# Forming set with infinite elements

1. Sep 21, 2011

### jagbrar

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

2. Sep 21, 2011

### gb7nash

Pairs? Do you mean direct/cartesian product?

3. Sep 21, 2011

### micromass

Now, one cannot. We need a seperate 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.

4. Sep 21, 2011

### Fredrik

Staff Emeritus
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.