Set theory MOC

Set

“Unter einer Menge verstehen wir jede Zusammenfassung von bestimmten wohlunterscheidbaren Objecten unserer Anschauung oder unseres Denkens (welche die Elemente von genannt werden) zu einem Ganze.”¹

A set is a Collection of different things, called elements or members, with the property that these elements may be compared by Propositional equality. #m/def/set In a material conception², two sets are said to be the same iff they have the same members, i.e.

which is the Axiom of Extensionality. See axiomatic set theory for different axiomatic treatments of the set.

Further terms

• It follows from extensionality that the Empty set is unique.
• Subset

Forming sets

In a material conception

• is the finite set with members
• is the set of all satisfying predicate , i.e.
• is the union of and
• is the intersection of and
• is the set difference of from

Foundation-agnostic usage

• The axiomatic set theories each give a notion of set.
  • In a -small set is an element of .
• In a Type theory with some notion of Equality, a set should be taken to be a type with Propositional equality.
  • In particular, in (extensions of) ITT, a set should be taken to be an h-set.

#state/tidy | #lang/en | #SemBr