This webpage is one of the best I have seen.
i am trying to understand the axiom of choice.
def from book: suppose that is a collection of nonempty sets. Then there exist a function. such that for each .
isnt this trivially obvious becouse ex.
if then
and if then and
have i understod this correct?.
now what i dont understand is that certain mathematician refuses to use this theorem, they think that this theorem cant be trusted.
why?
also about latex, when i tried to see what the code would look like when i posted it, stuff like f(< fontsize = \{1.... appeard.
so i thought something was off with the size, so i marked the text and clicked on 2. that removed the problem.
maby this is a bug, in this sites latex interpreter.
also when i clicked size, the [size] parameters did not appear.
This webpage is one of the best I have seen.
For finite C, the axiom of choice is indeed provable.
Do you mean f(x) = x? The f maps sets to sets, while for this C (which is a collection of sets of numbers), it should maps sets to numbers.
The expression does not makes much sense. First, f is defined on sets of numbers, not numbers themselves, so f(1) is not defined. Second, f(1) ∈ {1} is a proposition, i.e., something that is either true or false. Are you considering a set that contains one proposition?
Plato has already provided the link.
LaTeX on this site does not like when a formula has newline characters in it, i.e., the whole formula must be on a single line. Also, sometimes it helps to insert spaces every 50-60 characters.
here is a better example illustrating why the axiom of choice is viewed as "suspect".
suppose that is, C is a collection of open intervals that include every real number less than a particular real number.
it's not hard to see that .
the question is now: how do you define f? given the interval (-∞,a) we have to choose some real number in it. how do we do that? think about this for a while.
the axiom of choice in this setting essentially means: "well, we just pick one. who cares which one it is?"
of course, if you're clever, you might say:
ok, how about we define f(A) = floor(a). so, in a sense, this C is "still too simple". what if the elements of C are ANY subset of the real numbers? that is ?
put another way, when we define a function, do we have to "know something about it"? (like, how to compute f(x) given x). arbitrary functions f:X-->Y can be quite strange, and even when we DO know how to calculate f(x) = y, there may not be any clear way to choose a particular x that f maps to y as "distinguished". (example: pick some very large uncountable set, S. define the function f:S-->{1} by f(s) = 1, for all s in S. how do we pick an element in f^{-1}(1)?).
personally, i don't like to think about "arbitrary sets" because they ARE so arbitrary. but some people do, and are very good at it.