The non measurable set is formed by selecting one element from each equivalence class obtained by the relation x ~ y if x-y is rational.
but we suggest we do not accept this form of axiom of choice applied to collection of sets.
To form an arbitrary Cartesian product one needs an indexed family of sets and an indexing set is necessary.
In the above case it seems that the collection is not indexed . No indexing set and explicit indexing map is there.
So we can not form a nonmeasurable set. Thus Banach -Taraski paradox is absent.
we do not work with arbitrary collections but only indexed families and explicit indexing map.
how to avoid the paradox in measure theory
plane where axiom of choice is not needed is also open
Let the indexing set be the set of equivalence classes for that equivalence relation, R/~. Then, if C is such an equivalence class, define A_C =C. This gives an indexed family {A_C } for C in R/~. An element of the product will then select one element fro each class.
The point is that the axiom corresponding to non-empty products of indexed families of sets is equivalent to the usual axiom of choice.
thankful to Daniel Grub for a really apt answer.
now i am advocate of axiom of choice only for countable families of sets sets.
i dnot accept the general axiom of choice and i o not think the applicable Mathematics and most used pure mathematics can do without it.
we may not have the fact that mximal ideals or orthonormal sets exisst. but they exist in mosst concrete examples.
in the same vein. i disregard dealing with asubsset in which i can not pick up atleast one element explicitly.
I'd suggest considering some notions concerning functions.
So, for a function f:X->Y, the existence of a function g:Y->X such that the composition gof:X->X is the identity is the same as f being one-to-one. This is easily proven.
But, the similar result for *onto* functions, that f being onto is equivalent to the existence of a function g:Y->X with fog:Y->Y the identity, is equivalent to the full axiom of choice.
For more advanced considerations, the Tychonoff product theorem (that products of compact spaces are compact) is also equivalent to the axiom of choice (although the Hausdorff version is slightly weaker). This result is incredibly useful in functional analysis and operator theory. Also, such results as the Hahn-Banach theorem require some version of the axiom of choice.
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