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.