Relevance. Universities, as institutional organizations, are subject to the greatest reform in the transforming matrix of goals, values and norms of building the Future. In this regard, the problem of conceptualizing the ultra-new model of the University of the fourth generation (4.0) is the most relevant for science and social sciences.The purpose of this study is to consider and analyze the conceptual foundations reflected in the scientific manifestations of the formation of the fourth generation University model (4.0) as a subject of the “triple helix".Objectives. To analyze the concepts, ideologems and other units of the theoretical basis of the University 4.0 model presented in the scientific media space; to reflect the emerging concept of University 4.0 in the optics of the “triple helix” model, implemented in the context of the formation of a post-industrial society.Methodology. The study was carried out on the basis of synergetic principles of complexity theory, general provisions of system theory and interinstitutional and transinstitutional communication using system-structural and convergent network methods.Results. Today, new ontological continuums of social multi-subject practices are objectively “unfolding” based on the ethics of subjective integration and synergy of individual and collective, as well as machine and hybrid intelligences. Flexible models of collective management of the future should be synthesized on such university cognitive platforms 4.0, where even transinstitutional forms of relations, such as the “triple helix”, begin to be perceived as tools, rather than platforms for the implementation of new ideas and solutions that overcome historical and cultural predestination. The platform of the “cognitive turn of postindustriality” is the Universities of the fourth generation.Conclusions. The platform of the "cognitive turn of postindustriality" in the future is the Universities of the fourth generation. Then, in university activities, existing traditional subject (for example, institutional) ontologies act only as a tool for generating on their basis “packages” of understanding posteconomics in the spirit of “meta-subject forms of thinking”, such as collective, including hybrid human-machine intelligence. New ethics (bio-, techno-, etc.) are also being institutionalized and progressing.