The answer is no, if you allow me to adopt some weak-but-equivalent forms of the other axioms. And the reason is interesting:
- A shocking number of the axioms of set theory are true in the non-negative real line R+, with the usual order < being used to interpret set membership. (!)
Let's just check. For example, Extensionality holds, because if two real numbers have the same predecessors, then they are equal. The emptyset axiom holds, since there are no non-negative reals below 0. The Union axiom holds, since for any real x, the reals less than x are precisely the reals that are less than something less than x. (Thus, every set is its own union.) A weakened version of the Power set axioms holds, the Proper Power set, which asserts that for every x, there is a set p whose elements are the strict substs of x. This is because for any real number x, the reals below x are precisely the reals y (other than x), all of whose predecessors are less than x. (Thus, every real is its own proper power set.) An alternative weakening of power set would say: for every x, there is p such that y subset x implies y in p. This is true in the reals by using any p > x. A weakened pairing axiom states similarly: for every x,y, there is z with x ε z and y ε z. This is true in the reals by using any z above both x and y. The Foundation axiom is no problem, since 0 is in every nonempty set. Also, similarly AC holds in the form about families of disjoint nonempty sets, since this never occurs in this model. The Weak Collection Axiom holds since if every y < x has phi(x,y,w), then in fact any y in the same interval will work (since this structure has many automorphisms), and so we may collect witness with any B above x and w. Note that Separation fails, since, for example, {0} does not exist in this model. Also, Replacement fails for the same reason.
Similar interesting models can be built by considering the structure (ORD,<) built using only ordinals, or the class { Vα | α in ORD }. These also satisfy all of the weakened forms of the ZFC axioms without Separation, using Collection in place of Replacement.
Thus, part of the answer to your question is that it depends on what you mean by the "other axioms of ZFC".
Apart from this, however, let me say that the term Weak Collection is usually used to refer to the axioms that restrict the complexity of the formulas in the usual Collection scheme, rather than the axiom that you state. For example, in Kripke Platek set theory KP, a weak fragment of ZFC, one has collection only for Sigma_1 formulas, and this is described as a Weak Collection axiom. (What you call Weak Collection is usually just called Collection.) And there is a correspondingly weakened version of Separation in KP.
But I am happy to adopt your terminology here. You stated that AC plus Replacement implies Weak Collection, but this is not quite right. You don't need any AC. Instead, as your argument shows, what you need is the cumulative Vα hierarchy, which is built on the Power set axiom, not AC. That is, If you have Replacement and Power set and enough else to build the Vα hierarchy, then you get Collection as you described, even if AC fails. For example, ZF can be axiomatized equivalently with either Collection or Replacement.
Your question is a bit unusual, since usually Separation is regarded as a more fundamental axiom than Replacement and Collection, and more in keeping with what we mean by set theory. After all, if one has a set A and a property phi(x), particularly when phi is very simple, it is one of the most basic set theoretic constructions to be able to form { x in A | phi(x) }, and any set theory violating this is not very set-like. We don't really want to consider models of set theory where many instances of Separation fail (for example, Separation for atomic formula is surely elemental).
Incidentally, there is another version of a weakening of Collection in the same vein as what you are considering. Namely, consider the scheme of assertions, whenever phi(x,y) is a property, that says for every set A, there is a set B such that for every a in A, if there is b with phi(a,b), then there is b in B such that phi(a,b).
OK, let me now give a positive answer, with what I think is a more sensible interpretation of your question. I take what I said above (and Dorais's comment) to show that we shouldn't consider set theories where the Separation Axiom fails utterly. Rather, what we want is some very weak set theory, such as the Kripke Platek axioms, and then ask the relationship between Weak Collection and Replacement over those axioms. And here, you get the postive result as desired.
Theorem. If KP holds, then Weak Collection implies Replacement.
Proof. Assume KP plus (Weak) Collection. First, I claim that this is enough to prove a version of the Reflection Principle, since that proof amounts to taking successive upward Skolem hulls, which is what Collection allows. That is, I claim that for every set x and any formula phi, there is a transitive set Y such that x in Y and phi(w) is absolute between Y and the universe V. This will in effect turns any formula into a Delta0 formula using parameter Y.
