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Comment by calhoun137 | Hacker News Reader

I have been studying Category theory recently, on the surface, its merely a theory about dots that are connected by arrows.

For some reason, category theorists often claim that category theory is "more fundamental" than set theory, whatever that means. It's especially bizarre since many category theory books begin by saying they will define categories without using sets, and then proceed to say "let A be a collection of dots, and B a collection of arrows" but only God knows what the difference between a collection and a set is, from what I can tell...

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Russell's Paradox (https://en.wikipedia.org/wiki/Russell%27s_paradox) means you cannot have a set-of-all-sets. For the category of sets and functions between sets, that means you have to use the weaker idea of a 'collection'.

A positive side-effect of this avoidance of set axioms in the definition of a category is that we can more easily define categories with very different ideas of membership. For example, a category where membership is a real number (like in fuzzy logic). Categories with this idea of 'generalised membership' (plus a few extra bits and bobs) are called 'toposes', and can be used in much the same way as sets are. Importantly, we can relate different categories like this together using the same terminology and concepts and so more easily define ways to join classical code with fuzzy logic, or with neural networks (or anything else which satisfies the topos axioms) and know they're not insane.

Incidentally, you can define the axioms of a topos (and sets are a topos) in purely category theory terms. That is why some (and I'm one of them) view categories as a more useful foundation than set theory (where attempts to define the converse are much more messy).

Basically it's more fundamental because if you try to bootstrap mathematics from set theory you need a set of 10 axioms which seem unmotivated and disconnected. To bootstrap math from category theory invokes stepping through a rich series of theories starting from the merest notion of "combining things" and moving upward. It helps us see the foundations of mathematics as living within a large universe of alternative, slightly differing foundations and therefore recognize the somewhat "arbitrary" nature of standard math foundations. In many ways this is a more appealing sort of foundations. Category theory requires sets to "bootstrap" itself, but you can get by with a naive construction and work up from there.

Sets are more than a collection. They have axioms that are not needed to define categories.

Mathematicians used the "naive" concept of a set as "a collection of objects" for thousands of years, and it led to a major crisis in the foundations of mathematics that was only resolved by establishing a rigorous system of axiom's that define what a set is.

So I disagree that a "set is more than a collection", with the qualification that the concept of a "collection" that is not associated with any axioms whatsoever, is meaningless. Why do I say "meaningless"? Because this "naive" concept of a collection leads to a number of logical paradox's that can only be resolved by using a system of axioms to define these collections.

So it's true that you can define sets by various axiom systems that are not the same, but these systems all exist under the umbrella of "set theory".

Using the "naive" concept of a collection is fair enough, since we all "know what you mean", but only as long as your point isn't that category theory is more fundamental than set theory.

Actually, it didn't resolve the crisis. In fact, categories, computability theory, and most mathematical findings of the last 150 years are a direct result of stepping around the crisis, namely Russell's paradox.

My point is not that category theory is more fundamental than set theory (though set theory can be described in terms of categories), but rather that it isn't derivative. You don't need the formalisms of set theory to have category theory.

If you continue your studies that statement will be clarified.

I presume a collection can have multiple of the same thing, whereas a set cannot?

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