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text: '#Getting the Heavy Maths out the Way: Definitions#\nIntuitively, the [-axiom_mathematics axiom] of choice states that, given a collection of *[-5zc non-empty]* [-3jz sets], there is a [-3jy function] which selects a single element from each of the sets. \n\nMore formally, given a set $X$ whose [-5xy elements] are only non-empty sets, there is a function \n$$\nf: X \\rightarrow \\bigcup_{Y \\in X} Y \n$$\nfrom $X$ to the [-5s8 union] of all the elements of $X$ such that, for each $Y \\in X$, the [-3lh image] of $Y$ under $f$ is an element of $Y$, i.e., $f(Y) \\in Y$. \n\nIn [-logical_notation logical notation],\n$$\n\\forall_X \n\\left( \n\\left[\\forall_{Y \\in X} Y \\not= \\emptyset \\right] \n\\Rightarrow \n\\left[\\exists \n\\left( f: X \\rightarrow \\bigcup_{Y \\in X} Y \\right)\n\\left(\\forall_{Y \\in X} \n\\exists_{y \\in Y} f(Y) = y \\right) \\right]\n\\right)\n$$\n\n#Axiom Unnecessary for Finite Collections of Sets#\nFor a [-5zy finite set] $X$ containing only [-5zy finite] non-empty sets, the axiom is actually provable (from the [-zermelo_fraenkel_axioms Zermelo-Fraenkel axioms] of set theory ZF), and hence does not need to be given as an [-axiom_mathematics axiom]. In fact, even for a finite collection of possibly infinite non-empty sets, the axiom of choice is provable (from ZF), using the [-axiom_of_induction axiom of induction]. In this case, the function can be explicitly described. For example, if the set $X$ contains only three, potentially infinite, non-empty sets $Y_1, Y_2, Y_3$, then the fact that they are non-empty means they each contain at least one element, say $y_1 \\in Y_1, y_2 \\in Y_2, y_3 \\in Y_3$. Then define $f$ by $f(Y_1) = y_1$, $f(Y_2) = y_2$ and $f(Y_3) = y_3$. This construction is permitted by the axioms ZF.\n\nThe problem comes in if $X$ contains an infinite number of non-empty sets. Let's assume $X$ contains a [-2w0 countable] number of sets $Y_1, Y_2, Y_3, \\ldots$. Then, again intuitively speaking, we can explicitly describe how $f$ might act on finitely many of the $Y$s (say the first $n$ for any natural number $n$), but we cannot describe it on all of them at once. \n\nTo understand this properly, one must understand what it means to be able to 'describe' or 'construct' a function $f$. This is described in more detail in the sections which follow. But first, a bit of background on why the axiom of choice is interesting to mathematicians.',
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