Applying this, suppose we have a set A and every a in A has a unique b such that phi(a,b). By the Reflection Principle, let Y be a large set transitive containing A such that phi(a,b) and "exists b phi(a,b)" are absolute between Y and V. So Y has all the desired witnesses b for a in A. But also, now { b | exists a in A phi(a,b) } is a Delta0 definable subset of Y, since we can bound the quantifier again by Y. So the set exists. So Replacement holds. QED
I think we can get away with much less than KP. Perhaps one way to do the argument is to just prove Separation by induction on the complexity of formulas. One can collect witnesses by (weak) Collection, and this turns the formulas into lower complexity, using the new bound as a bound on the quantifiers.
There’s one issue underlying a lot of the discrepancies between people’s answers, I think:
How are we defining “$f$ is a function $s \to V$”, where $s$ is a set and $V$ is a (possibly proper) class?
(hence also, how we define subsequent things like “a small-category-indexed diagram of sets”) There are at least two main options here:
$f$ is a class of pairs, such that…
$f$ is a set of pairs, such that…
At least in most traditional presentations, I think it’s defined as the latter, but some people here also seem to be using the former. The answer to this question depends on which we take.
If we take the “a function is just a class” definition, then as suggested in the original question, and as stated in François’ answer, we definitely have some big problems without replacement: Set is no longer complete and co-complete, etc. (nor are the various important categories we construct from it); we can’t easily form categories of presheaves; and so on. Under this approach, we certainly get crippling problems in the absence of replacement.
On the other hand, if we take the “a function must be a set” definition, we get some different problems (as pointed out in Carl Mummert’s comments), but it’s not so clear whether they’re big problems or not. We now can form limits of set-indexed families of sets; presheaf categories work how we’d hope; and so on. The problem now is that we can’t form all the set-indexed families we might expect: for instance, we if we’ve got some construction $F$ acting on a class (precisely: if $F$ is a function-class), we can’t generally form the set-indexed family $\langle F^n(X)\ |\ n \in \mathbb{N} \rangle$.
This is why we still can’t form something like $\bigcup_n \mathcal{P}^n(X)$, or $\aleph_\omega$. On the other hand, such examples don’t seem to come up (much, or at all?) outside set theory and logics themselves! Most mathematical constructions that do seem to be of this form — e.g. free monoids $F(X) = \sum_n X^n$, and so on — can in fact be done without replacement, one way or another.
Now… I seem to remember having been shown an example that was definitely “core maths” where replacement was needed; but I can’t now remember it! So if we take this approach, then we certainly still lose something; but now it’s less clear quite how much we really needed what we lost.
(This approach is very close to the question “What maths can be developed over an arbitrary elementary topos?”.)
Best Answer
I think the main reason replacement is seen as an essential part of ZF is that it naturally follows from the ontology of set theory, as do the other axioms of ZF. The ontology of set theory is rooted in the idea that sets are obtained by an iterative process along a wellordered "ordinal clock", where at each step all the sets whose elements were generated earlier now appear. It is intuitively clear that, in order to be exhaustive, this process must go on for a long, long time. From this point of view, the replacement axiom can be intuitively stated: no set can be used as an indexing of a family of ordinals that reaches to the end of the ordinal timeline. This is a natural consequence of the idea that the iterative process should be exhaustive.
Another interesting aspect of replacement is that (in most formulations of ZFC) it is logically equivalent to reflection. (Informally: for any formula $\sigma(\vec{x})$ in the language of set theory, if $V \models \sigma(\vec{x})$ then there is an ordinal $\alpha$ such that $V_\alpha \models \sigma(\vec{x})$.) This is an extremely useful principle. One of its side effects is that ZFC is "self-justifying" in the sense that any finite fragment of ZFC is realized in a level $V_\alpha$ of the cumulative hierarchy. In other words, if one were to test set theory by examining a finite fragment of the axioms within the universe of sets, one would see that this finite fragment is not only consistent but that it has a model $V_\alpha$ that arises from the same iterative process that all sets do. In particular, ZFC comes very close to proving its own consistency even though we know this is not possible after Gödel. This feature makes ZFC very appealing as a foundational theory. (Note that PA has a similar self-justifying feature, but ETCS doesn't appear to have this.)
