Is the Axiom of Choice needed here, when looking into the properties of an element of a set?

Asked Sep 12 '21 at 16:45

Active Sep 12 '21 at 17:01

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I understand that the Axiom of Choice is used when making choices from an infinite set or making a choice each from an infinite number of sets. Would it be used in the following situation:

$A \cap B \neq \varnothing$. Therefore, there is some $x \in A \cap B$, meaning $x \in A$, $x \in B$.

My question is - if $A \cap B = \infty$, do you need the Axiom of Choice to be able to even consider $x$, or is this more of a hypothetical consideration of all elements that fall into this category without actually picking anything out?

More generally, how do I know for certain if the Axiom of Choice is being used?

EDIT: https://faculty.washington.edu/smcohen/120/Chapter12.pdf helped me get existential instantiation, which answered my question.

edited Sep 12 '21 at 17:01

asked Sep 12 '21 at 16:45

theChemist

2

You don't need the axiom of choice when you choose from one set. – Mark Sep 12 '21 at 16:48
You might want to look up existential instantiation. Elementary books on "Foundations of Mathematics" call this principle "Name It, then Use It". – Lee Mosher Sep 12 '21 at 16:52
Thanks Lee. This link helped a lot. – theChemist Sep 12 '21 at 16:59
No.... You are told that $A\cap B$ is not empty. That means it has at least one element. You may refer to an arbitrary element by a variable name (but you can't assume anything about the element other than it exists). That is not an application of the axiom of choice. – fleablood Sep 12 '21 at 18:01

Is the Axiom of Choice needed here, when looking into the properties of an element of a set?

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