SkipCaptcha
710
edits
Changes
Jump to navigation
Jump to search
Line 36:
Line 36: − + Line 42:
Line 42: + +
comment on usage
== Controversy ==
== Usage ==
Despite its usefulness, many mathematicians prefer to avoid the Axiom of Choice when possible. Stephen Willard, author of the textbook ''General Topology'', wrote the following after describing the Axiom of Choice:
Despite its usefulness, many mathematicians prefer to avoid the Axiom of Choice when possible. Stephen Willard, author of the textbook ''General Topology'', wrote the following after describing the Axiom of Choice:
The avoidance of the Axiom of Choice reflects a preference for [[constructive proof]]s. AC, by its very nature, is nonconstructive, since it merely asserts that a choice function exists, but does not give an explicit method for its construction. Since the Axiom of Choice is independent of the other axioms, assuming this axiom results in a more constrained system than otherwise. Thus results provable with it might not be provable or might not even be true, without it.
The avoidance of the Axiom of Choice reflects a preference for [[constructive proof]]s. AC, by its very nature, is nonconstructive, since it merely asserts that a choice function exists, but does not give an explicit method for its construction. Since the Axiom of Choice is independent of the other axioms, assuming this axiom results in a more constrained system than otherwise. Thus results provable with it might not be provable or might not even be true, without it.
The general situation is this: some results (for example, Tychonoff's theorem) simply can't be proved without assuming the axiom of choice. When proving theorems, mathematicians prefer to proceed using as few axioms as possible, and it is now understood that there are some theorems which can be proved without the axiom of choice, and some that can not. It is well-understood which theorems fall into which category. In some branches of mathematics, for example functional analysis, the use of the axiom of choice is ubiquitous because of reliance on things like the Hahn-Banach theorem (though this theorem can be proved using a strictly weaker axiom).
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