Theories in maths which do not use the axiom of choice.

In summary, there are many theories that do not use the axiom of choice, including discrete mathematics, real analysis, and most algebra. These theories typically deal with finite sets and do not require an algebraic closure. It is not just logicians who choose to use or ignore the axiom of choice, as it is not often necessary in most mathematical fields. However, there are some theories of sets that do not use AC.
  • #1
MathematicalPhysicist
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i read that there are some logicians who do not use the axiom of choice in their axioms systems. i wonder what is the math that isn't using the axiom of choice, or what theories do not use it?
 
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  • #2
A lot of theories do not use the axiom of choice. Pretty much any discrete mathematics (i.e. with finite sets only) won't use it. Most of real analysis has no need to invoke the axiom of choice. Most algebra gets by without needing the axiom of choice to be invoked. You don't really ever need an algebraic closure of anything, for instance: start with a ground field, decide what polys you need to have roots, this will normally only be a finite number, and add in the roots accordingly. The vast majority of algebra only ever deals in finite dimensional vector spaces.

Incidentally, it is not just logicians who chose to use or ignore the axiom of choice specifically. Completely unjustified statistic alert: you can probably get by doing 99% of maths without ever needing to invoke the axiom of choice.
 
  • #3
i should have been clearer, i meant theories of sets which do not use AC.
 

1. What is the axiom of choice and why is it important in mathematics?

The axiom of choice is a principle in set theory that states that given any collection of non-empty sets, it is possible to choose one element from each set to form a new set. It is important because it allows for the creation of new sets that may not have any defining properties, and it simplifies many mathematical proofs and constructions.

2. What are some examples of mathematical theories that do not use the axiom of choice?

There are many examples of mathematical theories that do not rely on the axiom of choice, such as constructive mathematics, intuitionistic mathematics, and finitism. These theories tend to reject the existence of infinite sets or objects and instead focus on constructive methods of proof and computation.

3. How do theories without the axiom of choice differ from those that do use it?

The main difference is in their treatment of infinite sets. Theories without the axiom of choice often have stricter rules for defining and constructing infinite sets, and they may reject the existence of certain infinite sets altogether. This can lead to different conclusions and results in various areas of mathematics.

4. Can the axiom of choice be proven or disproven?

No, the axiom of choice is an assumption or principle that cannot be proven or disproven within mathematics. Its validity depends on the underlying axioms and definitions of a given mathematical theory. However, it has been shown to be consistent with most commonly used axioms of set theory.

5. Are there any practical applications of theories without the axiom of choice?

Yes, there are practical applications in computer science and engineering, where constructive methods of proof and computation are often preferred over non-constructive ones. For example, intuitionistic logic is used in computer programming, and finitism is applied in the design of efficient algorithms and data structures.

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