Is the infinity of irrational numbers equal to the infinity of rational numbers? Or is one is greater than other? And what is the proof? I could not find out a rigorous proof about this.
P.S. I am interested in which set of numbers is more dense - rational or irrational. Other similar questions do not consider the density of the numbers.
There is no formal definition of "more dense", you would need to explain what you mean by that.
Anyhow, note that the rationals are countable while the irrationals are uncountable.
One way to "measure" how dense they are is the following: the rationals have Lebesgue measure 0, while the irrationals have full measure in every interval. Every set which has full measure in every interval is automatically dense, while sets of measure zero are rarely dense.
Also, if we remove subsets of the same cardinality for both sets, the rationals can lose the denseness property, while the irrationals will never do that [subsets of same cardinality imply countable subsets]...
But again, you need to explain what you mean by "more" dense.