We can define the law of composition with finite elements by induction. But there are some operations such as sum or union, which can be compute with infinitely many, or even uncountably many elements. I tried to define it using transfinite induction, but this idea didn't work well. So how can we make it?
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1If you are thinking about things like infinite sums of real numbers, keep in mind that these are defined analytically, not algebraically. That is, they rely on a notion of limit...something not typically available in algebraic settings. Even then, they are aren't generally defined, special conditions need to be satisfied. – lulu Jul 11 '24 at 12:57
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The following answer of mine might be relevant: https://math.stackexchange.com/a/1952747/350214 – Mitchell Spector Jul 12 '24 at 01:39
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Thanks for the comment! Actually I was realizing that infinite sum and finite sum are different operation in strict view, but the case of union or intersection kept making me question. Anyway, your answer was a great help! – Jul 12 '24 at 08:15