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> The axiom of choice is allowed. Why does this need to be stated explicitly? Why does the axiom of choice not enjoy the privilege of being a first-class axiom

> The axiom of choice is allowed. Why does this need to be stated explicitly? Why does the axiom of choice not enjoy the privilege of being a first-class axiom like other axioms? Why do many proofs need to state the assumption of this axiom explicitly before using it?

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 ago
  There 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 ago
    Iiuc your examples are undecidable rather than true in ZF. That's why we can add Axiom of Choice without getting contradictions.
    - hansvm 6y ago
      Well, 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.
qsort 6y ago
Proofs (at least when they are written "the usual" way) are human-to-human communication, not human-to-machine encoding. A proof depending on AC is a more interesting fact than it depending on, say, the axiom schema of specification, so it makes sense that the author might want to say "hey, pay attention to that".
lopsidedBrain 6y ago
It is partly historical notoriety. It allows one to prove certain statements that seem counter-intuitive. So proofs that employ it can often be viewed with some skepticism. Moreover, this axiom is _independent_ of the other axioms in ZFC. It is in fact possible to have entirely self-consistent "worlds" of mathematics, ones where axiom of choice is true, and ones where it is false. More details and examples of alternate axioms are in the Wikipedia article: https://en.wikipedia.org/wiki/Axiom_of_choice https://en.wikipedia.org/wiki/Axiom_of_choice If it seems weird that math can give you contradictory results, remember that the difference only shows up when you deal with some form of infinity (e.g. when performing an operation on an infinitely large set). For any usage of math in the real world, the truth or falsity of this axiom won't give you contradictory results.

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 ago
There 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 ago
  Iiuc your examples are undecidable rather than true in ZF. That's why we can add Axiom of Choice without getting contradictions.
  - hansvm 6y ago
    Well, 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.

qsort 6y ago

Proofs (at least when they are written "the usual" way) are human-to-human communication, not human-to-machine encoding. A proof depending on AC is a more interesting fact than it depending on, say, the axiom schema of specification, so it makes sense that the author might want to say "hey, pay attention to that".

lopsidedBrain 6y ago

It is partly historical notoriety. It allows one to prove certain statements that seem counter-intuitive. So proofs that employ it can often be viewed with some skepticism. Moreover, this axiom is _independent_ of the other axioms in ZFC. It is in fact possible to have entirely self-consistent "worlds" of mathematics, ones where axiom of choice is true, and ones where it is false. More details and examples of alternate axioms are in the Wikipedia article: https://en.wikipedia.org/wiki/Axiom_of_choice https://en.wikipedia.org/wiki/Axiom_of_choice If it seems weird that math can give you contradictory results, remember that the difference only shows up when you deal with some form of infinity (e.g. when performing an operation on an infinitely large set). For any usage of math in the real world, the truth or falsity of this axiom won't give you contradictory results.