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.”^[1895. Beiträge zur Begründung der transfiniten Mengenlehre. “By a set we understand any amalgamation of definite, well distinguished objects of our conception or our thought (which are called the elements of ) to a [single] whole.”]

A set is a Collection of different things, called elements or members with Propositional equality. set In a material conception¹, 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 materical 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 TG 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) MLTT, a set should be taken to be an h-set.

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Footnotes

1. Such a statement becomes vacuous in a structural theory like ETCS. ↩