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 [itex]\mathbb{N}[/itex] 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.

 

[blue_raver22]

Yes, it is possible to form sets with infinitely many elements using only the empty set, pairs, and unions. This can be achieved by using the concept of power set, which is defined as the set of all possible subsets of a given set. By starting with the empty set and taking its power set, we can create a set with one element (the empty set itself) and its power set will have two elements. By taking the power set of this set, we can create a set with three elements and so on. This process can be continued infinitely, resulting in a set with infinitely many elements. Additionally, by taking the union of these sets, we can also form larger sets with infinitely many elements. Therefore, using only the empty set, pairs, and unions, we can indeed form sets with infinitely many elements.