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Because it gets you applications like this (or Banach-Tarski), which make you wonder how much it deserves its place. To some extent this is more due to Tertium
by generationP 6y ago
Because it gets you applications like this (or Banach-Tarski), which make you wonder how much it deserves its place. To some extent this is more due to Tertium Non Datur than to the axiom of choice, but my impression is ZF doesn't know enough about infinite sets to get to the really counter-intuitive stuff.
- hansvm 6y agoThere are counter-intuitive results in ZF without choice too: - Collection of non-empty sets with empty cartesian product. - Infinite set without a countable infinite subset. - There is a pair of sets such that neither is equinumerous with a subset of the other. More or less, our intuition about seemingly obvious ideas is completely thrown off without choice. Banach-Tarski at least has the property that it probably doesn't directly apply to the real world (if you can split an object into probably physically impossible sets and then rejoin them correctly then you can double the volume of the physical object) and so doesn't really violate our intuition -- the premise doesn't apply in the real world, so no conclusion really matters. It's like claiming that every element of the empty set is a leprechaun with a pot of gold -- it's true, but it doesn't matter in any meaningful sense.
- im3w1l 6y agoIiuc your examples are undecidable rather than true in ZF. That's why we can add Axiom of Choice without getting contradictions.
- hansvm 6y agoWell, sure. Without choice there exist models of ZF where those properties hold. Adding choice precludes those models -- i.e., it restricts our axioms in a way which prevents certain kinds of unexpected results.