MHB About the “Axiom of Dependent Choice”

  • Thread starter Thread starter steenis
  • Start date Start date
  • Tags Tags
    Choice
AI Thread Summary
The "Axiom of Dependent Choice" states that for any nonempty set and entire binary relation, a sequence can be formed where each element is related to the next. This axiom is often implicitly used in mathematical proofs, particularly in functional analysis, such as in Baire's theorem, which is equivalent to the axiom. The discussion highlights that many practitioners may not directly confront the axiom but still rely on its implications in foundational theorems. Participants express a desire for deeper insights from those more familiar with the foundational aspects of choice axioms. Understanding and applying the axiom can enhance comprehension of various mathematical concepts and theorems.
steenis
Messages
312
Reaction score
18
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?
 
Physics news on Phys.org
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.
 
Last edited:
Back
Top