Hello everyone, I'm just beginning set theory and I am having trouble understanding the signifigance of the axiom of pairing. I understand the axiom, but why do we need it? Can some of you guys/girls just elaborate on the axiom in the least technical way possible? I would appreciate it. I do not have a very specific question. I find that I learn best by simply talking about the topic at hand.
Just in general, if you have two sets x and y, wouldn't you want to be allowed to form the set whose only two members are x and y?
Then, as you get down the road in set theory, even just a chapter or two away from where you are now, you'll see that certain mathematics formulated in set theory uses pairs of sets.
Just as a techical note (and the previous post somewhat alludes to this also): In ZF we don't need the pairing axiom, since it is derivable from the axiom schema of replacement. But of course, we do need (in some suitable sense of 'need') to be able to form pairs, whether as derived as a theorem from the replacement schema or, without the replacement schema, as an axiom itself.
EDIT: Perhaps by 'axiom of existence' you mean the principle that there exists at least one set. Sometimes such a principle is mentioned, however, from a technical point of view, it is superfluous, since by identity theory alone we get Ex x=x, and moreover, from the axiom schema of separation (either as an axiom or as derived from the axiom schema of replacement) we get ExAy ~yex, i.e., that there is at least one empty set.
To be even more specific: One instance of the axiom schema of replacement and the power set axiom together prove the pairing axiom. Moreover, we could even derive pairing if all we would had are an appropriate instance of the axiom schema of replacement and an "axiom" that there exist an x and y such that x not equal y.
The Wikipedia article to which Plato referred does articulate this. They do, however, indicate that the use of the axiom of replacement is restricted to sets of at least cardinality 2. The use of other axioms (existence/empty set, power set, or infinity) can be used to build up to sets of cardinality 2. That seems an important point, I believe.
The Wikipedia article points out the need for definable bijections. (Thus, the classes need to be "small enough" so the bijection defines a set.) If we have "paradoxical sets" that are too "large" you will end up with something like the "set of all sets." Hrbacek and Jech Introduction to Set Theory (3rd ed.) define it such that it "is intuitively obvious that the set F[A] is "no larger than" the set A," (p. 113). They go on to indicate any problematic definitions of such functions will produce proper classes, basically. Therefore, In ZFC we do end up having class functions that are definable bijections.
axiom of empty set as Hrbacek and Jech do (p. 7; it is an empty set axiom, but they call it Existence).
Maybe you mean that proving pairing with the axiom schema of replacement requires that we already have two different sets? Yes, I mentioned that above. From the axiom schema of replacement and the power set axiom, we do get that there exist at least two distinct sets.
But with the proof that there exists an empty set, we don't even need identity theory. The axiom schema of separation alone, with just first order logic (not even involving identity theory) proves ExAy ~yex. And again, the syntax of proof "matches" the semantics. Even without identity. Given ANY 1-place predicate symbol (or adjusted suitably for any n-place predicate symbol), we prove, for example, Ex(Px -> Px), and thus matching the semantics that requires that a universe for a model is non-empty.
Note that Halmos is not working in a formal context. However, sometimes authors who work even in a formal context do include such existence axioms. Some of those authors mention that they do that only for purposes of convenience since we really don't need such an axiom. Other authors who adopt such an axiom don't even bother discussing it much. But in any case, whether included as a formal axiom or not, it is superfluous, as is the empty set axiom, when we have the axiom schema of separation either as a theorem schema from the axiom schema of replacement or as its own independent axiom schema.Halmos does this in Naive Set Theory, making an official assumption that "there exists a set".