Consider ZFC with both powerset and infinity removed and collection included. The well-ordering principle is not assumed.

Does this theory prove that, for every set $X$, there is a set of all finite subsets of $X$?

Consider ZFC with both powerset and infinity removed and collection and $\in$-induction included. The well-ordering principle is not assumed.

Does this theory prove that, for every set $X$, there is a set of all finite subsets of $X$?

Consider ZFC with both Powerset and Infinity removed and Collection included. The Well-Ordering Principle is not assumed.

Does this theory prove that, for every set $X$, there is a set of all finite subsets of $X$?

Consider ZFC with both powerset and infinity removed and collection included. The well-ordering principle is not assumed.

Does this theory prove that, for every set $X$, there is a set of all finite subsets of $X$?