Ramsey showed that the axiom of reducibility could be replaced by treating propositional functions extensionally, collapsing the ramified hierarchy without sacrificing analytic results.

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Axiom of Reducibility(Principia Mathematica, ∗12; underlies the ramification of the theory of types): The axiom asserting that for an arbitrary function of any order there exists an equivalent predicative function true of exactly the same range of arguments
Extensionally(as used in logic and semantics): Treating something based on what it actually is or what group it belongs to, rather than how we describe it. If something can't be 'handled extensionally,' it means we can't just look at what it actually contains or is.
Ramsey: # Ramsey Ramsey theory is a branch of mathematics that studies how order and patterns inevitably emerge in large systems, even when things seem random. The basic idea is that if you have a large enough collection of objects (like numbers, points, or people), some organized structure or pattern will always appear somewhere within it. A famous example is the "Ramsey number," which answers questions like: "How many people do you need at a party to guarantee that some group of them are all mutual friends or all mutual strangers?"
analytic results

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