About the “Axiom of Dependent Choice”

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In summary, the "Axiom of Dependent Choice" is a mathematical principle that states for every nonempty set and entire binary relation on that set, there exists a sequence in the set where each element is related to the next by the binary relation. This axiom is closely related to Baire's theorem and is often used implicitly in functional analysis. While the speaker does not often directly use this axiom, they have indirectly used it when invoking foundational theorems in their field. They suggest that someone with a deeper understanding of foundations may have more insight to offer.
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steenis
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I learned something new today: the “Axiom of Dependent Choice”:

The axiom can be stated as follows: For every nonempty set $X$ and every entire binary relation $R$ on $X$, there exists a sequence $(x_n)_{ n \in \mathbb{N} }$ in $X$ such that $x_nRx_{n+1}$ for all $n \in \mathbb{N}$. (Here, an entire binary relation on $X$ is one where for every $a \in X$, there exists a $b \in X$ such that $aRb$.)

See Wikipedia: https://en.wikipedia.org/wiki/Axiom_of_dependent_choice

I want to ask here: what is your experience with this axiom? Did you ever use the “Axiom of Dependent Choice”, how and why?
 
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Thank you for this.

I am rarely directly confronted with AC and its weaker siblings such as DC, but reading the link made me realize that, yes, I must have been using DC implicitly, because DC is enough to prove Baire's theorem. (In fact, per this reference in the article, DC and Baire are equivalent.) Now, since the metric space version of Baire is used in three foundational theorems in functional analysis (open mapping, closed graph and Banach-Steinhaus), I have indeed implicitly used DC when I was invoking these theorems.

Surely someone closer to the foundations has a deeper insight to offer.
 
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FAQ: About the “Axiom of Dependent Choice”

What is the Axiom of Dependent Choice?

The Axiom of Dependent Choice is a mathematical principle that states that given a set of choices, each choice must be made in a way that is dependent on the previous choices.

Why is the Axiom of Dependent Choice important?

The Axiom of Dependent Choice is important because it allows mathematicians to make assumptions and deductions about infinite sets without having to consider every possible case individually, making proofs and calculations much more efficient.

How is the Axiom of Dependent Choice used in mathematics?

The Axiom of Dependent Choice is used in many branches of mathematics, including set theory, topology, and analysis. It is often used to prove the existence of certain mathematical objects and to show that certain properties hold for infinite sets.

Are there any criticisms of the Axiom of Dependent Choice?

Yes, there are some mathematicians who reject the Axiom of Dependent Choice, arguing that it leads to counterintuitive or paradoxical results in certain situations. However, it is still widely accepted and used in modern mathematics.

Can the Axiom of Dependent Choice be proven?

No, the Axiom of Dependent Choice is an axiom, which means it is a fundamental assumption that cannot be proven by other mathematical principles. It is simply accepted as a starting point for mathematical reasoning.

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