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I am reading through Kenneth Kunen's Foundations of Mathematics in which he invokes the Axiom of Replacement to justify the existence of Cartesian products for two sets. I have two problems, and I apologize if they are quite vague.

1) Can anyone here give an intuitive understanding of what exactly the Axiom of Replacement does? I'm just having an issue seeing it's necessity, especially here.

2) Is this axiom really needed for justifying the existence of the cartesian product (and thus relations and functions) between two sets? I have seen how in some other books, they note that the Cartesian product, as defined, would satisfy $A\times B \subset P(P(A\cup B))$, the latter of which is justified to exist by the Axiom of Power Set. In that case, why use replacement?

Thanks in advance for any response.

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    $\begingroup$ If you have a specific ordered pair in mind, like the Kuratowski pair, then you probably won't need replacement. But with the axiom of replacement, you can use any (definable) ordered pair you like. $\endgroup$
    – Zhen Lin
    Sep 28, 2015 at 19:45
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    $\begingroup$ I think the point of the approach is that this shows you can prove the existence of Cartesian products without needing to appeal to the power set axiom. This turns to be useful later as, for instance, you will encounter structures where power set definitely fails and yet the other axioms hold. You are correct, of course, in noticing that, had we access to the power set axiom, then replacement is not needed for this existence theorem. $\endgroup$
    – Andrés E. Caicedo
    Sep 29, 2015 at 1:45

1 Answer 1

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About (1): The Axiom of Replacement states a kind of "limitation of size" idea. We know that there are some classes which are too big to be sets (say, the class of all sets); on the other hand, if we have a set, then any class the same size as this set should also be a set. That's (more or less) what the Axiom of Replacement says; indeed, when Mirimanoff first proposed it, that was its original formulation: if a set $A$ exists, then so does any (well-founded) collection equipotent with it. Nowadays, the most common version of the Axiom of Replacement is in its function form: very roughly, if $z$ is a set and $F$ is a function (including class functions), then $F[z]$, the image of $z$ under $F$, is also a set.

About (2): As Zhen Lin says in the comments, that particular proof requires the specific definition of ordered pair as a Kuratowski ordered pair. The proof using Replacement has the advantage of being more general.

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