On Jul 2, 4:45 pm, MoeBlee <jazzm...@hotmail.com> wrote: > On Jul 2, 4:12 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com> > wrote: > > > On Jul 2, 9:28 am, MoeBlee <jazzm...@hotmail.com> wrote: > > > > On Jul 2, 12:06 am, "Ross A. Finlayson" <ross.finlay...@gmail.com> > > > wrote: > > > > > countable choice is a theorem of ZF. > > > > No, it's not. > > The definition of countable is there existing a bijective function > > between some set and the natural integers or here their ordinals. > > A definition of countable is ['w' stands for the set of natural > numbers]: > > x is countable iff (there exists a bijection from x onto a natural > number of there exists a bijection from x onto w) > > > A > > choice function is representable by any such function bijecting an > > ordinal as a set to a set. > > I have no idea what you're trying to say. > > ['P' stands for the power set operation:] > > A choice function for x is a function f whose domain is Px\{0} and > such that for all y in Px\{0}, we have f(y) in y. > > I guess sometimes people also refer to a choice function for x as > being a function whose domain is x\{0} and such that for all y in x > \{0} we have f(y) in y. > > Context should make clear which sense is intended. > > > Then, where countable choice means there > > are choice functions for countable sets, > > The axiom of countable choice is: > > For all x such that 0 is not in x, there exists a function f whose > domain is x and such that for all y in x we have f(y) in y. > > Since finite sets are not at issue anyway, we can narrow: > > For all x such that x is denumerable, there exists a function f whose > domain is x\{0} and such that for all y in x\{0} we have f(y) in y. > > > otherwise there are > > "countable" sets with no provable bijection to the naturals, > > You don't know what you're talking about. We don't require any choice > principles just to prove that there exists a bijection between a > countable set and either a natural number or the set of natural > numbers. The result follows simply by the DEFINITION of 'countable'. > > > where if > > that's undecideable then so is whether they're countable, not > > disproving the definition. > > Get help. Start with a book on working in the first order predicate > calculus. > > MoeBlee
Here a choice function is a bijective function from an ordinal (which is a collection of all lesser ordinals) to a set, or vice versa. Then for set X and ordinal O, f(x) is a well-ordering. I see that I am not using the same definition. I was presuming that a well-ordering was a choice function, which it is, not the existence of them in non-well- orderable set theories, like ZF.
So, my mistake, thanks.
Still, in terms of countable choice, if a set if countable but its elements are uncountable sets, in the process of building up the structures to represent that, various support structures then aren't countable. So, there are ways to distinguish between countable sets with recursively countable elements and those otherwise, in terms of ready choice functions that are easily constructed.
Maybe it is something about the definition along the lines of "the choice function selects an element from each subset of a set". For something like the naturals then a choice function is always surjective, never injective, in regular set theories that would always be so. Then point here is that it is known that for there to be as well that any subset has an element selected (in terms of a course-of- passage in induction over the elements generally however they are), any choice function f is a surjection f:P(X)->X. Consider then the cartesian product P(x) x X, or P(N) x N. There is an indicator value for each p in P(N) for only one value n in N. Now, these functions aren't "choice functions" but do select each element according to at least one subset of the set. The point here is that usual operations build up constrctively from the countable see no end of selection methods.
Where Feferman/Levy reiterates that a countable union of countable sets may not be, that would be a paradox with the same objects in regular set theories.
