Set theory MOC

Set

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

A set is a Collection of different things, called elements or members with Propositional equality. set In a material conception², two sets are said to be the same iff they have the same members, i.e.

(∀𝔐⁡𝐴,𝔐⁡𝐵)[𝐴=𝐵⟺(∀𝑥)[𝑥∈𝐴⟺𝑥∈𝐵]]

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

• 𝐴 ={𝑎1,𝑎2,…,𝑎𝑛} is the finite set with members 𝑎1,𝑎2,…,𝑎𝑛
• 𝐴 ={𝑥 :𝑃(𝑥)} is the set of all 𝑥 satisfying predicate 𝑃, i.e. 𝑥 ∈𝐴 ⟺ 𝑃(𝑥)
• 𝐴 ∪𝐵 ={𝑥 :𝑥 ∈𝐴 ∨𝑥 ∈𝐵} is the union of 𝐴 and 𝐵
• 𝐴 ∩𝐵 ={𝑥 :𝑥 ∈𝐴 ∧𝑥 ∈𝐵} is the intersection of 𝐴 and 𝐵
• 𝐴 ∖𝐵 ={𝑥 :𝑥 ∈𝐴 ∧𝑥 ∉𝐵} is the set difference of 𝐵 from 𝐴

Foundation-agnostic usage

• The axiomatic set theories each give a notion of set.
  • In TG a U-small set is an element of U.
• 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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tidy | en | SemBr

Footnotes

1. 1895. Beiträge zur Begründung der transfiniten Mengenlehre. “By a set we understand any amalgamation 𝑀 of definite, well distinguished objects 𝑚 of our conception or our thought (which are called the elements of 𝑀) to a [single] whole.” ↩

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