mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...

, 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, re ...

or relation defined by a

formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...

. 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

\beta

, we must have :

\lor

\lnot \beta (a, b) \land \lnot \beta (b, a) \land \lnot \beta(a, c) \land \lnot \beta (c, a) \land \lnot \beta (b, c) \land \lnot \beta (c, b) \,.

Historically, the identity of indiscernibles was one of the laws of thought of

Gottfried Leibniz Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Isaac Newton, Sir Isaac Newton, with the creation of calculus in ad ...

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 :

(\varphi (a, b) \land \varphi (a, c) \land \varphi (b, c) \lor \lnot \varphi (a, b) \land \lnot \varphi (a, c) \land \lnot \varphi (b, c))

and :

(\varphi (b, a) \land \varphi (c, a) \land \varphi (c, b) \lor \lnot \varphi (b, a) \land \lnot \varphi (c, a) \land \lnot \varphi (c, b)) \,.

More generally, for a structure

\mathfrak A

with domain

A

and a linear ordering

<

, a set

I\subseteq A

is said to be a set of

<

-indiscernibles for

\mathfrak A

if for any finite subsets

\\subseteq I

and

\\subseteq I

with

i_0<\ldots and j_0<\ldots and any first-order formula \phi of the language of \mathfrak A with n free variables, \mathfrak A\vDash\phi(i_0,\ldots,i_n) \iff \mathfrak A\vDash\phi(j_0,\ldots,j_n) . J. Baumgartner, F. Galvin,

Generalized Erdős cardinals and 0^#
. Annals of Mathematical Logic vol. 15, iss. 3 (1978).^{p. 2}

Applications

Order-indiscernibles feature prominently in the theory of Ramsey cardinals, Erdős cardinals, and zero sharp.

References

Citations

{{reflist Model theory

Examples

Generalizations

Applications

See also

References

Citations