# Interpreting ETCS in ZFC

I am currently doing a project on foundation of mathematics. Set theory and category theory are two different approaches to describe the core principles. ZFC is an axiomatic system which is inspired by set theory that is commonly accepted as the foundation of maths. Then there's also ETCS axiomatisation which has categorical origins.

I am looking into category theory as of now to have a better undertstanding of the context and motivation behing ETCS and looking to interpret ETCS in ZFC. However, i am very overwhelmed and not sure where to start my research into category theory and struggling to find any good reasources that shows the connection between the two axiomatic systems. So i would greatly appreciate if you guys could let me know some good introductory resources.

• "From my understanding there is not a formal foundation for mathematics." What exactly does that mean? Why does $\mathsf{ZFC}$ not count as a formal foundation for mathematics? Jan 28 at 19:41
• From what i have read so far ZFC commonly accepted as one of the foundation of maths. But theres many axiomatic systems that can also be used as the foundation. Jan 28 at 22:03
• But you wrote "there is not a formal foundation for mathematics," and that's false: there's one standard, and multiple alternatives. Jan 28 at 22:04
• I have now edited my post Jan 28 at 22:21