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Sep27-06, 09:44 AM
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Quote Quote by Oxymoron
For example. Suppose I choose my favourite non-empty, well-ordered set, X. Then I would not be able to define an explicit choice function for this set. However if I choose my favourite non-empty set, X, such that every element is a well-ordered set, then I would be able to define an explicit choice function because I could let my function choose the least element of each set of X. Is this correct reasoning?
At first glance, that looks good. But what is a well-ordered set? It is a set that has a well-order. But in general, a well-ordered set has many well-orders. So for each set you can't first just say "pick the least element" you first have to pick a well-order, then you can choose the least element. But the problem of picking a well-order for each set is tricky in itself. I have to think on how you'd do this.