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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.

Thank you in advance.

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    $\begingroup$ "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? $\endgroup$ Jan 28 at 19:41
  • $\begingroup$ 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. $\endgroup$
    – Elise
    Jan 28 at 22:03
  • $\begingroup$ But you wrote "there is not a formal foundation for mathematics," and that's false: there's one standard, and multiple alternatives. $\endgroup$ Jan 28 at 22:04
  • $\begingroup$ I have now edited my post $\endgroup$
    – Elise
    Jan 28 at 22:21
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The most introductory material related to ETCS is Trimble’s informal notes starting here: https://ncatlab.org/nlab/show/Trimble+on+ETCS+I

However, this may not be introductory enough, depending on your background, since you say you’re quite new to category theory. Leinster’s book Basic Category Theory is where’d I’d usually send an undergraduate (my best guess at your level, do let me know if I’ve misjudged) wanting to learn some category theory proper, which is likely to be necessary to understand much about ETCS. You would also likely find Lawvere and Rosebrugh’s book Sets for Mathematics a useful and initially gentle introduction to both category theory and how it thinks about set theory.

It might be worth noting a warning here that ETCS is not explicitly used as a logical foundation by any (as far as I know) working mathematicians in the way that ZFC is. Rather it purports to better explicate how such mathematicians actually assume sets to behave. Thus the purposes of the two systems are not perfectly comparable. Penelope Maddy’s papers on “believing the axioms” might be a good starting point if you hope to clarify your thoughts about what exactly a foundation is for.

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