2 ms·

"category theorists are happy to tell you how much they despise ZF set theory" Why do they despise ZF set theory?

auggierose 5y ago
There is for example the book "Sets for Mathematics" [0] by Lawvere, one of the inventors of category theory. Those sets he talks about in his book are category-theoretic sets. Which kind of implies that ZFC sets are NOT for Mathematics :-) [0]: https://www.cambridge.org/core/books/sets-for-mathematics/E899F592AD8FBA9A550B1ED3E1E61EC3 https://www.cambridge.org/core/books/sets-for-mathematics/E8...
joe_the_user 5y ago
The article describes several ways that various seemingly simple mathematical objects like integer are defined in set theory using complex combinations of sets and I believe the inelegance of such constructions is the main motivation. The thing is, any constructive definition of (almost)all mathematical objects is likely to share this kind of inelegance imo. But I don't think anyone has come up with an alternative to this.
IngoBlechschmid 5y ago
Working category theorist here. I wouldn't subscribe to that statement in this strong form, even though I am convinced that various flavors of type theory are a better foundation for mathematics. But this feature doesn't make me despise ZF set theory. In fact, ZF set theory is a beautifully elegant theory. I love it for studying sets. It's just not my favorite foundation. Additionally to the reasons listed by my siblings, many important results in category theory cannot be adequately formalized in ZF for size reasons. Too often, we have to or want to deal with results concerning categories which are too big to fit into sets (these collections are called "proper classes"). In ZF, we can often only formalize specific instances of our theorems but not the general theorem itself. This is explained in more detail by Mike Shulman here: https://arxiv.org/abs/0810.1279 https://arxiv.org/abs/0810.1279 Among other partial solutions, one is to extend ZF to a theory called ZFC/S where S is a very useful mathematical fiction came to live.

There is for example the book "Sets for Mathematics" [0] by Lawvere, one of the inventors of category theory. Those sets he talks about in his book are category-theoretic sets. Which kind of implies that ZFC sets are NOT for Mathematics :-) [0]: https://www.cambridge.org/core/books/sets-for-mathematics/E899F592AD8FBA9A550B1ED3E1E61EC3 https://www.cambridge.org/core/books/sets-for-mathematics/E8...

joe_the_user 5y ago

The article describes several ways that various seemingly simple mathematical objects like integer are defined in set theory using complex combinations of sets and I believe the inelegance of such constructions is the main motivation. The thing is, any constructive definition of (almost)all mathematical objects is likely to share this kind of inelegance imo. But I don't think anyone has come up with an alternative to this.

IngoBlechschmid 5y ago

Working category theorist here. I wouldn't subscribe to that statement in this strong form, even though I am convinced that various flavors of type theory are a better foundation for mathematics. But this feature doesn't make me despise ZF set theory. In fact, ZF set theory is a beautifully elegant theory. I love it for studying sets. It's just not my favorite foundation. Additionally to the reasons listed by my siblings, many important results in category theory cannot be adequately formalized in ZF for size reasons. Too often, we have to or want to deal with results concerning categories which are too big to fit into sets (these collections are called "proper classes"). In ZF, we can often only formalize specific instances of our theorems but not the general theorem itself. This is explained in more detail by Mike Shulman here: https://arxiv.org/abs/0810.1279 https://arxiv.org/abs/0810.1279 Among other partial solutions, one is to extend ZF to a theory called ZFC/S where S is a very useful mathematical fiction came to live.