  The existence of various physical phenomena stems from the concept called
asymptotic emergence, that is, they seem to be exclusively reserved for certain
limiting theories. Important examples are spontaneous symmetry breaking (SSB)
and phase transitions: these would only occur in the classical or thermodynamic
limit of underlying finite quantum systems, since for finite quantum systems,
due to the uniqueness of the relevant states, such phenomena are excluded by
Theory. In Nature, however, finite quantum systems describing real materials
clearly exhibit such effects. In this paper we discuss these apparently
``paradoxical'' phenomena and outline various ideas and mechanisms that
encompass both theory and reality, from physical and mathematical points of
view.
