If a mathematician wants to explore infinity, there are many options - for example by calculating pi, or the square root of 2, or dividing any number by 0.
For philosophers, the concept of infinity is fraught with enigmas.
Are some infinities larger than others? Are infinities created over time different from instant infinities? Is it necessary to have an infinite universe to accommodate infinities (going against the current opinion on the universe)? Is it possible to sub-divide something an infinite number of times (going against quantum theory)? If it's possible to 'prove' something by the use of logic, but the proof is infinitely long, is it proven?
Such questions are on record as having been considered for at least 2,000 years.
Further reading with examples The Internet Encyclopedia of Philosophy (IEP), University of Tennessee.