An elementary fact about language is that it is a system of discrete infinity whose source is recursion and reveals itself as Phrase Structure Grammar. This is the cause and effect of something that we might call unique to humans. We are the only species, as far as we can tell, that have evolved such a system, the only one whose signalling system shows both compositionality and combinatoriality. What is it that endows our linguistic system with discrete infinity/recursion? Is it a matter of the actual system or the cognitive underpinnings that allow it? Using Bayesian Iterated Artificial Language Models, the present work will show the evolution of Phrase Structure through cultural transmission in populations of Bayesian learners from a trade-off of the need for the constraining of the product of a stimulus-free mind with unbounded combinatorial mechanism and a need for simplicity and coverage in the constraints. What endows our species’ linguistic system with recursion and allows it to be a system of structural discrete infinity is cumulative culture. What makes this possible is an unbounded combinatorial capacity that allows the merging of any concept with any concept product of the relaxation of constraints of some domain-specific combinatory and endows our species with recursive processing. I will conclude that linguistic structure represented in PSG –including hierarchical and recursive structures– evolves to be learnable and it is a product of domain-general learning mechanisms but its evolution requires minds with the organic unbounded combinatorial capacity in the first place.