Do we really need the axiom schema of replacement?

Question

I'm repeating a bit of set theory these days and was a bit perplexed by the axiom schema of replacement, for which I can't really seem to find a substantial application. After all, if $F: A \to B$ is a function between sets, the image will exist due to separation.

I read on WP about some implications, but they all seem of limited practical interest for day-to-day maths. So why do we usually include this axiom?

Answer 1

Since all the answers went straight into the comment section, let me here summarise the arguments that were made in favour of the axiom schema of replacement. I want to begin with one that I just came up with. All the other items are taken from the comment section.

• It is not unreasonable that the "images" of "class functions" are also sets, because everything which is not a set is supposed to be "larger" than a set, but these "images" are "equal or smaller".
• Certain constructions (such as associating an element of $\operatorname{Ring}$ to each point $x$ of a topological space $X$) are easier if the axiom schema is being regarded as true
• The axiom schema allows for the construction of von Neumann ordinals, which we want because they exist in any universe set class
• Transfinite recursion is impossible without the axiom schema

I have to admit that the second item is the one that convinces me the most, and the others are more like "added bonuses". Constructing a suitable set of rings in such a situation seems unnecessarily inconvenient to me.

Answer 2

I've thought of it as: the other axioms are useful for "day-to-day math", but Replacement and Foundation are added because they make set theory more beautiful. Since the set theorists are the ones who came up with the axioms, they've earned a reward :)

Indeed, set theory without Replacement is ugly. It's interesting to see what happens in the set $R(\omega + \omega)$, the set of all sets of rank less than $\omega + \omega$, where all of ZC holds but Replacement fails. We have enough sets to do all of everyday arithmetic, analysis, etc, but we have the following facts, all of which are therefore consistent with the absence of Replacement:

• Every set admits a well ordering, but many of them (e.g. $\mathbb{R}$) are not bijective to any ordinal (since all ordinals are countable). So you lose the canonical representative of a well-ordering type.

• As a result, you also lose the canonical representative of a cardinality. You can't define $|A|$ as "the least ordinal in bijection with $A$" because there might not be any. So it's no longer clear what objects in your universe to use as your cardinal numbers.

• Hartogs' lemma is false: every ordinal injects into $\omega$.

• There exists exactly one limit ordinal, namely $\omega$. No transfinite hierarchy for you.

• Ordinal arithmetic no longer works, because you cannot add $\omega$ to $\omega$, let alone multiply.

• For analysis, you have $\mathbb{R}$, you have subsets and functions on $\mathbb{R}$, you have sets of sets and sets of functions, you have operators on function spaces, etc. But you can't take all those objects and put them into one single set; they are a proper class.

• Even analysis can get a little tricky. Let's take tensor products of Hilbert spaces, so that $H \otimes K$ is a vector space with a bilinear map from $H \times K$, and define higher tensor products recursively as $H \otimes K \otimes L = (H \otimes K) \otimes L$. Then, unfortunately, Fock space doesn't exist, so quantum mechanics becomes awkward. (The problem was that because of the recursion, the set $H^{\otimes k}$ has a rank that increases with $k$, and so we can't put all of them in a set. We could fix it with a non-recursive definition, but I don't think most analysts want to have to take that kind of care.)

• Goodstein's theorem is still true, but the proof no longer works.