- #1
Oxymoron
- 870
- 0
Does the choice function let me choose exactly one element from a set? The set which I am choosing from must be the domain of the choice function, correct?
Also, if I have a finite set, say something like [itex]X = \{a,b,c\}[/itex] where a, b, c are distinct non-empty sets, then can I construct a choice function for this set? Could this choice function simply say: "Pick one element of X, then pick another, then pick the third."?
Since there are only finite choices, well three actually, then does this mean that I do not have to appeal to the Axiom of Choice?
Could I say that I have defined a choice function without resorting to the axiom of choice?
What if my set is infinite, say the set of all integers? Must I then appeal to the Axiom of choice to define an explicit choice function?
What if my set is infinite, but it is well-ordered? Could I cheat and define a choice function without appealing to the Axiom of Choice?
Basically I want to know when and how to define Choice functions on sets without having to use the Axiom of Choice. Thanks for any help.
Also, if I have a finite set, say something like [itex]X = \{a,b,c\}[/itex] where a, b, c are distinct non-empty sets, then can I construct a choice function for this set? Could this choice function simply say: "Pick one element of X, then pick another, then pick the third."?
Since there are only finite choices, well three actually, then does this mean that I do not have to appeal to the Axiom of Choice?
Could I say that I have defined a choice function without resorting to the axiom of choice?
What if my set is infinite, say the set of all integers? Must I then appeal to the Axiom of choice to define an explicit choice function?
What if my set is infinite, but it is well-ordered? Could I cheat and define a choice function without appealing to the Axiom of Choice?
Basically I want to know when and how to define Choice functions on sets without having to use the Axiom of Choice. Thanks for any help.