Definition:Supremum of Set/Real Numbers/Propositional Function

Definition

Let $\family {a_j}_{j \mathop \in I}$ be a family of elements of the real numbers $\R$ indexed by $I$.

Let $\map R j$ be a propositional function of $j \in I$.

Then we can define the supremum of $\family {a_j}_{j \mathop \in I}$ as:

$\ds \sup_{\map R j} a_j := \text { the supremum of all $a_j$ such that $\map R j$ holds}$

If more than one propositional function is written under the supremum sign, they must all hold.

Let the fiber of truth of $\map R j$ be finite.

Then the supremum of $\family {a_j}_{j \mathop \in I}$ can be expressed as:

$\ds \max_{\map R j} a_j = \text { the maxmum of all $a_j$ such that $\map R j$ holds}$

and can be referred to as the maximum of $\family {a_j}_{j \mathop \in I}$.

$\ds \sup _{\map \Phi j} a_j$

where $\map \Phi j$ is a propositional function of $j$.

Suppose that there are no values of $j$ for which $\map \Phi j$ is true.

Then $\ds \sup_{\map \Phi j} a_j$ is defined as being $-\infty$.

This supremum is called a vacuous supremum.

This is because:

$\forall a \in \R: \sup \set {a, -\infty} = a$

Hence for all $j$ for which $\map \Phi j$ is false, the supremum is unaffected.

In this context $-\infty$ is considered as minus infinity, the hypothetical quantity that has the property:

$\forall n \in \Z: -\infty < n$