Forming set with infinite elements

[jagbrar]

using only the empty set, pairs, and unions can you form sets with infinitely many elements?

 

[gb7nash]

using only the empty set, pairs, and unions can you form sets with infinitely many elements?

Pairs? Do you mean direct/cartesian product?

 

[micromass]

using only the empty set, pairs, and unions can you form sets with infinitely many elements?

Now, one cannot. We need a separate axiom to be able to form infinite sets. Normally, one uses the axiom that $\mathbb{N}$ exists. But this is (usually) equivalent to asserting that an arbitrary infinite set exists.

Using only the empty set, pairs and unions, one cannot derive that infinite set exists.

 

[Fredrik]

Pairs? Do you mean direct/cartesian product?

The axiom of pair/pairs/pairing says that if x and y are sets, there's a set z such that x and y are both members of z. Link.