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If I have an uncountable series of elements $x_j$, what is the convention for listing them? It doesn't correct to write: $\{x_1, x_2, x_3, \ldots \}$ since that implies they can be indexed by $\mathbb{N}$. Maybe we don't list uncountable elements but that makes telling a story about them hard. (I couldn't find a better tag, sorry if notation is misleading.)

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  • $\begingroup$ If you have your uncountable indexing set $I$, and the elements are called $a_i$, then one fairly standard way is $\{a_i\}_{i\in I}$ $\endgroup$
    – Lubin
    Jan 16 at 2:06
  • $\begingroup$ @Lubin, thanks - does the job nicely. $\endgroup$
    – TonyK
    Jan 16 at 6:30

2 Answers 2

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You can't list an uncountable set. Hence the name uncountable.

The standard way of indexing is as follows: given set $A$ and indexing set $I$, such that there is a bijection $i\mapsto x_i$ that maps each $i\in I$ to some $x_i\in A$, we write $$\{x_i\}_{i\in I}$$

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  • $\begingroup$ Right. I think your original notation is fine. $\endgroup$
    – Lubin
    Jan 16 at 2:07
  • $\begingroup$ That is exactly what I need. Thank you all. $\endgroup$
    – TonyK
    Jan 16 at 3:30
  • $\begingroup$ That depends on what you mean by "list". Uncountable simply means that is not in bijection with the natural numbers, and hence cannot be counted using the natural numbers. Depending on the context, "list" can be understood as simply "well-ordering", in which case your first line is very wrong. $\endgroup$
    – Asaf Karagila
    Jan 16 at 11:56
  • $\begingroup$ I think calling a well-ordering a list out of context is very much a stretch, @AsafKaragila $\endgroup$ Jan 16 at 12:41
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There's actually no reason to "index" an uncountable set this way. Any time you feel the urge to write something like $$\bigcup_{i\in I}x_i$$where $I$ is an uncountable index set and $S=(x_i)_{i\in I}$ you can simply write $$\bigcup_{x\in S}x$$instead.

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