The following definition is a generalization of the concept of minimal elements in a set and of a well-order.

Definition: Well-founded Relation

Let $V$ be a set, and let $R\subseteq V\times V$ be a binary relation. The relation $R$ is called well-founded if every non-empty subset $S\subseteq V$ contains a minimal element $m$ with respect to $R$. In other words, no element $x\in S$ is in the left-sided relation $xRm.$ Formally:

$$\forall S\subseteq V\; \exists x\in S \Rightarrow \exists m\in S \quad \forall x\in S \quad (x,m)\not\in R.$$

Examples: 1

Definitions: 1
Examples: 2 3 4
Explanations: 5
Motivations: 6 7
Proofs: 8 9 10
Propositions: 11
Theorems: 12

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  1. Hoffmann, D.: "Forcing, Eine Einführung in die Mathematik der Unabhängigkeitsbeweise", Hoffmann, D., 2018