Publisher's Synopsis
In this book, I am going to talk about structural reflection. In section 1 I will introducethe notion of structural reflection and I am going to state a theorem of Joan Bagaria which asserts that structural reflection produces a proper class of supercompact cardinals and a proper class of extendible cardinals. When we relativize a class of structures that is Π1 definable in V to an inner model, we transcend 2 this inner model and so we have a richeruniverse. In section 2 I will introduce Bagaria's result [Bagaria 13] implying that structuralreflection with Π1-definable in V classes of structures relativized to L is equivalent to theexistence of 0]. In this section I will explain how to iterate structural reflection and I willintroduce the finite transcendental structural reflection hierarchy, which forms a sequence of inner models. Then I will introduce briefly the theory of 0+ and I will relativize structuralreflection to the model L[U]. Also in this case, when we relativize structural reflectionto L[U], we transcend this inner model, thus producing 0+. In section 5, I will focuson philosophical aspects concerning structural reflection. I will highlight that structuralreflection principles are natural principles of set theory. However, I am going to argue thatthey cannot be considered as intrinsic principles of set theory. Then I will apply structural reflection to a weak extender model and I will prove that we do not get transcendence over this inner model and that principles of structural reflection transfers down from V to a suitable extender model. At the end of this article I will discuss the philosophy of settheory. My thesis, that I am going to defend, is that Woodin's Ultimate L would be thetrue universe of mathematics, if the Ultimate L conjecture were true. Thus, in this case, the continuum hypothesis would be a true mathematical proposition. Furthermore, I amgoing to address the issue of necessity within the realm of mathematics. I will argue that ifthe ultimate L conjecture were true, the continuum would be a true necessary proposition. At the end of this brief book, I will apply german idealism to the foundational axioms of set theory.