Set theory and category theory

In summary, Category theory and set theory both aim to provide foundations for mathematics, but there have been suggestions that set theory has problematic paradoxes. However, category theory also leads to paradoxes and has been successful in some aspects of mathematics already. Category theory does have sets, but they are referred to as objects and morphisms are analogous to functions. While set theory tries to describe everything as a set with structure, this approach can be misleading, awkward, or not work in certain situations. Category theory, on the other hand, offers a different perspective and has been successful in certain areas of mathematics.
  • #1
tgt
522
2
They seem to be different fields but both try to underpin maths. There has been suggestions that set theory is problematic, where some paradoxes cannot be resolved. But how about Category theory? Any problems or paradoxes? Is it more promising then set theory?
 
Mathematics news on Phys.org
  • #2
paradoxes are a normal part of math and logic. they don't indicate that there is anything wrong. categories will also lead to paradoxes.
 
  • #3
How successful will category theory be as a foundations of maths? I can't believe it doesn't use any sets?
 
  • #4
I'm not an expert. I'm just as confused as you are about that.
 
  • #5
tgt said:
How successful will category theory be as a foundations of maths? I can't believe it doesn't use any sets?

It depends what you mean by "successful". Will it ever be as powerful as our set theories? Answer: yes, it already is. Will it ever be accepted socially as a superior alternative to set theory? Lawls, no way.

Really, category theory *does* have sets. It just call them (and everything else) by a different name. The analog for a set is essentially the object. Morphisms are analogous to functions. While functions map members of sets to member of other sets, morphisms map members of objects to other objects.

The big difference is that CT almost *never* talks about the "members of an object". They don't value the members. Instead, they focus their energy on finding relations between morphisms, especially when chained together through function composition.
 
  • #6
Tac-Tics said:
Will it ever be accepted socially as a superior alternative to set theory? Lawls, no way.
My view of the future isn't quite so pessimistic. :tongue:

Really, category theory *does* have sets. ... morphisms map members of objects to other objects.
A set is a member of a set-theoretic universe. Your statement is only true if we're studying some set-theoretic universe and the category happens to be a subcategory of it.

Yes, it's true that category theory has analogs of the idea of elements, and in some situations, the analogy can be effectively equivalent (e.g. in the Set, one of the categorical notions of 'element' of X is a function {{}}--->X). But the set-theoretic approach of trying to describe everything as a 'set with structure' is often misleading, awkward, or simply doesn't work.
 
  • #7
Hurkyl said:
But the set-theoretic approach of trying to describe everything as a 'set with structure' is often misleading, awkward, or simply doesn't work.

For the unillustrated :) , could you show an example? In which realms category theory do provide a better start point than set theory or other tradition?
 
  • #8
Wikipedia's article on the origins of topos theory is an interesting read.


Wikipedia also gives two examples of categories whose objects can't be viewed as sets with structure.

I should state a caveat, though -- if you add a 'large cardinal axiom' to your set theory, then while hTop cannot be viewed as 'small sets' with structure, you can represent them as 'large sets' with structure. However, the general recipe for such a construction is wholly unenlightening -- it's simply a restatement of the category structure. (It's directly analogous to the fact that any group can act on itself)
 

1. What is the difference between set theory and category theory?

Set theory and category theory are two branches of mathematics that deal with the concept of collections and relationships between them. The main difference between the two lies in their approach - set theory is primarily concerned with studying the properties of sets and their elements, while category theory focuses on the structure and relationships between different categories of objects.

2. What is a set in set theory?

In set theory, a set is a collection of distinct objects or elements. These objects can be anything - numbers, letters, or even other sets. A set is defined by its elements and is denoted by curly braces, such as {1, 2, 3}.

3. What is a category in category theory?

A category in category theory is a mathematical structure that consists of objects and arrows (also known as morphisms) that connect these objects. Categories are used to study the relationships between different types of mathematical structures and provide a framework for understanding abstract structures.

4. What is the importance of set theory and category theory in mathematics?

Set theory and category theory are fundamental branches of mathematics that serve as the foundation for many other areas, such as algebra, topology, and logic. They provide a rigorous framework for understanding relationships between mathematical structures and have applications in various fields, including computer science and physics.

5. How are set theory and category theory related?

Set theory is considered a special case of category theory, as sets can be seen as categories with a single object. Category theory provides a more abstract and general approach to understanding the structure and relationships between sets and other mathematical objects. Many concepts and techniques in set theory can be translated and applied in category theory, making it a powerful tool in studying mathematical structures.

Similar threads

  • General Math
Replies
2
Views
1K
  • General Math
Replies
1
Views
889
  • General Math
2
Replies
38
Views
2K
Replies
4
Views
492
Replies
0
Views
233
  • STEM Academic Advising
Replies
9
Views
379
  • Science and Math Textbooks
Replies
10
Views
1K
Replies
4
Views
909
Replies
33
Views
5K
  • Set Theory, Logic, Probability, Statistics
Replies
6
Views
1K
Back
Top