I'm glad you decided to follow up on my suggestion to look at Descriptive Set Theory! As I mentioned in my earlier answer, there are two flavors of Descriptive Set Theory: the "boldface theory" and the "lightface theory." As you observed, the boldface theory of sets of natural numbers is completely trivial since every subset of $\mathbb{N}$ is clopen, but the lightface theory is not trivial at all.

The lightface theory is completely analogous to the boldface theory except that all operations are required to be computable, or *effective* in the descriptive lingo. Let's go through the first few levels of the boldface and lightface Borel hierarchies in parallel. Let $X$ be a Polish space and let $(U_n)_{n=0}^\infty$ be a *nice* enumeration of a basis for the topology on $X$.

A boldface $\boldsymbol{\Sigma}^0_1$ set is simply an open subset of $X$, or a countable union of basic open sets. Similarly, a lightface $\Sigma^0_1$ set is an *effective* union of basic open sets, i.e. a union of the form $\bigcup_{n=0}^\infty U_{f(n)}$ where $f$ is a computable function. Well, not quite, since this informal definition technically excludes the empty set. A better definition is that the lightface $\Sigma^0_1$ sets are the sets $G_e = \bigcup_{n \in W_e} U_n$ where $W_e$ is the $e$-th computably enumerable set. This also gives a better listing of the lightface $\Sigma^0_1$ sets.

A boldface $\boldsymbol{\Pi}^0_1$ set is simply a closed subset of $X$, or the complement of a boldface $\boldsymbol{\Sigma}^0_1$ set. Similarly, a lightface $\Pi^0_1$ set is a the complement of a lightface $\Sigma^0_1$ set. Note that we have a nice enumeration of all lightface $\Pi^0_1$ sets, namely $F_e = X \setminus G_e$ where $G_e$ was defined above.

A boldface $\boldsymbol{\Sigma}^0_2$ set is a $F_\sigma$ subset of $X$, or a countable union of boldface $\boldsymbol{\Pi}^0_1$ sets. Similarly, a lightface $\Sigma^0_2$ set is an *effective* union of $\Pi^0_1$ sets. As above, the lightface $\Sigma^0_2$ sets are enumerated by $G'_e = \bigcup_{n \in W_e} F_n$ where $W_e$ is the $e$-th computably enumerable set and $F_e$ was defined above.

A boldface $\boldsymbol{\Pi}^0_2$ set is a $G_\delta$ subset of $X$, or the complement of a boldface $\boldsymbol{\Sigma}^0_2$ set. Similarly, a lightface $\Pi^0_2$ set is a the complement of a lightface $\Sigma^0_2$ set. Note that we again have a nice enumeration of all lightface $\Pi^0_2$ sets, namely $F'_e = X \setminus G'_e$ where $G'_e$ was defined above.

And so on, lightface $\Sigma^0_{n+1}$ sets are effective countable unions of lightface $\Pi^0_n$ sets, which in turn are complements of lightface $\Sigma^0_n$ sets and at each step we have a nice enumeration of all of these sets.

Just as for the boldface Borel hierarchy, there is no reason to stop at $\omega$; this process can be continued well into the transfinite, but there are a few roadblocks along the way. Because we want the hierarchy to be effective, we can only go up to the Church-Kleene ordinal $\omega_1^{CK}$, which is the smallest non-computable ordinal. (Also, the hierarchy is defined using ordinal notations instead of actual ordinals.) In the end, we get the family lightface Borel sets, which is a countable subfamily of the usual Borel sets.

The analytic sets, or boldface $\boldsymbol{\Sigma}^1_1$ sets, also have a lightface analogue. The boldface $\boldsymbol{\Sigma}^1_1$ sets are precisely the projections onto the $X$-axis of the boldface $\boldsymbol{\Pi}^0_1$ subsets of the Polish space $X \times \mathbb{N}^\mathbb{N}$, where $\mathbb{N}^\mathbb{N}$ denotes Baire space. Similarly, the lightface $\Sigma^1_1$ subsets of $X$ are the projections onto the $X$-axis of the lightface $\Pi^0_1$ subsets of $X \times \mathbb{N}^\mathbb{N}$. Note that since we have a nice enumeration of the lightface $\Pi^0_1$ sets, we also have a nice enumeration of the lightface $\Sigma^1_1$ sets. There are similar lightface analogues of the entire projective hierarchy. A lot of the results of the boldface theory transfer to the lightface theory. For example, the Suslin Theorem that the boldface $\boldsymbol{\Delta}^1_1$ sets are precisely the Borel sets. A theorem of Kleene says that the same holds for the lightface $\Delta^1_1$ sets, namely these are precisely the lightface Borel sets described above.

In the case of the discrete (and Polish) space $\mathbb{N}$, the natural choice of basis are the singletons $U_n = \{n\}$. Working through the definitions, we see that the lightface $\Sigma^0_1$ subsets of $\mathbb{N}$ are precisely the computably enumerable subsets of $\mathbb{N}$ and the lightface $\Pi^0_1$ sets are the complements thereof. Hence, the $\Delta^0_1$ subsets of $\mathbb{N}$ are precisely the computable sets. In fact, the lightface $\Sigma^0_n$, $\Pi^0_n$, $\Delta^0_n$, perfectly match the arithmetic hierarchy. The lightface Borel subsets of $\mathbb{N}$ are commonly called hyperarithmetic sets.

Obviously, there are lots and lots of interesting properties of subsets of sets of natural numbers!

thisquestion to be either speculative or foundational: it's just very, very soft. – Pete L. Clark Feb 8 '10 at 18:27