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The Axiom of Choice 
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#1
Feb913, 09:34 PM

P: 376

When I read the AC, "that the ∏ of a coll. of non∅ sets is itself non∅" I understand its meaning, yet I come short from understanding its cardinal importance in Axiomatic set theory.
I have no exposure "yet" in ZFC but I was hoping if someone could clarify to me why is it that AC is such an important axiom especially that Zermelo used it to formulate the wellordering theorem. Being also that Set Theory is regarded as the foundation of Mathematics. (Disregarding Godel's work of course) Thank you 


#2
Feb1013, 03:51 AM

Sci Advisor
P: 839

AC has a large number of equivalent statements. You touched on wellordering but there is also
"every surjective function has a right inverse", "every nontrivial unital ring has a maximal ideal", "every vector space has a basis", and "two set either have the same cardinality or one is greater than the other." 


#3
Feb1013, 07:09 AM

Mentor
P: 18,345

 A function [itex]f:\mathbb{R}\rightarrow \mathbb{R}[/itex] is continuous at a fixed point x (from the [itex]\varepsilon\delta[/itex] condition) if and only if for each sequence [itex]x_n\rightarrow x[/itex] holds that [itex]f(x_n)\rightarrow f(x)[/itex]. (fun fact: if we change the "at fixed x" by "at every point x", then we don't require AC anymore!)  For every set X holds that X is either finite or there exists an injection [itex]\mathbb{N}\rightarrow X[/itex]  [itex]\mathbb{N}[/itex] is Lindelof: every open cover of [itex]\mathbb{N}[/itex] has a countable subcover  [itex]\mathbb{R}[/itex] is not a countable union of countable sets  Any two bases in a vector space must have the same cardinality  The HahnBanach theorem  The AscoliArzela theorem  The existence of the CechStone compactification  Lebesgue measure is [itex]\sigma[/itex]additive  Every unbounded subset of [itex]\mathbb{R}[/itex] contains an unbounded sequence 


#4
Feb1113, 09:06 PM

P: 376

The Axiom of Choice



#5
Feb1113, 09:11 PM

P: 376

For instance with respect to the right inverse? Where is its role? Thanks 


#6
Mar513, 10:38 AM

P: 376




#7
Mar513, 12:44 PM

Sci Advisor
P: 1,172

Check too, the construction of a nonmeasurable set using AC. Still, I believe nonmeasurable subsets can be constructed in theories that do not use AC.
There is also Tikunov's theorem, LowenheimSkolem, which allows you to construct models of the reals of any infinite cardinality (have you heard of the nonstandard reals?). For more, see, e.g: http://plato.stanford.edu/entries/ax.../#MatAppAxiCho 


#8
Mar513, 12:56 PM

Mentor
P: 18,345




#9
Mar513, 01:04 PM

Sci Advisor
P: 1,172

Actually, AFAIK, using forcing, you can come up with models that satisfy ZF+ ~AC, and these models contain nonmeasurable subsets. But I have not seen this in a while, and it would take me a while to produce more arguments.



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