# Does ZF minus infinity imply collection?

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.

• What is "the axiom of Transitive Containment"? – Wojowu Dec 12 '20 at 23:07
• What is the axiom scheme of Collection? Is it provable in ZF? – bof Dec 12 '20 at 23:13
• Perhaps see Kaye, Richard; Wong, Tin Lok. On Interpretations of Arithmetic and Set Theory. Notre Dame J. Formal Logic 48 (2007). projecteuclid.org/euclid.ndjfl/1193667707 – jeq Dec 12 '20 at 23:20
• – Wojowu Dec 12 '20 at 23:26
• @Wojowu, the axiom of Transitive Containment says that everything belongs to a transitive set. It is provable in ZF. But Mancini provided a model of ZF minus Infinity where it does not hold: projecteuclid.org/euclid.ndjfl/1054837937 – Paul Blain Levy Dec 13 '20 at 1:00

## 1 Answer

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.

• One needs to be careful about that inductive argument. It is internal to the model. – Asaf Karagila Dec 13 '20 at 16:32
• What model? Elliot didn't mention any model. – Paul Blain Levy Dec 13 '20 at 22:58