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Line 26: − − == Instruction == − − In many undergraduate mathematics curricula, the Axiom of Choice is introduced early as a tool for solving problems. It often goes under one of its other guises, such as [[Zorn's Lemma]] or the [[Well-Ordering Theorem]]. Students are taught it as a proof method, but often with little mention of the questionable nature of the axiom. Often, a full discussion of the [[paradox]]es that accompany the axiom is left to a higher (graduate) level special topics course in [[logic]], despite the fact that mathematicians in numerous fields may be tempted to use it and thus should be aware of its controversy. Undergraduate courses also neglect to adequately mention when the axiom is being invoked, giving a false sense that use of the axiom is incidental, rather than documenting its widespread nature in recent mathematical proofs. Line 52:
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reorder sections: let's introduce the axiom before discussing its status in classrooms
It has been shown that neither the Axiom of Choice nor its negation will contradict the other axioms of set theory. Thus assuming the Axiom of Choice and assuming its negation will each result in differing, consistent set theories. It is the choice of the mathematician whether to believe the axiom or believe its negation; many people choose to not take a position on this issue at all because the majority of mathematics can be proven without invoking the axiom altogether.
It has been shown that neither the Axiom of Choice nor its negation will contradict the other axioms of set theory. Thus assuming the Axiom of Choice and assuming its negation will each result in differing, consistent set theories. It is the choice of the mathematician whether to believe the axiom or believe its negation; many people choose to not take a position on this issue at all because the majority of mathematics can be proven without invoking the axiom altogether.
== Use of the Axiom of Choice ==
== Use of the Axiom of Choice ==
The formulation of a constructive Axiom of Choice is one of three major problems which challenge 21st century logicians.<ref>Edna Ernestine Kramer, "The Nature and Growth of Modern Mathematics", Princeton University
The formulation of a constructive Axiom of Choice is one of three major problems which challenge 21st century logicians.<ref>Edna Ernestine Kramer, "The Nature and Growth of Modern Mathematics", Princeton University
Press, 1982. p. 595.</ref>
Press, 1982. p. 595.</ref>
== Instruction ==
In many undergraduate mathematics curricula, the Axiom of Choice is introduced early as a tool for solving problems. It often goes under one of its other guises, such as [[Zorn's Lemma]] or the [[Well-Ordering Theorem]]. Students are taught it as a proof method, but often with little mention of the questionable nature of the axiom. Often, a full discussion of the [[paradox]]es that accompany the axiom is left to a higher (graduate) level special topics course in [[logic]], despite the fact that mathematicians in numerous fields may be tempted to use it and thus should be aware of its controversy. Undergraduate courses also neglect to adequately mention when the axiom is being invoked, giving a false sense that use of the axiom is incidental, rather than documenting its widespread nature in recent mathematical proofs.
== Sources ==
== Sources ==