Another, more practical, use of replacement is to obtain "cheap universes". Grothendieck universes have proven useful for handling large objects. Unfortunately, one cannot prove their existence in ZFC. It is nevertheless often true that a "reasonable theorem" proven using Grothendieck universes is actually provable in ZFC. The reason is that the proof often doesn't make full use of all the features of Grothendieck universe, a finite fragment of those features often suffices and in such cases reflection provides a set $V_\alpha$ with all the necessary features to make the argument work. ETCS doesn't appear to have a good way of obtaining "cheap universes". This also hints at an alternative to replacement, which is to have a hierarchy of universes similar to those used in dependent type theory.
Operations on dependent families is where the need for replacement arises most. It's actually really hard to even talk about dependent families in the language of ETCS. The main issue with ETCS isn't necessarily that it can't prove the existence of coproducts like $\coprod_{n \in \mathbb{N}} \mathcal{P}^n(\mathbb{N})$, but that it has a hard time even talking about the family of all $\mathcal{P}^n(\mathbb{N})$ in the first place. Introducing universes would be an interesting way to get around that problem but there are other means, all of which are likely to make the need for replacement-like principles clear.
As for the proposed workaround, it's unclear you would get much more by this kind of process. Rather than ETCS, I'll work in BZC extended with terms for powersets and union and a constant symbol for $\omega$. The exponential-bounded formulas are defined like bounded formulas except that the bounding terms can involve powerset and union.
Fact. If $\phi(x,y)$ is an exponentially-bounded formula such that BZC proves that $\forall x \exists y \phi(x,y)$ then there is a standard number $n$ such that BZC proves that $\forall x \exists y(\phi(x,y) \land |y| \leq |\mathcal{P}^n(x \cup \omega)|)$.
Proof. Find a model $M$ of BZC and consider $x \in M$. Let $M' = \bigcup_n \{z \in M : |z| \leq |\mathcal{P}^n(x \cup \omega)|\}$ where $n$ ranges over the standard numbers only. Note that $M'$ is model of BZC that contains $x$. By hypothesis, there is a $y \in M'$ such that $M' \models \phi(x,y)$. Note that exponential-bounded formulas are absolute between $M'$ and $M$ since the only sets that we need to look at to figure out that $\phi(x,y)$ is true are in $M'$. Thus $M \models \phi(x,y)$ and also $M \models |y| \leq |\mathcal{P}^n(x \cup \omega)|$ by definition of $M'$. Now the fact that there is a fixed standard $n$ that provably works for all $x$ follows by a compactness argument. $\square$
The examples $\omega, \mathcal{P}(\omega), \mathcal{P}^2(\omega), \ldots$ and $V, V^{\ast}, V^{\ast\ast},\ldots$ are definable by a formula of the form $\exists z\phi(x,y,z)$ where $\phi(x,y,z)$ is exponential-bounded. Because of the nice biinterpretation between ETCS and BZC, for these and similar examples, either ETCS doesn't prove that the $n$-th iterate exists for every natural number $n$, or ETCS already proves that replacement holds in this particular instance.
Let me also address one aspect of the footnote, which states that replacement would be "the most likely culprit" if ZFC were found to be inconsistent. The "standard objection" to axiomatic set theory is actually with comprehension. If you think about it, comprehension is a rather bold statement: every formula in the language of set theory can be used to define a subset of a given set. The issue is that formulas can be complex beyond (human) understanding, it's hard to justify the use of comprehension for formulas we can't understand. In fact, it's not clear that comprehension is fully justified by the ontology of set theory described above. (Note that the same kind objection applies to PA, where one asks for induction to hold for arbitrary formulas.)