In
mathematical logic, indiscernibles are objects that cannot be distinguished by any
property or
relation defined by a
formula. Usually only
first-order
In mathematics and other formal sciences, first-order or first order most often means either:
* "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of hig ...
formulas are considered.
Examples
If ''a'', ''b'', and ''c'' are
distinct and is a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of indiscernibles, then, for example, for each binary formula
, we must have
:
Historically, the
identity of indiscernibles 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 they ...
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 mat ...
.
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.
Le ...
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 th ...
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
*
Rough set
References
* {{cite book , last1=Jech , first1=Thomas , author1-link=Thomas Jech , title=Set Theory , edition=Third Millennium , publisher=
Springer-Verlag , location=Berlin, New York , series=Springer Monographs in Mathematics , isbn=978-3-540-44085-7 , year=2003 , zbl=1007.03002
Model theory