1
$\begingroup$

In the course I am taking of axiomatic set theory we've defined a family of sets $F$ indexed by $I$ as any function satisfying $dom(F)=I$, where there are no assumptions about its image. This definition confuses me a little because I've read, in other contexts, that families of sets differ from sets because the first can be proper classes while the later obviously can't, but with this definition $Im(F)$ must be a set since it is a subset of $\bigcup \bigcup F$

Could anyone clarify it? Thanks

$\endgroup$
1
$\begingroup$

What you defined as 'family of sets' is more commonly known as a 'sequence'. Let's write $F_i := F(i)$ for each $i \in I$. Then your family of sets $F$ is precisely the sequence $F = (F_i \mid i \in I)$ and you're right in that, that - following your definition and working in a sufficiently strong theory, e.g. ZFC - this is always a set.

Other authors use the term 'family of sets' differently and to them it means just 'a collection of sets' that does not neccessarily have to be a set itself. An example is the collection of all groups or the collection of the von Neumann hierarchy $\{ V_\alpha \mid \alpha \in \operatorname{On} \}$.

Definitions are sensitive to their context and the same name may mean very different things even to the same author in different contexts. A famous example is the term 'regular' which is widely used in all sorts of contexts with entirely different meanings.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.