Large cardinals and structural reflection

The axiomatic system of first-order $ZFC$ set theory constitutes one of the most prominent bases for mathematics; at least for classical ones. However, after the ``discovery" of Cohen's forcing technique, a plethora of mathematical problems have been proved to be independent of these axioms, thereby suggesting that the search for new axioms for mathematics is an issue of paramount importance. One of the most prominent categories of such axioms are the so-called large cardinal axioms which, up to this day, are playing a pivotal role in ``eliminating" some of these independence phenomena. Moreover, not only there have been unveiled deep connections between such axioms and various areas of mathematics, but also it has been observed that these postulates form a hierarchy which can be used to ``measure" the consistency strength of several other axioms that have been proposed. Now, in this thesis we first make a brief introduction to the theory of large cardinals, outlining that way the basic concepts and tools we will be using, as well as commenting upon some intriguing related issues. Subsequently, we follow the work of Bagaria in \cite{Bagaria} and we focus our attention on the notions of (some) $C^{n}$-cardinals; especially on that of $C^{n}$-extendibles. Moving to the final part and the core of our study, we investigate the area in between supercompact cardinals and Vop\v{e}nka's Principle, where a level-by-level correspondence between the hierarchy of $C^{n}$-extendible cardinals and strata of Vop\v{e}nka's Principle is uncovered, as presented in Section 4 of \cite{Bagaria}.