 I would like to express my immense gratitude to Lev Beklemishev, who was so kind as to write me a long letter including many literature references to recent developments in provability logic as seen from his perspective; it was a great pleasure to read about all these intriguing new developments. Thanks are also due to Albert Visser, Sara Uckelman, Rajeev Goré, Giovanni Sambin, Vítek Švejdar, and Sara Negri for their suggestions of improvements and additions.