TITLE: Positive logical definition of finiteness
QUESTION [6 upvotes]: In his debate with Cassirer and in some other writing, Heidegger tries to define positively the concept of finiteness for the beings. It is a novel and difficult approach as traditionally, in theology and metaphysic, finiteness of beings is negatively defined as an imperfection or a negation, the absence of infiniteness, as opposed to some perfect and infinite being like god.
I feel that this difficulty is strangely reflected in set theory. I reckon we can naturally associate positive definition with $\Sigma_1$ sets (it abides to our definition if there is one set such that... ) and negative definitions with $\Pi_1$ sets (it abides to our definition if there is not set such that...).
Now for example Dedekind definition of an infinite set is a positive definition, a set is infinite if there is a bijection between itself and a proper subset. Then the natural way of defining finite sets would be non-infinite set which is $\Pi_1$ or a negative definition.
We could also use $\omega$, the smallest set satisfying the axiom of infinity and say that another set is finite if there is no injection from $\omega$ in this set ; but once again its a negative definition.
So formally, what this boils down to is the following :

Is the definition
of finiteness in set theory $\Pi_1$ or $\Delta_1$ ? I've just seen that
it is $\Delta_1$ according to wikipedia, then what is a $\Sigma_1$
definition of it ?Also, have the philosophical implications of this already been discussed ?

And for a subsidiary questions :

Are there models of set theory that "think" one of its element is infinite when it is actually finite ?
Akin to what happens between countable and non-countable sets with the Löwenheim-Skolem theorem.

REPLY [3 votes]: Let me add a remark to Noah's excellent answer.
While $\sf ZFC$ prove that "finite" is equivalent to "Dedekind-finite", and therefore both are $\Delta_1^{\sf ZFC}$ notions; this is not true for $\sf ZF$. Namely, the formula $\varphi(x)$ stating that $x$ is Dedekind-finite is not $\Delta_1^{\sf ZF}$.
If $M$ is a countable transitive model of $\sf ZFC$, then there is a generic extension of $M$, $M[G]$, and an intermediate model of $\sf ZF$, $N$, such that:

$M\subseteq N\subseteq M[G]$ are all transitive.
$x\in N$ and therefore $x\in M[G]$.
$N\models x$ is Dedekind-finite, but $M[G]\models x$ is countable (and therefore Dedekind-infinite).

So, if "Dedekind-finite" was $\Delta_1^{\sf ZF}$, it would be absolute between the transitive models $N$ and $M[G]$. But that's not true, of course. So it's worth keeping track of our assumptions when going between equivalent definitions.

Let me also throw in another $\Pi_1$ definition of finiteness, due to Tarski: $x$ is finite if and only if $(\mathcal P(x),\subseteq)$ is well-founded. Which is to say, for every non-empty $y\subseteq\mathcal P(x)$, there is some $z\in y$ which is $\subseteq$-minimal. Writing this definition out carefully will be $\Pi_1$.