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chingkui
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Is the existence of basis in all vector space equivalent to the axiom of choice?
What is P(C)? The power set of C? I don't understand. And isn't this theorem supposed to rely on Zorn's lemma?Hurkyl said:Every element of B is a linear combination of finitely many elements of C, and this defines a function f:B --> P(C)
Apparently, you need the ultrafilter principle, which follows from the AC, but is not equivalent to it. So I was right that the uniqueness of cardinality of basis doesn't follow from ZF, but wrong in assuming that it requires choice. See Schechter HAFHurkyl said:I'm pretty sure you don't need the axiom of choice to prove that if you have two bases, their cardinalities are equal.
the support of a function is the closure of the set where the function is nonzero. unless we're talking about topological vector spaces, I don't think we can apply that term.Hurkyl said:Is "support" the word I'm looking for? I.E. "f(b) is the support of b in the basis C." (or something like that?)
The Basis and Axiom of Choice are two fundamental concepts in set theory. The Basis is a collection of elements that are used to construct other objects, while the Axiom of Choice is a principle that allows for the selection of one element from each set in a collection of non-empty sets.
The Axiom of Choice is important because it allows for the creation of infinite sets, which are essential in many areas of mathematics and science. It also helps in proving the existence of certain mathematical objects, such as well-orderings, which cannot be constructed without the Axiom of Choice.
The controversy surrounding the Axiom of Choice stems from the fact that its use can lead to counterintuitive results, such as the Banach-Tarski paradox, which states that a solid ball can be disassembled into a finite number of pieces and rearranged to form two identical copies of the original ball. Some mathematicians argue that the Axiom of Choice should not be considered as a valid principle, while others believe that it is necessary for the development of mathematics.
The Axiom of Choice is one of the axioms in the Zermelo-Fraenkel set theory, which is the most commonly used foundation for mathematics. It is included as the fourth axiom in the Zermelo-Fraenkel set theory, and its acceptance has been a subject of debate among mathematicians since its introduction.
No, the Axiom of Choice cannot be proven within the Zermelo-Fraenkel set theory. It is an axiom that is accepted as true without proof, and its validity is based on its usefulness in mathematics and the lack of any known contradictions that arise from its use.