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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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 $$\{x_i\}_{i\in I}$$
Rushabh Mehta
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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. – Asaf Karagila Jan 16 '22 at 11:56
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I think calling a well-ordering a list out of context is very much a stretch, @AsafKaragila – Rushabh Mehta Jan 16 '22 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.
David C. Ullrich
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