Question

The philosopher Quine famously said that second-order logic is set theory in sheep's clothing. However, what if it is really the other way around? Is set theory part of mathematical logic, or even the same thing as mathematical logic? I am not talking merely about first-order logic, but second(and higher)-order logic. Perhaps set theory is the same thing as higher-order logic. So, has any philosopher argued for this point of view, and if so, can I see some of their papers?

Answer 1

While philosophers examine the interplay between set theory and mathematical logic, there is not a broad consensus on whether set theory is simply a form of higher-order logic.

Step-by-step explanation:

The question of whether set theory is part of mathematical logic, or even the same thing, touches on a deep philosophical inquiry into the nature of logic and mathematics. The idea that second-order logic may actually be 'set theory in sheep's clothing', as suggested by Quine, or vice versa, is not widely addressed in contemporary literature as a direct claim, but many philosophers of logic and mathematics explore the foundations and boundaries of these disciplines and their interconnections. One angle into this inquiry is the work of German philosopher Gottlob Frege, who demonstrated the capability to translate natural language sentences into a formal, symbolic language, which is the basis of modern logic.