Can the concept of infinity be rigorously defined in the framework of modern set theory?

The idea of infinity has fascinated mathematicians and philosophers for centuries, yet it remains an elusive and somewhat abstract concept. In modern mathematics, infinity is often introduced through the lens of set theory, particularly with the help of Cantor's work on cardinality and the different sizes of infinity. However, the notion of infinity extends beyond mere cardinal numbers and enters realms such as limits, sequences, and calculus. This raises the question: in what ways can the concept of infinity be rigorously defined and understood when considered from the perspective of modern set theory? By examining models, paradoxes such as Hilbert's Hotel, and various mathematical definitions, we can explore how set theory attempts to provide a solid foundation for this seemingly nebulous idea.