This work shows the connections between conceptual metaphors and unconscious, tacit models that benefit our understanding of mathematical infinity in the university classroom. From the perspective of cognitive linguistics, it is argued that conceptual metaphors play a key role in explaining how this mathematical concept is grounded in our experience, simultaneously providing a mechanism to address these tacit models in a more conscious way. Moreover, it is shown that conceptual metaphors can be built from the conflicting cognitive structures underlying these models, specifying obstacles and difficulties that teachers must consider when designing activities aimed at achieving an adequate understanding of mathematical infinity. This type of study allows us to improve our teaching practice, making us aware and stimulating students to become aware, to reflect on the inconsistencies of their own thoughts and intuitions regarding this mathematical concept. At the same time, it allows us to show the validity of these inconsistencies by revealing how our cognitive processes are constrained by bodily-grounded experiences determined by the complexity of our human nervous system. The use of technology would also engage students in these reflections and could also help them by fostering new ways of thinking about mathematical infinity. This perspective would be of interest in the current context of digital technology in mathematics education research.