We offer a new angle to explain the comprehension of mathematical infinity. We use Action-Process-Object-Schema (APOS) theory to explore the construction of different aspects of the cognitive Object related to this notion. We focus our attention on different ways of interacting with infinity, and how to act on infinite entities. Our aim is to shed light on the nature of mental constructions that take place when individuals deal with mathematical infinity. An infinite union of finite subsets of N provides the context for our theoretical analysis and the design of interviews conducted with graduate students and instructors. In this study we identify three types of Actions with different complexity levels. When these Actions are performed, they lead to the construction of different facets of the associated Object. We question the construction of an Object conception in infinity-related situations and offer an explanation that might change the way we think about the learning of certain mathematical concepts. We discuss the implications of our research for the advancement of APOS theory. We also offer pedagogical suggestions related to the comprehension of mathematical infinity.