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.
Best Answer
Yes, finiteness is $\Delta_1$ in set theory.
A set $x$ is finite iff there are $f,\alpha$ such that:
$f$ is a bijection from $x$ to $\alpha$;
$\alpha$ is a nonzero ordinal; and
every ordinal $<\alpha$ has an immediate predecessor.
This is a $\Sigma_1$ definition, the only subtle point being that ordinalhood is $\Delta_1$: to see this, use the "hereditarily transitive set" characterization of ordinals.
Note meanwhile that unlike the $\Sigma_1$ definition above, the $\Pi_1$ definition of finiteness coming from the $\Sigma_1$ definition of infiniteness you give assumes "Dedekind-finite = finite" (the definition of "finite" in $\mathsf{ZF}$ is "in bijection with an ordinal $<\omega$"). While easily provable in $\mathsf{ZFC}$ (or indeed vastly less), this is not a theorem of $\mathsf{ZF}$ alone.
Here's a $\Pi_1$ characterization of finiteness which works in $\mathsf{ZF}$ alone:
The point is that in $\mathsf{ZF}$, a set is infinite iff every finite ordinal injects into it. In $\mathsf{ZF}$ this falls short of the existence of an injection from $\omega$ (= Dedekind-infiniteness), but is still fundamentally of the same "shape." EDIT: As Asaf shows, Dedekind-finiteness is not $\Delta_1$ in $\mathsf{ZF}$ alone, so we really have divergent complexity behaviors here.
As a coda, note that your question is in terms of the Levy hierarchy. Other notions of complexity can yield more complicated situations - see e.g. here.
As to your subsidiary question, models of $\mathsf{ZFC}$ can only be confused about finiteness in one way: $M\models\mathsf{ZFC}$ may have elements it thinks are finite which are actually infinite, but it cannot have elements it thinks are infinite which are actually finite.
The former point is a consequence of the compactness theorem, and the argument is identical to the existence of nonstandard models of arithmetic. To see the latter point, just note that $M$ is correct about the finiteness of all sets of size $0$ (there's only one of those) and if $M$ is correct about the finiteness of all sets of size $n$ (for $n$ finite) then $m$ is correct about the finiteness of all sets of size $n+1$ (think about what $M$ knows about the impact on cardinality of adding a single element to a set).