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Consider these two (informally stated) axioms:

  1. (Small Axiom of Union) For any two sets $A,B$ there exists the set $A \cup B$.
  2. (Great Axiom of Union) For any set $A$ there exists its union $\bigcup A$.

It seems to me, that the small axiom of union, together with the other ZFC axioms, is enough to construct function spaces $X^Y$ and finite cartesian products $X \times Y$ for all sets $X,Y$. And if $(X_i)_{i \in I}$ is a family of subsets of a given set $X$, its union $\bigcup_{i \in I} X_i$ exists already by using the axiom of separation.

The question: What situations are there in mathematics, where the great axiom of union is really needed?

Two things come to my mind:

  1. Constructing infinite cartesian products via the definition $$ \prod_{i \in I} X_i := \{ f \in (\bigcup_{i \in I} X_i)^I \ \vert\ \forall i \in I : f(i) \in X_i \} $$ and hence proving the category of sets to be complete.
  2. Most applications of the axiom of Replacement like constructing limit ordinals or the von-Neumann stages $V_\alpha$ for $\alpha$ a limit ordinal, first use Replacement to construct a set (e.g. a set containing all lower von-Neumann stages) and then, well, unite it.

Am I true in regarding the (great) axiom of union here to be necessary? Both these situations are not important to develop fundamental analysis and algebra, arent they? So are there more common situations, maybe in abstract algebra, where Union is needed? I think of constructing projective resolutions, algebraic closures, or something like that.

Lucina
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  • Another important technical application of "Great Union" in set theory is the existence of transitive closures. But I cannot think of any situation in "ordinary mathematics" where "Great Union" is needed at the moment. – Eric Wofsey Jan 31 '19 at 22:14
  • Note that Small Union can be proved from the other axioms of ZFC (see https://math.stackexchange.com/questions/498256/can-we-prove-the-existence-of-a-cup-b-without-the-union-axiom), as can any application of Great Union to a family of sets of bounded cardinality. But this relies heavily on Replacement, and I think one can say more strongly that "ordinary mathematics" can get away without using either Great Union or Replacement, if you assume Small Union. – Eric Wofsey Jan 31 '19 at 22:37
  • I cannot find any relation between your question and derived functors. Why does your question have that tag? – Hanul Jeon Feb 03 '19 at 01:44
  • @Hanul Jeon: I guess that homological algebra is full of situations, where the great axiom of union is needed, so that an expert for derived functors has an example right at hand. – Lucina Feb 03 '19 at 21:43

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