The standard approach in foundations of mathematics is to consider a special first order theory called ZFC, it describes sets, whose elements are themselves sets. Inside this theory you can encode all other mathematics using sets for example by the Von Neumann construction of ordinals. Then you can restrict yourself to the finite ordinals and verify the Peano axioms, including the principle of induction which you can now formulate using sets. So everything turns out to be unique and pinned down inside your set theory.
What about pinning down your set theory? Well most mathematicians don’t worry about set theory. The set theorists seem to be quite fine with it not being pinned down, but are sometimes need to be careful about inside which model they are working. A very useful consequence of set theory not being pinned down is a construction called forcing, which allows you to prove the independence of ZFC from the continuum hypothesis (there not being a set of reals which can’t be bijected into neither the naturals nor the reals). What you do in this construction is that you work inside a model of set theory which is countable, which allows you to define a special kind of subset that does not exist in the original model but can be used to create a new model where certain properties fail to hold.
Some people want to use second order logic, talking about properties as primitive objects, to get this kind of pinpointing instead. The standard criticism to this is that you need to formalise what you mean by properties through axioms and rules of inference, giving you something quite similar to set theory. I’m not very familiar with second order logic so can’t elaborate on the differences or similarities, but it does look like the next post in this sequence will be about it.
The thing about ZFC is that it doesn’t feel like “the definition” of a set. It seems like the notion of a “set” or a “property” came first, and then we came up with ZFC as a way of approximating that notion. There are statements about sets that are independent of ZFC, and this seems more like a shortcoming of ZFC than a shortcoming of the very concept of a set; perhaps we could come up with a philosophical definition of the word “set” that pins the concept down precisely, even if it means resorting to subjective concepts like simplicity or usefulness.
On the other hand, the word “set” doesn’t seem to be as well-defined as we would like it to be. I doubt that there is one unique concept that you could call “the set of all real numbers”, since this concept behaves different ways in different set theories, and I see no basis on which to say one set theory or another is “incorrect”.
The standard approach in foundations of mathematics is to consider a special first order theory called ZFC, it describes sets, whose elements are themselves sets. Inside this theory you can encode all other mathematics using sets for example by the Von Neumann construction of ordinals. Then you can restrict yourself to the finite ordinals and verify the Peano axioms, including the principle of induction which you can now formulate using sets. So everything turns out to be unique and pinned down inside your set theory.
What about pinning down your set theory? Well most mathematicians don’t worry about set theory. The set theorists seem to be quite fine with it not being pinned down, but are sometimes need to be careful about inside which model they are working. A very useful consequence of set theory not being pinned down is a construction called forcing, which allows you to prove the independence of ZFC from the continuum hypothesis (there not being a set of reals which can’t be bijected into neither the naturals nor the reals). What you do in this construction is that you work inside a model of set theory which is countable, which allows you to define a special kind of subset that does not exist in the original model but can be used to create a new model where certain properties fail to hold.
Some people want to use second order logic, talking about properties as primitive objects, to get this kind of pinpointing instead. The standard criticism to this is that you need to formalise what you mean by properties through axioms and rules of inference, giving you something quite similar to set theory. I’m not very familiar with second order logic so can’t elaborate on the differences or similarities, but it does look like the next post in this sequence will be about it.
The thing about ZFC is that it doesn’t feel like “the definition” of a set. It seems like the notion of a “set” or a “property” came first, and then we came up with ZFC as a way of approximating that notion. There are statements about sets that are independent of ZFC, and this seems more like a shortcoming of ZFC than a shortcoming of the very concept of a set; perhaps we could come up with a philosophical definition of the word “set” that pins the concept down precisely, even if it means resorting to subjective concepts like simplicity or usefulness.
On the other hand, the word “set” doesn’t seem to be as well-defined as we would like it to be. I doubt that there is one unique concept that you could call “the set of all real numbers”, since this concept behaves different ways in different set theories, and I see no basis on which to say one set theory or another is “incorrect”.