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Proof by induction in sets
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[QUOTE="andrewkirk, post: 5222091, member: 265790"] Since Fredrik and Ray have answered your main question, I'll take the liberty of replying to your ancillary question, which is the above. The formal interpretation of ##\bigcup_{i=1}^\infty A_i## is as follows. First, for it to make sense at all, we need a set ##\mathscr{C}##, which we can think of as a 'collection' of sets, and a function ##f:\mathbb{N}\to\mathscr{C}##, which is our 'indexing' function. We denote ##f(i)## by ##A_i##, which is the ##i##th set in collection ##\mathscr{C}##. We also need a master set that is a superset of all the ##A_i##. That is, we need a set ##M## with the property that ##\forall i\in\mathbb{N}:\, A_i\subseteq M##. With these formalities completed, we can define $$\bigcup_{i=1}^\infty A_i\equiv \{x\in M|\exists i\in\mathbb{N}:\,x\in A_i\}$$ The existence of this set is guaranteed not by the Axiom of Union, as one would expect, as that axiom is only useful for [I]finite [/I]unions. Instead it is guaranteed by the Axiom Schema of Separation (see the heading 'Separation Schema' under [URL='http://plato.stanford.edu/entries/set-theory/ZF.html']this link[/URL]). A more compact and - to me - more aesthetically pleasing notation is available whereby we discard the index function and just write: $$\bigcup_{A\in\mathscr{C}} A$$ This is the same set as defined above. I prefer this notation as it is more 'purist' but most people find the numeric index notation more intuitive as they are used to indexing things with integers. However, once we start dealing with uncountable collections we are forced to give up the numeric index notation. [/QUOTE]
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