Explore a groundbreaking lecture on a finitary formal meta-theory of meta-mathematics that merges the work of Cantor, Frege, and Zermelo using pure set theory. Delve into the development of a formal system with specific instances of Frege's Logic Rules and Zermelo's Postulates, emphasizing formal deductions and predicative definitions. Contrast this approach with Kleene's claim about intuitive meta-theory, and discover how the lecture restricts methods to finitary techniques. Examine the syntactic and semantic development of ZFC and first-order logic within this framework. Investigate novel definitions of finiteness, natural numbers, and arithmetic without relying on infinity, challenging traditional approaches by Dedekind and Frege-Russell. Consider the implications for the axiom of infinity and explore a reinterpretation of logical truth based on an analysis of Quine's definition.