In this talk, I will study a change that took place in the 1960's within the field of set theory. Here, the introduction and development of model-theoretic methods, such as ultrapower constructions, inner models and forcing, sparked an explosion of results, clarified long-standing questions and opened up new areas of research. Most importantly, these methods provided a unified methodology for set theory's two main epistemic functions: serving as a foundational theory on the one hand, and studying mathematical infinity on the other.

I will analyse this change considering the introduction of forcing to set theory. Not only did this resolve one of the most pressing problems of that time, the question about the independence of the Continuum Hypothesis, but also led to establishing $ZFC$ as the foremost axiomatization fruitful for metamathematical and mathematical results. All in all, it was a major step in
resolving a fundamental flaw of set theory at that time, namely the lack of a unified methodological basis. The new basis led to an abandonment of earlier methods and explanatory patterns, as well as changing the general understanding of fundamental concepts and research questions.

Raising-to-powers is an expansion of ACF proposed by Zilber. The idea is to construct the theory of the complex field with "power functions" using the Hrushovski construction. Following his work, Bays, Kirby, and Gallinaro proved quasiminimality of this structure. In this talk, I will present my attempt to apply the Hrushovski construction to ordered fields — constructing a similar expansion of RCF that axiomatizes the real field with "power functions" on the unit circle. The resulting theory possesses, instead of quasiminimality, an o-minimal open core.

Reflection principles within set theory are generally thought of as sharing little but the name with reflection principles in proof theory. Within a broader context of reflection phenomena, we can see that they do have more in common, but the challenge is to make this precise. In this talk I will explore similarities, both structural and motivational, between set theoretic reflection principles on the one hand and proof theoretic reflection principles on the other. This will also lead to an analysis of what makes these two families of principles different in type.

I discuss some new methods in logic and proof theory in connection with fundamental problems in propositional complexity. References: https://arxiv.org/abs/2005.00809, https://arxiv.org/abs/2311.17939.