ETD

• downward directed grounds hypotesis
• ground models
• hyper-huge
• large cardinals
• mantle
• multiverse
• multiverse truth
• set-theoretic geology

The thesis explores set-theoretic geology, a discipline that reverses the traditional method of forcing. Instead of constructing extensions from a base model, it investigates whether the mathematical universe V can be viewed as a generic extension of a smaller inner model, known as a ground. The work reviews the foundational concepts and results of the field, before presenting the recent theorems by Toshimichi Usuba. A central theme is the proof of the Downward Directed Grounds (DDG) hypothesis, which states that for any family of grounds, there exists a deeper common ground contained in all of them. This allows for the analysis of the mantle, defined as the intersection of all grounds of the universe, proving that it is itself a model of ZFC. Finally, the text investigates the unexpected correlation between the existence of hyper-huge cardinals and the geological structure of the universe. It demonstrates that the existence of a hyper-huge cardinal implies that the universe possesses only set-many grounds, and that the mantle constitutes the solid bedrock, namely the absolute minimum ground.