Recursively defining a set

Question

I have what may seem a very trivial question, but how it is answered may affect how a proof of mine is structured. It pertains to formatting and convention. When 'recursively' defining a function does it make sense to use quantifiers?

For example would:

$ 5 \in R $

If $ r \in R $, then $ \forall s \in \mathbb Z, r + s \in R $

be an acceptable substitute for:

$ 5 \in R $

If $ r \in R, $ then $ r + 1 \in R $ and $ r - 1 \in R $

Or would using quantifiers in the former definition violate some fundamental rule about how recursive functions are supposed to be considered?

Anyways, thanks for any help!

Thanks,

Tuba09

Answer 1

What you're missing is that you want the smallest $R$ that satisfies your constraints. Note also that you are not actually defining a function recursively here. It will, however, turn out that you can define $R$ in terms of a recursively defined function. Do you see how?

Answer 2

You should probably explicitly say that you are 'inductively' defining the set $R$ or that $R$ is the 'smallest' set satisfying your constraints. So

"Let $R$ be the smallest subset of ${\mathbb R}$ satisfying (1) $5 \in R$; (2) for all $r \in R$, $r - 1, r + 1 \in R$."

Or in the other formulation.

"Let $R$ be the smallest subset of ${\mathbb R}$ satisfying (1) $5 \in R$; (2) for all $r \in R$ and all $s \in {\mathbb Z}$, $r + s \in R$."

Well, for this particular case you could also say "Let $R$ be ${\mathbb Z}$", but I guess the point is how you formulate this type of definitions in general.