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The axiom of choice is not really about assigning names to objects using finite strings of symbols. In this example, you take R, remove countably infinitely man
by voldacar 4y ago
The axiom of choice is not really about assigning names to objects using finite strings of symbols. In this example, you take R, remove countably infinitely many elements, and end up with a set with uncountably many elements. From a set theoretic perspective, it is not any more difficult or unintuitive to select an element from this set using AoC than it would be for R. When we use AoC, we are not really assigning a name to anything or doing any computation. We merely claim that a function exists which returns some element from the set. AoC doesn't compute or describe this function.
- horsawlarway 4y ago> We merely claim that a function exists which returns some element from the set. Isn't this where the challenge lies, though? What if you have a set for which there is no function which returns some element from the set, but the set is non-empty?
- drdeca 4y agoFor any set S which is non-empty, there is an element x of that set. For any set S and any x, the set S * {x} is a function which sends every element of S to x. Therefore, for every non-empty set S, there is a function with domain S, and range a single element of S. Did you mean a set S of sets such that there is no f : (A in S) -> A ?
- hgsgm 4y agoIt's only a challenge if you want to actually do it. Mathematicians figured out that it's impossible to actually do, so they decided to pretend it's possible, because nothing else breaks if they just pretend it's possible. Mathematicians like pretending.