In ZF with the axiom of infinity removed, is the axiom scheme of collection provable?


Note that Collection does follow from the axiom of Transitive Containment, which states that everything belongs to a transitive set. Mancini gave a model of ZF minus Infinity where Transitive Containment fails: see the end of Section 3 in


<https://projecteuclid.org/euclid.ndjfl/1054837937>


Mancini's model validates Collection, so this does not answer my question.



Does ZF minus infinity imply collection?
ZF - Inf does imply Collection. Fix a set $X$ and a property $P$ (which can be formalized in terms of a formula and a parameter). Since we have separation, we may assume for all $x \in X,$ there is $y$ such that $P(x,y).$ Suppose $X$ is finite, with cardinality $n.$ A standard inductive argument shows there is a set $Y$ such that for all $x \in X,$ there is $y \in Y$ satisfying $P(x,y).$


Now suppose $X$ is infinite. Then $\omega$ exists, by replacing every element of $\mathcal{P}\_{\text{fin}}(X)$ with its cardinality. Thus ZF holds, and this instance of Collection is justified by the standard argument.


