What is the rigorous definition of set?

[guitarphysics]

Hi all, first math post here. I was just wondering- after having read from quite a few textbooks that intuitively, a set is a collection of objects- if there's a rigorous definition of the concept of set. It's just out of curiosity- I mean, is a rigorous definition even necessary? I guess I'm not completely clear on the intuitive definition, which is why I'm looking for a more rigorous one. What exactly is meant by objects? Is anything an object?
If anyone could give me a more precise definition of set, that would be great.
Thanks

 

[chiro]

Hey guitarphysics.

An object can be anything, but typically you define a set by a rule, explicit set of objects, or by the use of set operations involving intersections and unions of other sets.

Sets also don't repeat elements and typically follow some kind of axiomatic framework like ZFC theory. The reason that you need to define these axioms is due to things like Russels Paradox which look at the idea of X e X (X is an element of X) which generates a contradiction.

The intersections and unions are very well defined. The rule based definition is basically a symbolic way of defining sets from all kinds of structures that are represented symbolically.

In terms of defining numbers, you have things like the Peano axioms which define natural numbers using set builder notation. There are also ways of looking at the real numbers in relation to sets as well.

The object itself is just a label for something. It could refer to a set and you could have a complex hierarchy of sets within one object. You could think of the object like a number - What a number refers to is completely contextual and its interpretation is entirely based on the context just like any linguistic symbol. We need context to interpret what an object is and without it, it's basically just a nice drawing or some kind of marker.

 

[R136a1]

Hi all, first math post here. I was just wondering- after having read from quite a few textbooks that intuitively, a set is a collection of objects- if there's a rigorous definition of the concept of set. It's just out of curiosity- I mean, is a rigorous definition even necessary? I guess I'm not completely clear on the intuitive definition, which is why I'm looking for a more rigorous one. What exactly is meant by objects? Is anything an object?
If anyone could give me a more precise definition of set, that would be great.
Thanks

The usual axiomatics of mathematics are the ZFC axioms. In this framework, we do not define what a set is. So no, there is no precise definition of a set. We are given some axioms though. So we could say that anything which satisfies those axioms can be called a set.
When working in ZFC, everything we work with is a set. That means that things like $\pi$ or vector spaces must be defined as sets. So, by convention, everything in mathematics is a set. The ZFC axioms then state that some obvious constructions like power sets are also sets.

You will find the same thing in other axiomatic theories. For example, in geometry, we usually don't define what a line or a point is. We just give axioms that it should satisfy.

 

[guitarphysics]

Thanks! I hadn't realized- I have no way to explain what a point is. I guess we have to start out with a few things taken for granted, and then move forward to rigorously define other concepts that follow logically...

 

[R136a1]

Thanks! I hadn't realized- I have no way to explain what a point is. I guess we have to start out with a few things taken for granted, and then move forward to rigorously define other concepts that follow logically...

Yeah, the point is that there are always undefined concepts. Of course, if you have set theory, then you can define what points and lines are. But then you'll need to take sets for granted. Likewise, it is very much possible to be able to define sets using some other concepts, but you'll always need to take things for granted.

The origin of the axiomatic method is of course Euclid. He did in fact try to define everything. For example, points are defined as: "A point is that which has no part." (see http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html). This is a rubbish definition from modern point of view. But we can't really do it better. The modern approach is just to leave it undefined and actually use the axioms to define the object in question. This is the approach of model theory. In there, you are given some axioms. Then you give some "model" that satisfies the axioms. For example, the model of a geometry can consist of $\mathbb{R}^3$ with lines defined as equations $ax + by + cz = d$. But one can have much stranger models. Hilbert remarked that if "plane = table", "line = beer mug" and "point = beer" satisfies the axioms, then this is a valid interpretation of the axioms (although a highly nonstandard one). So this is also a model.

Now, of course, we always have some intuition for mathematics. We see intuitively that a set is "some collection of objects". Keeping this intuition in mind is very very useful. You should not just rely on the axioms, but try to see things intuitively as well. After all, all axioms try to do is to make the intuition rigorous. However, when you actually write down and check the proof, then there can be no place for intuition since that might lead you in a wrong direction.

 

[HallsofIvy]

Thanks! I hadn't realized- I have no way to explain what a point is. I guess we have to start out with a few things taken for granted, and then move forward to rigorously define other concepts that follow logically...

I would not say "taken for granted" but rather simply "undefined". Think of an undefined term as a "template". A reason why mathematics can be applied to so many different fields is that we can assign different meanings to undefined terms in different applications.

 

[AlephZero]

I mean, is a rigorous definition even necessary? I guess I'm not completely clear on the intuitive definition, which is why I'm looking for a more rigorous one.

A rigorous definition certainly is necessary. Is a set an "object"? If so, is "the set of all sets" a set? Does it contain itself? What about "the set of all sets that do not contain themselves"? If it contains itself, then it doesn't contain itself. But if it doesn't contain itself, then it does contain itself.

A different version of the same logical paradox: Write on one side of a piece of paper, "The sentence on the other side of this paper is true". On the other side, write "The sentence on the other side of this paper is false".

This goes back long before set theory was invented. There is a version of it in the bible, attributed to St Paul: "A Cretan told me that all Cretans are liars".

Either you have to change your intuitions about what "true" and "false" mean, or you have to define a set such that situations like this are excluded.

Or, you "get out of jail" a different way, by refusing to define was a set is, and instead define what a set does, i.e. what properties it has. Then, make the explicit assumption that "At least one set exists", even though you never actually nailed down what a set looks like.

 

[nomadreid]

One point that the usual definitions of set seem to skirt is the following:
Relations are defined using ordered n-tuples; so the membership relation, as a binary relation, is a class of ordered pairs. But to define ordered pairs one uses a definition such as <a,b> = {a,{a,b}} or something similar, but to build this you need the membership relation.
What is the way to break this vicious circle?

 

[R136a1]

One point that the usual definitions of set seem to skirt is the following:
Relations are defined using ordered n-tuples; so the membership relation, as a binary relation, is a class of ordered pairs. But to define ordered pairs one uses a definition such as <a,b> = {a,{a,b}} or something similar, but to build this you need the membership relation.
What is the way to break this vicious circle?

Membership is defined axiomatically. So the notions of membership and set are left undefined. So membership is not defined as a relation, although that is possible a posteriori.