refers implicitly to this weaker notion. In our example of binary formulas, to say that the triple (''a'', ''b'', ''c'') of distinct elements is a sequence of indiscernibles implies
:
Applications
Order-indiscernibles feature prominently in the theory of
Ramsey cardinal In mathematics, a Ramsey cardinal is a certain kind of large cardinal number introduced by and named after Frank P. Ramsey, whose theorem establishes that ω enjoys a certain property that Ramsey cardinals generalize to the uncountable case.
L ...
s,
Erdős cardinal In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by .
The Erdős cardinal is defined to be the least cardinal such that for every function there is a set of order type tha ...
s, and
zero sharp
In the mathematical discipline of set theory, 0# (zero sharp, also 0#) is the set of true formulae about indiscernibles and order-indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the integers (using Gödel ...
.
See also
*
Identity of indiscernibles
The identity of indiscernibles is an ontological principle that states that there cannot be separate objects or entities that have all their properties in common. That is, entities ''x'' and ''y'' are identical if every predicate possessed by ''x'' ...
*
Rough set
References
* {{cite book , last1=Jech , first1=Thomas , author1-link=Thomas Jech , title=Set Theory , edition=Third Millennium , publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 ...
, location=Berlin, New York , series=Springer Monographs in Mathematics , isbn=978-3-540-44085-7 , year=2003 , zbl=1007.03002
Model theory