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As I understand the set of extended real numbers ($\bar{\mathbb{R}}_+\left[0, \infty \right]$) has the property that any subset of it has a unique least upper bound.

In the following $[n]$ with $n \in \mathbb{N}$ denotes $\{ k \in \mathbb{N} \mid k < n \}$

By a collection I mean a set $X$ together with a function $f : X \to \bar{\mathbb{R}}_+$

I define a subset $S_f \subset \bar{\mathbb{R}}_+$ as follows: $x \in S_f$ iff there exists $n \in \mathbb{N}$ and an injective function $m : [n] \to X$, such that $x = \sum_{i \in [n]} f(m(i))$

Then the least upper bound of $S_f$ can be considered the sum of the collection $(X, f)$

My question is: would this argument run into size issues, if the set $X$ is sufficiently large?

Edit: By sufficiently large I mean: what happens if X is, for example, an inaccessible cardinal?

Necrosovereign
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  • I don't understand what you are asking. Maybe it's me, but I think you should reformulate your question (the question, not notations) in a more precise way. – user126154 Mar 29 '20 at 14:42

1 Answers1

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There aren't any issues with this argument. If there are too many elements in $X$, then the sum is simply infinity.

Aphelli
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