It is known that there are an infinite number of worlds, simply because there is an infinite amount of space for them to be in.
It is thought by many cosmologists that the Universe is infinite due to theories like inflation, it is not known whether it is or not.
However, not every one of them is inhabited. Therefore, there must be a finite number of inhabited worlds..
Incorrect, even assuming a tiny probability like only one in 10^20 worlds have intelligent like, that results in an infinite number of worlds. The likely reason for us not seeing any evidence is that intelligent life is rare, space between stars is large and there is a maximum speed limit c which takes a stupendous amount of energy to approach.
An intelligent life form would I hope not be silly enough to create von-Neumann probes because evolution would likely mean that in cosmological time they would evolve to create a hegemonizing swarm which would be impractical to eradicate.